Yeah. That's why we didn't throw out relativity 6ish years ago when it muons were measured moving faster than light. It turned out that literal bird shit had caused the error, it was on the sensors.
I think it’s a bit loose. Evidence should determine our prior beliefs and new evidence should update them. Thus, evidence should determine belief generally.
I don't normally comment on youtube videos. But I must say this 15-minute video has helped me to grasp Bayes' Theorem so deeply that i was able to solve all the Bayes' Theorem-related questions in my recent math exam intuitively, with minimal plugging of formulas! It feels like magic. I am deeply grateful.
@@yashaswikulshreshtha1588 update: 6 months after diving deeper into math. I have come to find that applying and understanding the intuition is just the start. I’ve learnt that in fact, I don’t know much 😅
thinking of events as "H" (hypothesis) and "E" (evidence) instead of random variables (A, B, C,...) is definitely game changing. personally, it made the theorem much more immersive and useful. also, brilliant demonstration!
The follow-on video mentioned here did not, er, end up getting finalized and published. At least not yet! I have a bad tendency to do this with probability videos, where there are always plans and drafts for more, but they often don't quite feel "there" once they're more fully mapped out.
Well, what were the chances of it being made? I think with this knowledge, we can look in retrospect and update our views on the chances of it occurring.
Dang - but thanks for the heads up! I was about to go searching for it and I'd probably have wasted way too much time looking since I assumed the likelihood the video existed was close to 1.0. Now I need a model for how to update my beliefs given an unknown probability! ;)
First time in my life at age 55, I really understood Bayes Theorem. The link between the tree diagram and this box explains why the probabilities on successive branches of the tree diagram are multiplied. This is brilliant.
As Asanda said, the actual maths teacher has to manage the 50% of the class who don't give a shit about anything, no matter how well it is presented! Then there's the student who will put up their hand and ask "Is this examinable?". Then there's the parent-teacher meeting where you get accused of going "off-track". There are many hurdles to prevent inspired teaching.
@@garethb1961 I think you're missing the point though; all those hurdles will still exist but the maths teacher would not be a hurdle which is definitely not the case for a lot of students unfortunately.
@@stretch8390 I don't think I missed the point at all. That boring maths teacher who can't teach for shit and disincentivizes students may have been good before the system wore him down.
I would suggest you consider a less scientifically rigid discipline if you expect to be more than a high school teacher with your phd. His hypothesis literally demands you consider datasets that are not presented then guestimate those datasets. Good luck with that gym teacher career
There are certain channels on youtube which have this extraordinary quality of content consistently in all of their video's. 3Blue1Brown is definitely one of those and the content on this channel is worth gold. These kind of channels should somehow be recognized by UA-cam and be rewarded.
Agreed. However….. keep in mind that on cable tv, there’s a thing called The Learning Channel. And that channel has now become a relentless purveyor of crappy ‘reality’ shows. Point being that the mass market never will dig this sorta thing.
What about all the professors from the early 2000s who put javascript simulations on their html websites with white background and times new roman as the only font.
It makes such a significant difference to one’s comprehension when something is explained in a certain way. This is one such example, in particular, the square diagram as opposed to the usual Venn diagram usually cited.
Soo true. When I was attempting a question, the venn diagrams weren't reflecting the actual data given so I ended up with a diagram similar to his. Needless to say I clicked on this thumbnail with the quickness! lol
The cool thing about Bayes' theorem as practice is that it isn't even necessarily important that your estimates are correct or accurate, but rather that the simple act of going through the motions allows for more refined guesswork.
Wow - if UA-cam had a love button that depicted a greater appreciation of a video than the like button, I would be pressing it right now. I loved how this not only explained a seemingly complex probability concept, but also challenged the way we approach probability through visualisations. Thank you.
This brought me to tears. I've seen Bayes theorem so many times, and just plugged in the numbers. I finally have an intuitive understanding of this now. Thank you so much.
This is a REALLY nice presentation. I think that Bayes' theorem should be a mandatory subject in all schools and put in a wider context of epistemology. Even if you don't do the math all the time, just knowing the principles behind Bayesian inference changes the way you think. It is an awesome thinking tool!
Everyone tries to model the world. Those with the capacity to model with Bayes' theorem but not doing so are inefficient in their modelling, and the resulting errors are horrifying.
It is the most abused bit of math ever. Probability is taught in math and it should be taught as a mathematical concept. Applying math to philosophy and belief is guaranteed to cause misunderstanding between what is true and what is believed.
@@Uhlbelk It would take a very strong argument to convince me you are right about that. You could say that my prior belief is very low. You need a lot of independent evidence to make me update my belief enough to really make a difference ;-)
@@Baekstrom Yes, my belief has been updated by many many independent measurements of Bayes being used correctly and incorrectly and this is my current belief and would require a lot of new data to change.
I had to watch this twice to get it because of the pace but this is fantastic. Bayes theorem is usually taught as a recipe. You just go through the motions of setting up the equation and solving it not knowing how it was put together in the first place. Being able to picture the probabilities is so powerful.
I wonder if the misunderstanding in the question about Linda is simply a matter of language. Many people likely assume that option 1 excludes option 2, ie it's implied to say "Linda is a bank teller who is not active in the feminist movement". In that sense it may become almost a trick question for people who are not trained in logic.
that's almost definitely the case. I wonder if the second version of the question made that fact click for the questioned or if they still thought about it as mutually exclusive options.
Interesting thought! I’d be keen to question these 85% of people that gave an impossible answer and try to understand how they interpreted the question! Because for me I read it as “what’s more likely, A or A&B?”, which is so easy it barely counts as a question!
Yes, I think that is the point though. To show what kind of thinking process people apply depending on the situation and how problems are pressented to them.
I at least misunderstood it as that. Only on second thought did I consider the rigorous interpretation of answer 1 not excluding her being a feminist. And I'm a mathematician & have the context of the video around it being about Bayes theorem. In a different context and without mathematical training, I certainly would have chosen answer 2 because of the misleading language rather than inability of thinking about probabilities.
That was after all your Videos of Algebra and Maxwells Equations for Electrodynamics the toughest one for me! I always was just putting numbers into bayes without having a feeling for what im doing. It took me 5 hours now, several selfmade exercises and a lot of swearing but finally it made click in my head ! Thank you once more for your amazing Video! Honestly your offer of amazingly intuitiv math content makes us better students. Greetings from a Electrical Engineering student from Germany.
thank you for your insights. I'm currently in these 5 hours but getting closer. Nothing better than getting an intuitive explanation like here and then testing yourself with real exercises - loads of exercises; goes to show what is wrong with our educational system. Greetings from a Swiss economics graduate
I remember, when studying mathematics so many years ago, noticing how one of the top maths students would often use pictures, diagrams, and graphs to express formulae or other problems. From then on, I also did this and it made so many things easier across this field of all things mathematical.
Great visualisation, as always! One thing about the Linda- example: This is rather a psychologic or even linguistic effect. If you give people the choice of "people with property A" and "people with property A and property B", they will interpret it as: "people with property A but not B" and "people with property A and B"
Not even that: I see "person with property A/property A+B". There is no 'people'. It's only later all these other bank tellers are conjured up to make us who literally are focused on *person* (for that's the scenario) feel stupid. (Am seriously annoyed at 3blue1brown for this.)
Agreed. It’s a linguistic thing. When someone talks like that, the implication is that you should only pick “bank teller” if the distinct absence of “feminist” more accurately describes the state of the world. It gets translated by your mind as “JUST a regular bank teller”, as opposed to the logical superset of all bank tellers.
@@johnnyblackrants7625 my thinking is that the "or" of natural language is not the "OR" of logic, that is, ⊕ is not ∨. I haven't looked much into the study myself but I heard that the people in STEM answered the Linda question "incorrectly" more often than social science majors, which is otherwise a strange thing.
My boards examination are from this February and Bayes theorem bugged me since SO LONG because i could never make an intuitive sense out of it. I'm so happy right now that YOU made a video on that! Love from India, Grant! ❤️
@@dev0_018 you could make a rough guess what they'd say, based on their username(hate to be racist but ive read too many such yt comments from such usernames. You could say it's my bayesian estimate 💀)
@@anonymousfry well, hate it to break it to you and face you with facts but your Bayesian estimate is pretty terrible and didn't estimate anything 💀, since i hold similar name and same belief that this name derives from
Pretty much like battleship, they deduced it (if i remembered correctly) into squares (actually circles but easier to understand in squares as shown in video) and searched a perimeter and ticked off squares as they went, the ship had a given size to which it could be deduced into a probability of multiple squares (they gained evidence of where it was not AND gained evidence as they found wreckage pieces) and in se gave a higher power of finding a higher probability to find the ship in a set square in a set range. Ofcourse they assumed the last position the ship was seen as a baseline. This is what I remembered when I had it lectured to me quite a few years back. Greetz!
They found someone who knew where the ship was. Then they tied him to a chair in a cellar and said "The next person to come into this room will be a shy, meek man named Steve. He will be the one who beats you to death with this hoe if you don't tell us where the gold is. Do you want to have a guess whether he's more likely to be a librarian or a farmer? Or would you prefer to just tell us where the gold is right now?"
Best teacher I never had. You have an uncanny knack for talking about the question that just occurs to me as a result of something you just explained. Incredibly helpful. Thank you!
11:00 The first version of this question is in regular English, while the second is not. As such, the first version implies it should be interpreted in good faith, while the second implies it should be interpreted literally. And the good-faith interpretation of "which of these is more likely" is that the options are mutually exclusive; as such, if the first option is "a" and the second "a and b" it implies that the first is really trying to say "a and not b" and the writer was simply sloppy. And given that interpretation, the answer is indeed reasonable. So, I'm not convinced this actually says anything about people's abilities regarding logic or proabilities, since the results are easily understandable by assuming that the parsing rules for incoming information are chosen based on the form of said information, which is in fact perfectly reasonable behavior. In short: it's a trick question where the reasonable and literal interpretation result in opposite conclusions.
Agreed. If you said "What's more likely: Linda is a bank and a feminist, or that Linda is a bank teller and either a feminist or not a feminist" I think a lot more people would get it right.
If the question is asked by a physiologist, it appears that one can assume that the question is _always_ phrased in bad faith, with trick parts of the question that anyone rational will fixate on, but then the physiologist then dismisses as completely irrelevant. The farmer question is relevant here: how many farmers have little interest in the world of reality? Excuse me? What the heck do you think _farmers_ do? They work with real world things like dirt, animals, mortgages, and conniving scientists and anti-farm activists every day of their lives. You are telling me that successful farmers aren't interested in reality? Bullshit. So then as a physiologist you simply skip that most important part of the statement and then say, "no, it says he is meek, and that works for either farmers and librarians, so you are completely wrong."
@@lwilton the farmer question is a trick because its something we fall for. i asked myself whats the most likely result and caught myself thinking yes or no. when i noticed the sliding %bar in the video and i couldn't give a reason why it might be 55% - 45% over something close like 60% - 40% judging their character i moved on and asked how many libraries compared to farms are there. recognising relevant data was part of the experiment and even though its a trick question it still answers the study. it just implies you work with what you're given i think. but there are much better examples of how to get it wrong using intuitions and show rational thinking is a skill we need to practice
agreed. for the farmer the sample set is implied to be "types of people the question author has thought of" and not "the actual population of the world". An A.I. might have guessed farmer, and been wrong on the majority of texts that would take the time to describe an individual in this way. Steve is almost certainly a fictitious character, so the correct answer is actually "the author is probably thinking of a librarian." I do, however, think it's relevant that arm-chair researchers take into account to what extent real world data might experience this issue. I think the farmer/librarian question could be better phrased as something like "you are a data scientist studying random facebook profiles that have been constructed by an A.I., and see this profile of Steve. If you had to guess that he was either a farmer or a librarian, which would you guess?"
I've been using Bayes theorem at school for over 2 years. This video finally helped me understand Bayes theorem from an intuitive perspective, and completely changed my understanding of it
In the case of our bank teller friend Linda, I think linguistic ambiguity, and not irrationality, is responsible for the weird result: Though the answer doesn't explicitly say so, the fact that the second answer is "Linda is a bank teller and is active in the feminist movement" creates the implicit notion that the first response "Linda is a bank teller" means "Linda is a bank teller and is NOT active in the feminist movement". Since the later examples where people were asked to estimate populations of bank tellers and of bank tellers who were active feminists came to rational conclusions, it is my hypothesis that the people conducting the study didn't realize what question the original group was actually answering. If the answers had been "Linda is a bank teller who may or may not be an active feminist" and "Linda is a bank teller and is certainly an active feminist", we might get more rational answers. Better still, if we had three answers ("Linda is a bank teller", "Linda is a bank teller and is NOT an active feminist", and "Linda is a bank teller and an active feminist") that might produce the best results overall, though there is still ambiguity in how people choose to read the meaning of the answers.
Exactly. The fact that people don't always interpret questions literally, or the way a logician would, isn't a fault of human reasoning. It reflects our ability to make assumptions about context in which we're being asked things. I wouldn't fault anyone for assuming that option 1 excluded option 2, thinking that this must be the intended meaning since it would be a ridiculous question otherwise. Just another example of psychologists drawing grand conclusions from linguistic ambiguity.
@@antiawarenessawarenessclub Because the second way of asking it asks a fundamentally different question; I think any person with a basic grasp of numbers would know that you can't have a subset of a group larger than the group. It further removes some ambiguity by parameterizing the group; we're explicitly told that 100 people fit the description, and to dead-reckon how many are bank tellers and, of those bank tellers, how many are active feminists. BUT - and the video didn't address this, so I wonder if the study did then, too - if we follow up our population estimates by asking the original two questions, we still have the problem where the first question implies "...and is not an active feminist." Based on the answers given in the study, if that ambiguity is in play, you wind up with the same non-Bayesian error: 8 people in the group are tellers and of those 5 are feminists, so it's more likely that Linda is an active feminist bank teller rather than an apathetic one. In the case of the population-estimating version of the question, what we really have to ask to eliminate ambiguity is "Out of 100 people, what are the odds that Linda is a bank teller?" (8%) and "Out of 100 people, what are the odds that Linda is a bank teller AND an active feminist?" (5%). Then when asked which statement is more likely, the ambiguity of which population groups we're discussing is clear ("all bank tellers total vs those tellers who are active feminists", rather than "all bank tellers who are not active feminists vs those tellers who are active feminists").
@@antiawarenessawarenessclub Because of the way most people are conditioned to approach multiple choice questions. On a multiple choice test, generally 1 answer is THE correct answer, and the rest are considered wrong (even if they are factually accurate), if more than one seems applicable we are taught to choose the one that is most accurate. So people are likely to ignore the bank teller portion of both options and focus on the difference between them to decide which is more accurate: is she an active feminist or is she not? The second form of the question doesn't have this ambiguity because there's no multiple choice to trick us into seeking a single best answer, and instead we have 2 separate and open questions. Even if you remove the "out of 100 people" part of this question and ask for percentages or probabilities you're likely to get the same rational results simply because they are now 2 separate questions instead of 2 competing choices to the same question.
Actually there isn't an ambiguity in language, "Linda is a bank teller" includes all bank teller possibilities. People just mentally interpreted that "Linda is a bank teller" means "Linda is a bank teller and is NOT active in the feminist movement," which is a flat out _wrong_ interpretation.
I was on the dean's list in my undergrad engineering major, and graduated with high honors in my 'brand name' MBA program. This is one of the BEST explanations of a fundamental tool of analytical thinking and insight, whether for business, medicine, law, sports,online dating(!), or just clear thinking I've ever seen. I learned and (explicitly & implicitly) used Bayes for decades, yet your preservation has given me another window into understanding/reminding me of its value in everyday thinking ... wish u were one of my prof's. Godspeed, the world needs more of your talent.
I am studying for the actuary P exam and I worked through all of my practice problems by making these diagrams. Thank you! I now understand Bayes Theorem.
I just love how you manage to visualize mathematical concepts! I've been drawing rectangles with subrectangles to help intuitively understand problems involving probability since before I was taught probability in secondary education, but I've never tried to represent Bayes' Rule so elegantly.
This is hands down the best explanation of an abstract mathematical concept that I have ever witnessed in my 33 years of living and learning. Absolutely brilliant display of mastery in teaching and explaining.
I'm told that many medical doctors do not understand Bayes Theorem, and it can be threatening to peoples' health. Example: There is a test for a very rare disease, and the test correctly gives a positive result for 95% of the people who have the disease. Your test comes back positive. What is the probability you have the disease? Unfortunately, a lot of people, including some MDs think the answer is 95%. The actual probability you have the disease can be much smaller if the false positive rate of the test is high and the fraction of people taking the test who do not have the disease is high. BTW, when I worked at FICO (the credit scoring company) we used Bayes Theorem so often they gave all of the employees shirts with the formula embroidered on the sleeve.
The way I think of it is that Bayes Theorem gives you a way to turn some measurements you can make, but which aren't really all that interesting, into something you can't measure, but which you're really interested in knowing. Like in your example, you can turn the probability of getting a positive test result for anyone who actually has a disease (which is measurable and is interesting, I guess, but not of huge importance to most people) into the probability of actually having the disease, given that you got a positive test result, which is not directly measurable but is going to be of extreme interest to anyone who gets a positive test result. The false positive rate doesn't have to be very large for a positive result to be largely meaningless. For anything rare, the odds that a positive result is meaningful is going to be small unless the false positive rate is similar to the rate of the condition in the whole population because there will be far more false positives than real positives.
@@parthashah9257 UK medic here... Check out more docs over the next few years, then update your beliefs :).. in the UK, 40% eligible health staff do not have free flu' jabs, because of false beliefs, mostly "I had the flu' straight after, once..", which appear impervious to the new evidence which in a rational system would update their beliefs :)
Gerd Gigerenzer studied this, and the approach of thinking about absolute numbers instead of probabilities (like in the video) seems to help in practice.
Thank you. For an aged brain this is one of the most accessbile and comprehensible explanations I've found. As Andyg2g commented below, for me the phrase "rationality is not about knowing facts, it's about recognizing which fact are relevant" lit up my understanding!
This gentleman is a master in teaching, he makes difficult things easy to understand in a variety of different topics. I have been watching his videos about different subjects and he is really amazing. Congratulations Sr. !.
what 8-9 hrs of watching several videos and tutorials, reading various texts could not explain me why is baye's formula the way it is was explained by this channel in just starting 5 mins without even showing the formula Brilliant!!
I imagine most people interpreted the bank teller question as "1) She is a bank teller not active in the feminist movement, 2) She is a bank teller active in the feminist movement". That was the first thought when I interpreted it anyway.
Yeah. They're basically telling us that she 100% IS a bank teller. So the only question left is whether she's an activist or not. I get what he meant to say but the question doesn't really fit.
No they’re not. They’re saying which is more likely? Not, given that they’re a teller, which is more likely? And these are very different things. Not sure how you’d justify interpreting A or (A and B) as meaning the first A was A and not B either, in response to the initial post
I agree. I had the same interpretation, and I think the problem lies on the difference between verbal language and mathematical language in terms of precision. It requires some "fluency" in math to convert the problem mathematically. (Sorry about my English haha)
@@kellmano1 Well they're asking: 1) A (without B) 2) A with B The way I understand her description she's more likely to be a bank teller activist rather than only a bank teller.
@@ervindark Haha, that's definitely not what they are asking. 1) Is she a bank teller ( including activist /not activist ) 2) Bank teller and an activists, the first actually includes the second options hence the propability is bigger, is it clear now? you added information wrongly to the 1) that "she is not an activist"
Immanuel Kant already figured that one out back in the 1700s so we're a few centuries late with it. He wasn't very good at writing snappy quotes though
@@francescocraighero5392 I'm sorry to say that Daniel Kahneman in his book Thinking fast and slow debunks most of the things that says that guy in his blog.
@@Ucedo95 In the last months I encountered that book many times, I think it's definitely time to read it. I don't know where WBW made wrong assumptions, but I think that the contribution that Tim gave by visualizing this topic will still be worth a read
By the way, the current decade will end on 31st December 2020, as there will be 202 decades since Christ was born, allegedly on the 25th December. Considering a decade for 2010-2019 is 10 years ok, but is misleading as one of the previous decades in history must be 9 years only. Because the year 0 does not exist for historians. So the first decade in history was not 0-9 but 1-10.
Once again you're explanation of something I have struggled with has made it intuitive. Your method of visualisations (with it's own language) is changing the way that math can be taught - by relying less on formulas and text and more in intuitive, visual ideas that our brains can retain so much more effectively. Thank you for your work Grant!
"New evidence does not completely determine your beliefs in a vacuum; it should update prior beliefs." That's the quote i'll take for my life after this video. Ty Blue, discovered your channel yesterday and been mesmerized by it since then.
This is an amazing video, but I'd like to point out that human speech doesn't occur in a vacuum. More specifically, people give answers that are useful to the addressee more often than answers that are technically true; after all, that's why people communicate (think: 'there's a shovel in the shed if it snows'; does the shovel cease to exist if it doesn't?). In the case of Linda, for example, it is more useful to say that Lind is a bank teller who is involved in the feminist movement (assuming that her description matches being a feminist more than not), given that the addressee seems to know, or at least have assumed, that Linda is a bank-teller already (answering that Linda is not a bank teller is not an option). Again, this video was amazing, but I think it's worth pointing out that a large and useful(!) part of human communication does not hinge on mathematical truth but on interspeaker convenience and we really shouldn't strive to 'correct' human judgments or label them as necessarily wrong.
This is an important point. We think that the text is informative, so the added information must provide some additional effect for the consequences we draw from the text, otherwise the speaker probably would not have provided it. Especially in a task like that where the hole point is to draw consequences. The idea that the pieces of information given in a cooperative conversation should be relevant goes back to the philosopher H Paul Grice, his “Maxime of quantity” and of “relevance”. There is lots of articles written about the Linda fallacy but as far as I know nothing makes this point.
Humans like to embellish their answers with fiction as it gives the impression of knowledge even if it is unsupported or fanciful. . The question was which was the more probable. That is why in courts the lawyers often ask the question and insist on a yes or no answer to cut through the irrelevant waffle!
That's the point. Just because we think like that for most questions doesn't mean it's the way to think in this specific context. And such lack of reevaluation of belief can lead to silly situations at best, big mistakes and their consequences at worse.
I love how you bring in the part about objections to Kahneman & Tversky's research. Gives us a very thorough understanding about context around the topic!
None of these objections are objections against Bayes Theorem used for updating beliefs however. They only propose that in the specific experiment more steps of updating the belief to get a different prior probability would be needed.
Please include in future discussions the relationship between bayesian inference and the scientific method and how all these things are related to deductive and inductive reasoning. Your content is amazing! Thank you!
I think the use of that second prompt actually reveals yet another mistake in human cognition: assuming humans are concise rule followers. 85% of people are getting the bank teller question wrong, not because they aren't thinking about the set of sets, but rather because they're inherently correcting for the perceived mistake you've made. They read the question, distilled, as "is she more or less likely to be a part of the feminist movement, than to not be." The reason for this, is that asking such a question of someone doesn't make any sense, since it's 'intuitively obvious', so they assume you've made an error and correct for it. In your rephrasing of the question, that presumed error goes away, because you're asking the percentage of generic people filling particular categories, and the question actually makes sense to ask, since an rational person can come up with genuinely different answers for each. In the previous example, one cannot answer any differently than a bank teller, which triggers their instinct that you've made a mistake in writing your question. You can see this very thing at work when people read articles with misspellings, or read texts with words that don't make sense in context. They'll automatically fix the spelling when reading, or find a word close in spelling that does make sense contextually. The assumption that people are like machines, doing things wholly within the defined ruleset, whether that's the rules of English, of culture, of whatever, is a fallacy. People are intuitive thinkers, they don't follow a prescribed set of rules as defined, they follow what they perceive or believe the rules are intended to be. That's why we can read the same set of rules and come up with different interpretations, because we have different priors and knowledge before reading and attempting to interpret said rules, despite the words we both read being identical.
Same thing with the librarian bit. I would be thinking about who wrote the description and would guess that 95% of people would describe a librarian as someone organized and with a farmer they would say something about nature. This perceived probability strongly over rules any % of farmers and librarians in the population. It's not that the farmers don't fit the description, many probably would, but it wouldn't be the first and only thing you say about them.
@@dp2404 Another point with the librarian bit is that it reads like "am I, the writer of this question, thinking of a librarian or a farmer when I made up this character Steve"? If it was phrased like "select a random individual from the actual population of the USA, with these traits" then it would naturally lead to thinking about real-world proportions of farmers and librarians.
@@dp2404...but the description includes the fact that "Steve" has "very little interest in the world of reality" -- and that fits precisely 0% of all the farmers in the world. It might fit a non-zero percentage of bankrupt ex-farmers, but working farmers depend on "the world of reality" for everything they do... The video as a whole is excellent, but that one phrase in the description in the beginning really broke my immersion.
@@s_m_we addressed this kind of thing at 3:05. The point is that most people don't even consider the prior probabilities. If you do, and come to the conclusion that this evidence makes the probability of being a farmer extremely low, you've still applied the theorem correctly. But this is not how most people get to their judgement in this question.
This geometrical approach to probability is gold worth. I struggled a lot to conceptualize a certain task we had, because our teacher introduced new variables instead of using the "not" symbol. I understood the steps but i don't not understand it intuitively. THIS, this worked excellent, i just need to add a couple of squares together and i get the result. It made it super easy to understand the task at hand as well, thanks a lot.
Years of bashing my head against the wall trying to build something resembling an intuition on Bayes Theorm, and in 15 minutes everything has now fallen into place. You are amazing, Grant! Thank you so much!!!!!!!
If you want to intuitively understand Bayes theorem, forget about Steve, leave the Bayes Church and adopt the frequency interpretation of probabilities (or at least accept that is exists). And then write down the theorem in a symmetric form. It becomes an almost trivial and intuitive statement about sets. But, of course, writing it in an unintuitive unsymmetric form and rejecting the interpretation of probabilities as predicted frequencies makes it somehow deep and mysterious. So you have to choose a side. You cannot have it both. Either you can intuitively understand it, or you can stick to the believe that there is something non-trivial behind it.
I'll just add that the explanation is GREAT! A pure example of how if you really understand something you can explain it with simple words to everyone. Thanks
This is a way of getting people to be much more rational in their beliefs. And 3Blue1Brown is a great teacher putting up stuff for free for us all to learn from and if everyone saw this and took the time to understand it we would have a better world. This guy is amazing!
11:12 I believe this too is a problem with the education system. In MCQ type questions, if multiple options are correct, we are expected to choose the "more correct" option. As an example: Q is a gaseous element that reacts with oxygen to create common water. 1. Q is an element in the periodic table 2. Q is the first element of periodic table Even though, 2 is a subset of 1, I can say with utmost certainty that the majority of students will answer 2.
but given the prior that the students know it's the first element, the probability for both is equal, so there is no answer more correct than the other. Once you know that Q is hydrogen, which happens before the MCQ, then all you need to do to reach this conclusion is to evaluate the truthfulness probability of the choices by plugging the answer, and this becomes What is the probability for each of the following statements being true? Hydrogen is an element in the periodic table Hydrogen is the first element of the periodic table They are both true, so none of the answer is more correct than the other, since you already knew the answer before the question being asked. The result above can also be determined with the formula explained. Say that we want to test answer 1, which becomes the first tested hypothesis, H1. The formula, as presented in the video is P(H|E) = (P(E|H)*P(H))/(P(E|H)*P(H)+P(E|~H)*P(~H)) To calculate P(H1|E) we need all the above terms, but let's start with the easy ones P(H1) = ? , or in other words, what is the probability that Q is an element? Obviously this depends on how we define our space, but let's use the periodic table as a space, so then P(H1)=1 P(~H1) = ? , or in other words, what is the probability that Q is NOT an element? Obviously, P(~H1)=0 P(E|H1) = ? , or in other words, what is the probability that Q is a gaseous element that reacts with oxygen to create common water given that Q is an element in the periodic table? Well, once again, we know there's exactly only hydrogen out of all the elements, so the answer is P(E|H1) = 1/n , where n is the number of elements in the periodic table. Let's simplify and say that we only discovered the first 100 elements, so n=100 P(E|H1) = 1/100 P(E|~H1) = ? , or in other words, what is the probability that Q is a gaseous element that reacts with oxygen to create a common water given that Q is not an element in the periodic table? Obviously 0, although I suspect the more appropriate answer is undefined. P(E|~H1) = 0 If we plug all these in, we get P(H1|E) = ((1/100)*1)/((1/100)*1+0*0 = 1 P(H1|E) = 1 Same about H2 P(H2) = ? , or in other words, what is the probability that Q is the first element of periodic table? Considering the same space of the periodic table, then P(H2)=1/100 P(~H2) = ? , or in other words, what is the probability that Q is NOT the first element? Obviously P(~H2)=99/100 P(E|H2) = ? , or in other words, what is the probability that Q is a gaseous element that reacts with oxygen to create common water given that Q is the first element of periodic table? Well, we know there's exactly only hydrogen to be first P(E|H2) = 1 P(E|~H2) = ? , or in other words, what is the probability that Q is a gaseous element that reacts with oxygen to create a common water given that Q is not the first element of periodic table? Obviously 0 P(E|~H2) = 0 So, we get P(H2|E) = (1*(1/100))/(1*(1/100)+0*(99/100)) = 1 P(H2|E) = 1 P(H1|E) = P(H2|E) , so both answers should be accepted as being the most correct answers. This problem is different than the Linda problem in the video. To make it equivalent, assume that you personally know that Linda is a bank teller and that she is active in the feminist movement. The 2 problems are also equivalent if you eliminate the priors, and then everyone would give the more inclusive answer. Say that you're only asking your question to people who don't know that Q is hydrogen, or that do not see any correlation between being active in the feminist movement and the evidence presented in the question. Then these people would pick the more likely answer, meaning the first, each time. This means that people are very selective with the priors they use. Moreover, people are often tricked by the fact that hypotheses overlapping, but people guess they are distinct (and sometimes complementary, and sometimes equal). So given H1 and H2, people make the following assumption P(H1|H2)=P(H2|H1)=0 (and sometimes P(H1)+P(H2)=1, and sometimes P(H1)=P(H2)=0.5), which means that the only possible way of testing actual knowledge using MCQ is for choices to hold as manu natural assumptions as possible, but at least the first, P(H1|H2)=P(H2|H1)=0 . So the proper choices for your question should be: 1. Q is an element on an odd position in the periodic table 1. Q is an element on an even position in the periodic table This way, the student will only perform better than chance if they truly know the exact answer.
I think the discrepancy in the Linda part can be that people see the two options they juxtapose them and intuitively take "a bank teller" to mean "just a bank teller and nothing else". Thinking fast and slow is pretty good, just about finished with it. I'd highly recommend it. Really changes your brain.
11:30 the problem with this question is the answer "banker" implies "Banker but not feminist", making them disjoint. It doesnt mathematically mean that, but in English it does. Consider if you had toast for breakfast, or toast and and egg, the first doesnt explicitly say you didnt have an egg, but it is implied by the second option
"Rationality is not about knowing facts, it’s about recognizing which facts are relevant." I would like to know if Mr. Sanderson himself wrote this line or someone else. It took me three weeks to fully absorb this. It helped me with my analytical ability, and is now one of the constructive pillars of my discussions.
We use Bayes Theorem when we assess and control risks on a day-to-day basis. It can even be common sense one could say. When we see something unsafe we are wondering about the likely hood and the consequences of continued action or inaction. When something might be unsafe there is a primal need to explore and gather more information in order to make better choices. This is displayed time and time again in nature and society. The snap judgements humans and other animals make seem to suggest an almost innate ability to see probabilities. Love the line in this about rational thinking being the ability to know which information is relevent. Keep up the good work!
A year later, watching again. Still good! This also gives good advice on how to argue with people who hold beliefs that are not backed by evidence. A lot of people target the likelihood, getting bogged down in trying to adjust the person's percentages. We forget to take into account the size of their prior.
Came across this video on a whim and I gotta say, I studied computer science in college with multiple classes touching on this subject and this is by far the best explanation I’ve ever seen. Fantastic teaching
Please make a series on probability like u have done for linear algebra and calculus. They have helped me a lot to visualize the topic but also to appreciate what I'm learning. Thank u for your work
Can't Thank you enough for the illustrations that make everything clear and easy to recall. Also, the fact that it is not just about teaching the formula but the concept and the notion of it is what we all need. Thanks a million.
Where Y is a subset of X, perhaps asking if she is more likely "an X or an (X and a Y)" is being interpreted as given that she is an X, is she more likely: A: (X and Y) B: (X and not Y) This is the same as swapping out the "or" for an "xor"? The two are used interchangeably, often the wrong way round in plain English! "It's this or that?" usually means "It's this xor that?".
Yeah I'm highly sceptical about the psychological import of these experiments. I feel like it's mostly explained by the vagueness in the word "likely". As soon as you put the problem in context, the incorrect answers disappear. Which totally makes it sound like a communication issue rather than a psychological flaw.
X and not Y is a subset of X. Therefore P(X) = P(X and Y) + P(X and not Y) which implies that P(X and Y) < P(X) which means Lynda is more likely to be bank teller than a bank teller who is part of the feminist movement.
@@Karthik-lq4gn You've misunderstood. Yes, P(X and Y) < P(X) is always true, but whether P(X and Y) < P(X and not Y) is not known, which is how Alan is saying people are interpreting the question.
@@gorgolyt Yeah, these experiments are no longer considered valid in as far as the original conclusions that were made, but are still important in that they provide good data showing that how a question is phrased can change the way a person interprets a question, and therefore how they will answer it (which is really important in any country where the citizens vote).
I think there's an additional piece of information when considering the Steve problem, which is that being told about the personality traits makes you think that his personality is what's most relevant to the correct answer. Basically it's like when you're given a formula or a hint on a test question, the fact that you're given the formula at all automatically makes you think that you have to use it somewhere in your solution (even if it's totally irrelevant and the examiner is just being a dick)
It's not a dick move to test how well you actually understand the problem. If the irrelevant formula trick works on you, it illustrates that you understand the material so poorly that you have no idea what pieces of information you need to solve the problem. Likewise, if you're so foolish as to think that completely irrelevant information about Steve is relevant just because somebody mentioned it, that's still an illustration of your irrational tendencies. These tendencies can and likely will be exploited in your day to day life by sociopaths such as in politics. We can't go through our lives expecting examiners to wipe our asses for us.
@@tj12711 Humans are not superrational machines and things like proximity and wording will inevitably affect how they think. This has been scientifically proven hundreds of times and telling people to "be more rational" does not change this. This is how you can get drastically different responses in a poll just by changing the wording around, even if semantically, both polls ask the same thing. If you're telling yourself that you're so rational that you don't have any cognitive biases, you definitely aren't self-examining close enough. If you're exploiting cognitive biases on exam questions to make the exam harder, you should probably ask yourself whether that's the most effective way to test the students' understanding of the material.
@@tj12711 However, provided that the examiners did not remind the students about the relevant info (like the proportion of librarians to farmers, or the fact that the options were not mutually exclusive), the examiners capability of proving their hypothesis about the students was impaired, since the students probably only used the available context to answer both questions. So, yeah, in your everyday life, and specially in politics, it is better to consider statistics more than anedoctal evidence, but in a test (specially written ones) it is not feasible to expect students to consider every piece of possible relevant information without proper context, specially in Mathematics with its formulas.
if it is true that certain personality types are more likely to frequent an occupation, then knowing that person's personality type does make it more likely to better place them. being told something does not make it more likely, but the correlation between two similar things can make it likelier. if we are told steve hates working outside or getting dirty we can assume he is probably less likely to work as a farmer. at the same time that doesn't make it impossible (steve may not have a choice and is desperate for a job)
Brains are trained to compare similarities (which is really useful), not probabilities (which is an artificial concept invented by mathematicians). So when you speak mathematician-gibberish to an average person, they still run the intuitive similarity matching in their head.
This music is very calming. I can't tell if it's distracting, but it's very calming. The content is very good too. Will keep coming back to it until the knowledge sets in a little better.
I wish every teacher would take students from understanding a mathematical theorem conceptually into reasoning with them to the actual formulas with this order, Truly remarkable!
I had to watch the video about 5 times, bring in some rough paper and pen for some visualization but I learnt a lesson for a lifetime. Thank you for such amazing explanation, its just amazing how new evidence can shape our prior perspectives and affect our mindsets completely. Kudos ❤
I feel like the hard problem here is recognizing when you have missed something important like how people missed that the ratio of librarians to farmers, was something they should have taken into consideration. Most people, if given a story problem, will reflexively self limit themselves to only the evidence in the stated problem.
I work at a high-tech company and you have just saved me a lot of pain! Now I can finally quantify my believes, present and update them! Thank you so so much!!!
This is why it's so important in machine learning to have a balanced dataset. If a model is programmed to reply 'farmer' every time, the model has an accuracy of 95.4%. This sounds both great and horrible at the same time.
Actually it is way more important to have a relevant loss function than a balanced dataset. If your goal is to minimise the number of errors then replying 'farmer' every time would be not so bad at all. Things change drastically when every wrong answer takes $100 from you and right ones give you just $1
@@dmitrynovikov5844 that wouldn't be enough. With such extremes in the dataset and an overcompensating loss function, it will likely make the distinction between 'farmers' and 'non-farmers' considering it has so many farmers to work with. This means that in practice it would classify atypical farmers that it didn't train on as librarians. It would see cats and firemen as librarians as well. In practice you'd want the model to return a low confidence on both librarian and farmer when a cat goes through the model. But that depends on what you'd want from the model. It might be right in most cases, but it's just not as elegant.
@@deldarel Yes, having a balanced dataset is very important if possible, but in cases where it isn't, there are other ways of evaluating the performance of an algorithm besides accuracy - (number of correct answers) / (number of total examples). Such as: dividing up all the answers into 4 categories: True positives, false positives, true negatives, and false negatives instead of just whether it predicted positive vs negative can help. Let's go with your example of an algorithm that gets 95.4% accuracy by always predicting "Farmer." If you know your dataset is skewed, and you notice in your test results that it predicts "Farmer" 100% of the time, instead of accuracy you can use a formula like (# of correctly predicted librarians) / (actual # of librarians) to evaluate your algorithm's performance. This formula basically asks the question "out of the number of people we predicted to be librarians (all predicted positives), how many are actually librarians (true positives)?" This alternate way of evaluating can quickly reveal an error, as if your algorithm is predicting "Farmer" every single time it would quickly be revealed that it never predicts "Librarian" because (# of correctly predicted librarians) in the numerator is equal to 0 so the performance would be 0%.
Bae's theorem: The probability that your bae is hungry, provided that she is angry is equal to the probability that your bae is hungry and angry divided by the probability that your bae is angry.
Thank you Grant. You were my weekend movie for 1 year in 2020. And now, you will be my pastime, my hobby and weekend entertainment for 2 years at least.
The problem with Lisa is in my opionion the phrasing. Intuitively, people might assume that "Linda is a bank teller" is more like some prior knowledge. Or, to phrase it differently, the question sounds more like P(Bank Teller, Feminist | Story) vs P(Bank Teller, NOT Feminist | Story) which essentially boils down to P(Feminist | Story) vs P(NOT Feminist | Story), which clearly edges for second answer.
Same with Steve. I think the point was a lot of people jump straight to: "Which group of people is more likely to be shy and withdrawn?" i.e. straight to the likelihood, neglecting the prior.
But that's the whole point of it. The question was entirely unambiguous but people managed to misinterpret it and arrive at a nonsensical conclusion _because_ they're grappling with unfamiliar, abstract mathematical ideas. Re-framing the question in more familiar, concrete terms resulted in a very different outcome.
Another example they cite in the book (Thinking Fast And Slow) , that follows the exact same structure as the Linda example is: Which alternative is more probable? Mark has hair. Mark has blond hair. by your logic people assume the options are (Mark has hair, excluding blond hair) and (Mark has blond hair) are the actual assumptions people have because these follow the exact same logic structure as the Linda problem.
@@raafaytariq9132 If blond hair was exceedingly common (given a story), then under that interpretation B would be correct like maybe the setup is that "mark's hair often shines bright in the day / mark's hair is a light tone" or something like that Given the interpretation that A v A & B implies the question is A & -B v A & B
@@MagicGonads I completely agree that if the given context was one that assumed blond hair was more common, it changes the story. However, it doesn't make sense to say that you can both infer A &-B v A & B because of context of the wording, yet you can't infer context of the question. You can either take it at straight value, and infer no context. Or infer your frame of context for both the logical argument and context of the argument.
I need some time to reprogram my feelings after this video, it was a hell of an exciting ride and i'm not even that interested in probabilities !! OH MY GOD THIS WAS SO GOOD.
My Intro to Statistical Reasoning prof assigned this video for us to watch and take a quiz on. What a fascinating topic! You are making seemingly complicated math accessible to everyone. Thank you so much!
As always, another wow moment. I'm waiting for the day when the intuition behind solving partial differential equation will be explained. Especially about CF and PI and how you interpret them physically on a graph
I just wanna take the moment to present my gratitude to you. I really appreciate the work that you put in to make us all understand such important and not so intuitive concepts. Thank You.
13:38 I actually thought farmer was more likely because I share a lot of personality traits with Steve so I approached it thinking "what job would I more likely have ended up in" and since I live in the country, farming was way more likely.
This is so weird... When presented geometrically like this, I'm having a hard time understanding the context. But I knew Bayes theorem well enough. It's just my brain can't comprehend it when you present it using geometry like this.
I had a similar issue. I guess I'm just not used to thinking about it that way. I learned conditional probability using contingency tables and tree diagrams; so, maybe is just a matter of not being used to seeing things this way. Still, it's cool to see other forms of thinking about Bayes theorem and conditional probability and getting a different perspective. Plus, the explanation was great.
@@rjsr03 I agree, the explanation was great. I wonder if the first exposure I had about the theorem is using this kind of explanations, then maybe the way I think about and use the theorem will also be different too.
The same thing happened with me in Calculus series. I understood differentials and integrals quite well, but then Grant came out with visuals for those, and it seemed kinda unnecessary to my knowledge. However I do think that we should learn to look at one thing through different angles.
I think that, because the topic is complex, you have to slow down the video, go back a few times, but eventually you'll get it. I think this is a fantastic video!
“Rationality is not about knowing facts, it’s about recognizing which facts are relevant.”
I felt this.
Isn’t that Wisdom?
And recognizing which fact matters and which one doesn't is the challenge.
Marcelino Deseo that’s Wisdom
@Lo Po yes, but all facts are not relevant in all situations.
Residuals and PCA anyone?
"Evidence should not determine beliefs, but update them."
This is pure gold!
so then why arent you talking about race and IQ?
Yeah. That's why we didn't throw out relativity 6ish years ago when it muons were measured moving faster than light.
It turned out that literal bird shit had caused the error, it was on the sensors.
I think it’s a bit loose. Evidence should determine our prior beliefs and new evidence should update them. Thus, evidence should determine belief generally.
Stubborn
@@biggieboomboom the calculated posterior can be seen as the updated prior. that's why this sentence is gold.
I don't normally comment on youtube videos. But I must say this 15-minute video has helped me to grasp Bayes' Theorem so deeply that i was able to solve all the Bayes' Theorem-related questions in my recent math exam intuitively, with minimal plugging of formulas! It feels like magic. I am deeply grateful.
What does it mean if I still don't understand this theorem intuitively or deep as you?
@@yashaswikulshreshtha1588 doesn't mean anything, everyone learns differently
@@yashaswikulshreshtha1588 update: 6 months after diving deeper into math. I have come to find that applying and understanding the intuition is just the start. I’ve learnt that in fact, I don’t know much 😅
@@justlooking9802 good to know i m not alone
@@justlooking9802 that's the start of wisdom when you realize this
thinking of events as "H" (hypothesis) and "E" (evidence) instead of random variables (A, B, C,...) is definitely game changing. personally, it made the theorem much more immersive and useful. also, brilliant demonstration!
factsssss. i actually understand the math and can visualize while working instead of just using some formula and plugging stuff in
totally agree, our teacher thought this to us with A and B, but this is way more intuitive.
The follow-on video mentioned here did not, er, end up getting finalized and published. At least not yet! I have a bad tendency to do this with probability videos, where there are always plans and drafts for more, but they often don't quite feel "there" once they're more fully mapped out.
That begs the question, "What is the probability you'll actually do it?" lol
Well, what were the chances of it being made? I think with this knowledge, we can look in retrospect and update our views on the chances of it occurring.
Thanks for all your hard work and the excellent quality of the content. Look forward to the next release on Bayes.
Dang - but thanks for the heads up! I was about to go searching for it and I'd probably have wasted way too much time looking since I assumed the likelihood the video existed was close to 1.0. Now I need a model for how to update my beliefs given an unknown probability! ;)
You should square off on this one more time..think false binary..inputs
First time in my life at age 55, I really understood Bayes Theorem. The link between the tree diagram and this box explains why the probabilities on successive branches of the tree diagram are multiplied. This is brilliant.
Dude, imagine every child had a math teacher as good as you... Congrats.
There's still going to be ones that fail. A subject only makes sense if you're interested in it or have some intuition of what is happening.
As Asanda said, the actual maths teacher has to manage the 50% of the class who don't give a shit about anything, no matter how well it is presented! Then there's the student who will put up their hand and ask "Is this examinable?". Then there's the parent-teacher meeting where you get accused of going "off-track". There are many hurdles to prevent inspired teaching.
@@garethb1961 I think you're missing the point though; all those hurdles will still exist but the maths teacher would not be a hurdle which is definitely not the case for a lot of students unfortunately.
@@stretch8390 I don't think I missed the point at all. That boring maths teacher who can't teach for shit and disincentivizes students may have been good before the system wore him down.
I just want to say that fortunately, any child with youtube and curiousity can have him as a math teacher :D
hey you finally did the probability thing
It was bound to happen.
What are the chances, right?
@@DharminShah09 good one
@@DharminShah09 *Shakes Magic 8-ball* ...... "All signs point to you being gay".
Yeah i thought it was probably not gonna happen
I'm an astronomy PhD student and this is hands down the best explanation of BT I've seen on the internet. Well done.
I would suggest you consider a less scientifically rigid discipline if you expect to be more than a high school teacher with your phd. His hypothesis literally demands you consider datasets that are not presented then guestimate those datasets. Good luck with that gym teacher career
Well I'm an observational astronomer so not really planning on doing anything terribly theoretically rigid! What is your PhD in?
@@andrewjolly319 😂good question....
good reply!
I'm a calisthenics athlete, and this is one of the best BT explanation I've ever seen.
There are certain channels on youtube which have this extraordinary quality of content consistently in all of their video's. 3Blue1Brown is definitely one of those and the content on this channel is worth gold. These kind of channels should somehow be recognized by UA-cam and be rewarded.
What are the others?
Can you suggest some names?
@@Naklibatuta Check the Channels column of this channel.
Nominate them for a Webby and vote!
that can be your job. good suggestion.
Agreed. However….. keep in mind that on cable tv, there’s a thing called The Learning Channel. And that channel has now become a relentless purveyor of crappy ‘reality’ shows. Point being that the mass market never will dig this sorta thing.
You will be known in the future as the father of visual mathology.
Aye that thee will
What about all the professors from the early 2000s who put javascript simulations on their html websites with white background and times new roman as the only font.
he spent so much emphasis of visualization
Couldn't agree more .. his visualisations show such attention to detail , it's awe inspiring
TapOnX they were the primitives
It makes such a significant difference to one’s comprehension when something is explained in a certain way. This is one such example, in particular, the square diagram as opposed to the usual Venn diagram usually cited.
Soo true. When I was attempting a question, the venn diagrams weren't reflecting the actual data given so I ended up with a diagram similar to his. Needless to say I clicked on this thumbnail with the quickness! lol
The cool thing about Bayes' theorem as practice is that it isn't even necessarily important that your estimates are correct or accurate, but rather that the simple act of going through the motions allows for more refined guesswork.
AGI needs this.
SCWAG (Dr. O J Curry ) scientificly computed wild ass guess
“Rationality is not about knowing facts, it’s about recognizing which facts are relevant.”
pure facts
I'm always blown away by how good these videos are, especially when I look back to how I was taught these concepts. Keep them coming!!
I was taught like that the equation was written on the board and then said "tomorrow we will have an exam on this". Sad.
Wow - if UA-cam had a love button that depicted a greater appreciation of a video than the like button, I would be pressing it right now. I loved how this not only explained a seemingly complex probability concept, but also challenged the way we approach probability through visualisations. Thank you.
This brought me to tears. I've seen Bayes theorem so many times, and just plugged in the numbers. I finally have an intuitive understanding of this now. Thank you so much.
Yes, nick. Me too
We need this kind of intuitive thinking. I wanted to study maths in this manner, how he teaches is brilliant.
Drawing out the table truly is a wonder
This is a REALLY nice presentation. I think that Bayes' theorem should be a mandatory subject in all schools and put in a wider context of epistemology. Even if you don't do the math all the time, just knowing the principles behind Bayesian inference changes the way you think. It is an awesome thinking tool!
It's taught in India in 12th grade.
Everyone tries to model the world. Those with the capacity to model with Bayes' theorem but not doing so are inefficient in their modelling, and the resulting errors are horrifying.
It is the most abused bit of math ever. Probability is taught in math and it should be taught as a mathematical concept. Applying math to philosophy and belief is guaranteed to cause misunderstanding between what is true and what is believed.
@@Uhlbelk It would take a very strong argument to convince me you are right about that. You could say that my prior belief is very low. You need a lot of independent evidence to make me update my belief enough to really make a difference ;-)
@@Baekstrom Yes, my belief has been updated by many many independent measurements of Bayes being used correctly and incorrectly and this is my current belief and would require a lot of new data to change.
I had to watch this twice to get it because of the pace but this is fantastic. Bayes theorem is usually taught as a recipe. You just go through the motions of setting up the equation and solving it not knowing how it was put together in the first place. Being able to picture the probabilities is so powerful.
I wonder if the misunderstanding in the question about Linda is simply a matter of language. Many people likely assume that option 1 excludes option 2, ie it's implied to say "Linda is a bank teller who is not active in the feminist movement". In that sense it may become almost a trick question for people who are not trained in logic.
that's almost definitely the case. I wonder if the second version of the question made that fact click for the questioned or if they still thought about it as mutually exclusive options.
Interesting thought! I’d be keen to question these 85% of people that gave an impossible answer and try to understand how they interpreted the question! Because for me I read it as “what’s more likely, A or A&B?”, which is so easy it barely counts as a question!
Yes, I think that is the point though. To show what kind of thinking process people apply depending on the situation and how problems are pressented to them.
I at least misunderstood it as that. Only on second thought did I consider the rigorous interpretation of answer 1 not excluding her being a feminist.
And I'm a mathematician & have the context of the video around it being about Bayes theorem. In a different context and without mathematical training, I certainly would have chosen answer 2 because of the misleading language rather than inability of thinking about probabilities.
One assumes the question is not so blindingly easy
That was after all your Videos of Algebra and Maxwells Equations for Electrodynamics the toughest one for me! I always was just putting numbers into bayes without having a feeling for what im doing. It took me 5 hours now, several selfmade exercises and a lot of swearing but finally it made click in my head ! Thank you once more for your amazing Video! Honestly your offer of amazingly intuitiv math content makes us better students.
Greetings from a Electrical Engineering student from Germany.
thank you for your insights. I'm currently in these 5 hours but getting closer. Nothing better than getting an intuitive explanation like here and then testing yourself with real exercises - loads of exercises; goes to show what is wrong with our educational system. Greetings from a Swiss economics graduate
I remember, when studying mathematics so many years ago, noticing how one of the top maths students would often use pictures, diagrams, and graphs to express formulae or other problems. From then on, I also did this and it made so many things easier across this field of all things mathematical.
Great visualisation, as always!
One thing about the Linda- example: This is rather a psychologic or even linguistic effect. If you give people the choice of "people with property A" and "people with property A and property B", they will interpret it as: "people with property A but not B" and "people with property A and B"
Not even that: I see "person with property A/property A+B". There is no 'people'. It's only later all these other bank tellers are conjured up to make us who literally are focused on *person* (for that's the scenario) feel stupid. (Am seriously annoyed at 3blue1brown for this.)
Agreed. It’s a linguistic thing. When someone talks like that, the implication is that you should only pick “bank teller” if the distinct absence of “feminist” more accurately describes the state of the world. It gets translated by your mind as “JUST a regular bank teller”, as opposed to the logical superset of all bank tellers.
@@johnnyblackrants7625 my thinking is that the "or" of natural language is not the "OR" of logic, that is, ⊕ is not ∨.
I haven't looked much into the study myself but I heard that the people in STEM answered the Linda question "incorrectly" more often than social science majors, which is otherwise a strange thing.
My boards examination are from this February and Bayes theorem bugged me since SO LONG because i could never make an intuitive sense out of it. I'm so happy right now that YOU made a video on that!
Love from India, Grant! ❤️
Veritasium also did a good video on it, but not as good as this.
@@LeoStaley Yeah! It was good too but this is better.
@@LeoStaley Veritasium's video took me on a ride XD
ah a fellow Indian. You probably know how probability is taught here lmao
I have my board exams too XD
@@arhmlmao how did the pre boards go man ;-;
This is the absolute best and most comprehensive bayes theorem explanation i have ever seen and i have a mathematics degree 😮 you sir are amazing
I've always wondered, what jobs to math majors do exactly, other than research?
No it's not
@@mohammadabdulla8601 ok then who has explained better ?
@@dev0_018 you could make a rough guess what they'd say, based on their username(hate to be racist but ive read too many such yt comments from such usernames. You could say it's my bayesian estimate 💀)
@@anonymousfry well, hate it to break it to you and face you with facts but your Bayesian estimate is pretty terrible and didn't estimate anything 💀,
since i hold similar name and same belief that this name derives from
But I wanted to know how they used Bayes theorem to find the sunken gold
Zach Star has something on it if I remember correctly.
Me too ... LoL
Pretty much like battleship, they deduced it (if i remembered correctly) into squares (actually circles but easier to understand in squares as shown in video) and searched a perimeter and ticked off squares as they went, the ship had a given size to which it could be deduced into a probability of multiple squares (they gained evidence of where it was not AND gained evidence as they found wreckage pieces) and in se gave a higher power of finding a higher probability to find the ship in a set square in a set range. Ofcourse they assumed the last position the ship was seen as a baseline. This is what I remembered when I had it lectured to me quite a few years back. Greetz!
They found someone who knew where the ship was. Then they tied him to a chair in a cellar and said "The next person to come into this room will be a shy, meek man named Steve. He will be the one who beats you to death with this hoe if you don't tell us where the gold is. Do you want to have a guess whether he's more likely to be a librarian or a farmer? Or would you prefer to just tell us where the gold is right now?"
@@labibbidabibbadum you forgot the feminist bank teller
A brilliant demonstration! I just love how the author converts formulas to pictures, either in this video or in others, it really always help a lot.
Dude you are not a teacher. You are a wizard, that's some next level way of explaining things. Great video.
Best teacher I never had. You have an uncanny knack for talking about the question that just occurs to me as a result of something you just explained. Incredibly helpful. Thank you!
11:00 The first version of this question is in regular English, while the second is not. As such, the first version implies it should be interpreted in good faith, while the second implies it should be interpreted literally. And the good-faith interpretation of "which of these is more likely" is that the options are mutually exclusive; as such, if the first option is "a" and the second "a and b" it implies that the first is really trying to say "a and not b" and the writer was simply sloppy. And given that interpretation, the answer is indeed reasonable.
So, I'm not convinced this actually says anything about people's abilities regarding logic or proabilities, since the results are easily understandable by assuming that the parsing rules for incoming information are chosen based on the form of said information, which is in fact perfectly reasonable behavior.
In short: it's a trick question where the reasonable and literal interpretation result in opposite conclusions.
Agreed. If you said "What's more likely: Linda is a bank and a feminist, or that Linda is a bank teller and either a feminist or not a feminist" I think a lot more people would get it right.
If the question is asked by a physiologist, it appears that one can assume that the question is _always_ phrased in bad faith, with trick parts of the question that anyone rational will fixate on, but then the physiologist then dismisses as completely irrelevant.
The farmer question is relevant here: how many farmers have little interest in the world of reality? Excuse me? What the heck do you think _farmers_ do? They work with real world things like dirt, animals, mortgages, and conniving scientists and anti-farm activists every day of their lives. You are telling me that successful farmers aren't interested in reality? Bullshit. So then as a physiologist you simply skip that most important part of the statement and then say, "no, it says he is meek, and that works for either farmers and librarians, so you are completely wrong."
Thank you for articulating my exact same impression.
@@lwilton the farmer question is a trick because its something we fall for. i asked myself whats the most likely result and caught myself thinking yes or no. when i noticed the sliding %bar in the video and i couldn't give a reason why it might be 55% - 45% over something close like 60% - 40% judging their character i moved on and asked how many libraries compared to farms are there. recognising relevant data was part of the experiment and even though its a trick question it still answers the study. it just implies you work with what you're given i think. but there are much better examples of how to get it wrong using intuitions and show rational thinking is a skill we need to practice
agreed. for the farmer the sample set is implied to be "types of people the question author has thought of" and not "the actual population of the world". An A.I. might have guessed farmer, and been wrong on the majority of texts that would take the time to describe an individual in this way.
Steve is almost certainly a fictitious character, so the correct answer is actually "the author is probably thinking of a librarian." I do, however, think it's relevant that arm-chair researchers take into account to what extent real world data might experience this issue. I think the farmer/librarian question could be better phrased as something like "you are a data scientist studying random facebook profiles that have been constructed by an A.I., and see this profile of Steve. If you had to guess that he was either a farmer or a librarian, which would you guess?"
I've been using Bayes theorem at school for over 2 years. This video finally helped me understand Bayes theorem from an intuitive perspective, and completely changed my understanding of it
In the case of our bank teller friend Linda, I think linguistic ambiguity, and not irrationality, is responsible for the weird result: Though the answer doesn't explicitly say so, the fact that the second answer is "Linda is a bank teller and is active in the feminist movement" creates the implicit notion that the first response "Linda is a bank teller" means "Linda is a bank teller and is NOT active in the feminist movement". Since the later examples where people were asked to estimate populations of bank tellers and of bank tellers who were active feminists came to rational conclusions, it is my hypothesis that the people conducting the study didn't realize what question the original group was actually answering. If the answers had been "Linda is a bank teller who may or may not be an active feminist" and "Linda is a bank teller and is certainly an active feminist", we might get more rational answers. Better still, if we had three answers ("Linda is a bank teller", "Linda is a bank teller and is NOT an active feminist", and "Linda is a bank teller and an active feminist") that might produce the best results overall, though there is still ambiguity in how people choose to read the meaning of the answers.
Exactly. The fact that people don't always interpret questions literally, or the way a logician would, isn't a fault of human reasoning. It reflects our ability to make assumptions about context in which we're being asked things. I wouldn't fault anyone for assuming that option 1 excluded option 2, thinking that this must be the intended meaning since it would be a ridiculous question otherwise. Just another example of psychologists drawing grand conclusions from linguistic ambiguity.
But when they asked about the “100 people”, nobody interpreted this statement with ambiguity, even though many did with “Linda”. Why is that?
@@antiawarenessawarenessclub Because the second way of asking it asks a fundamentally different question; I think any person with a basic grasp of numbers would know that you can't have a subset of a group larger than the group. It further removes some ambiguity by parameterizing the group; we're explicitly told that 100 people fit the description, and to dead-reckon how many are bank tellers and, of those bank tellers, how many are active feminists.
BUT - and the video didn't address this, so I wonder if the study did then, too - if we follow up our population estimates by asking the original two questions, we still have the problem where the first question implies "...and is not an active feminist." Based on the answers given in the study, if that ambiguity is in play, you wind up with the same non-Bayesian error: 8 people in the group are tellers and of those 5 are feminists, so it's more likely that Linda is an active feminist bank teller rather than an apathetic one.
In the case of the population-estimating version of the question, what we really have to ask to eliminate ambiguity is "Out of 100 people, what are the odds that Linda is a bank teller?" (8%) and "Out of 100 people, what are the odds that Linda is a bank teller AND an active feminist?" (5%). Then when asked which statement is more likely, the ambiguity of which population groups we're discussing is clear ("all bank tellers total vs those tellers who are active feminists", rather than "all bank tellers who are not active feminists vs those tellers who are active feminists").
@@antiawarenessawarenessclub Because of the way most people are conditioned to approach multiple choice questions. On a multiple choice test, generally 1 answer is THE correct answer, and the rest are considered wrong (even if they are factually accurate), if more than one seems applicable we are taught to choose the one that is most accurate. So people are likely to ignore the bank teller portion of both options and focus on the difference between them to decide which is more accurate: is she an active feminist or is she not?
The second form of the question doesn't have this ambiguity because there's no multiple choice to trick us into seeking a single best answer, and instead we have 2 separate and open questions. Even if you remove the "out of 100 people" part of this question and ask for percentages or probabilities you're likely to get the same rational results simply because they are now 2 separate questions instead of 2 competing choices to the same question.
Actually there isn't an ambiguity in language, "Linda is a bank teller" includes all bank teller possibilities. People just mentally interpreted that "Linda is a bank teller" means "Linda is a bank teller and is NOT active in the feminist movement," which is a flat out _wrong_ interpretation.
I was on the dean's list in my undergrad engineering major, and graduated with high honors in my 'brand name' MBA program. This is one of the BEST explanations of a fundamental tool of analytical thinking and insight, whether for business, medicine, law, sports,online dating(!), or just clear thinking I've ever seen.
I learned and (explicitly & implicitly) used Bayes for decades, yet your preservation has given me another window into understanding/reminding me of its value in everyday thinking ... wish u were one of my prof's.
Godspeed, the world needs more of your talent.
How did the first sentence help me?
I am studying for the actuary P exam and I worked through all of my practice problems by making these diagrams. Thank you! I now understand Bayes Theorem.
I just love how you manage to visualize mathematical concepts! I've been drawing rectangles with subrectangles to help intuitively understand problems involving probability since before I was taught probability in secondary education, but I've never tried to represent Bayes' Rule so elegantly.
We didn't include what's the possibility of a farmer having Steve as a name vs librarian having that same name...(laughs in Bayesian)
Super valid point
This is hands down the best explanation of an abstract mathematical concept that I have ever witnessed in my 33 years of living and learning. Absolutely brilliant display of mastery in teaching and explaining.
I'm told that many medical doctors do not understand Bayes Theorem, and it can be threatening to peoples' health. Example: There is a test for a very rare disease, and the test correctly gives a positive result for 95% of the people who have the disease. Your test comes back positive. What is the probability you have the disease?
Unfortunately, a lot of people, including some MDs think the answer is 95%. The actual probability you have the disease can be much smaller if the false positive rate of the test is high and the fraction of people taking the test who do not have the disease is high.
BTW, when I worked at FICO (the credit scoring company) we used Bayes Theorem so often they gave all of the employees shirts with the formula embroidered on the sleeve.
The way I think of it is that Bayes Theorem gives you a way to turn some measurements you can make, but which aren't really all that interesting, into something you can't measure, but which you're really interested in knowing. Like in your example, you can turn the probability of getting a positive test result for anyone who actually has a disease (which is measurable and is interesting, I guess, but not of huge importance to most people) into the probability of actually having the disease, given that you got a positive test result, which is not directly measurable but is going to be of extreme interest to anyone who gets a positive test result.
The false positive rate doesn't have to be very large for a positive result to be largely meaningless. For anything rare, the odds that a positive result is meaningful is going to be small unless the false positive rate is similar to the rate of the condition in the whole population because there will be far more false positives than real positives.
I agree on your comment about doctors. I am a med student in India and I believe that quite a lot of physicians don't know this well. It's sad.
@@parthashah9257 UK medic here... Check out more docs over the next few years, then update your beliefs :).. in the UK, 40% eligible health staff do not have free flu' jabs, because of false beliefs, mostly "I had the flu' straight after, once..", which appear impervious to the new evidence which in a rational system would update their beliefs :)
@@tim40gabby25 LMAO
Gerd Gigerenzer studied this, and the approach of thinking about absolute numbers instead of probabilities (like in the video) seems to help in practice.
Thank you. For an aged brain this is one of the most accessbile and comprehensible explanations I've found. As Andyg2g commented below, for me the phrase "rationality is not about knowing facts, it's about recognizing which fact are relevant" lit up my understanding!
for a rushed overworked young brain too
This gentleman is a master in teaching, he makes difficult things easy to understand in a variety of different topics. I have been watching his videos about different subjects and he is really amazing. Congratulations Sr. !.
I literally started research for a paper on Bayesian search theory yesterday and then you release this video? This is godsend.
It's so well done! On lecturer once said that Bayes treats all potential events and their likelihoods as independent from each other.
what 8-9 hrs of watching several videos and tutorials, reading various texts could not explain me why is baye's formula the way it is was explained by this channel in just starting 5 mins without even showing the formula
Brilliant!!
I imagine most people interpreted the bank teller question as "1) She is a bank teller not active in the feminist movement, 2) She is a bank teller active in the feminist movement". That was the first thought when I interpreted it anyway.
Yeah. They're basically telling us that she 100% IS a bank teller. So the only question left is whether she's an activist or not.
I get what he meant to say but the question doesn't really fit.
No they’re not. They’re saying which is more likely? Not, given that they’re a teller, which is more likely? And these are very different things.
Not sure how you’d justify interpreting A or (A and B) as meaning the first A was A and not B either, in response to the initial post
I agree. I had the same interpretation, and I think the problem lies on the difference between verbal language and mathematical language in terms of precision. It requires some "fluency" in math to convert the problem mathematically.
(Sorry about my English haha)
@@kellmano1 Well they're asking:
1) A (without B)
2) A with B
The way I understand her description she's more likely to be a bank teller activist rather than only a bank teller.
@@ervindark Haha, that's definitely not what they are asking. 1) Is she a bank teller ( including activist /not activist ) 2) Bank teller and an activists, the first actually includes the second options hence the propability is bigger, is it clear now? you added information wrongly to the 1) that "she is not an activist"
Quote of the day (or probably decade): Rationality is not about knowing facts, it’s about recognising which facts are relevant.
Immanuel Kant already figured that one out back in the 1700s so we're a few centuries late with it. He wasn't very good at writing snappy quotes though
Am glad to see someone else picked that up too...😇👍
@@francescocraighero5392 I'm sorry to say that Daniel Kahneman in his book Thinking fast and slow debunks most of the things that says that guy in his blog.
@@Ucedo95 In the last months I encountered that book many times, I think it's definitely time to read it. I don't know where WBW made wrong assumptions, but I think that the contribution that Tim gave by visualizing this topic will still be worth a read
By the way, the current decade will end on 31st December 2020, as there will be 202 decades since Christ was born, allegedly on the 25th December. Considering a decade for 2010-2019 is 10 years ok, but is misleading as one of the previous decades in history must be 9 years only. Because the year 0 does not exist for historians. So the first decade in history was not 0-9 but 1-10.
Once again you're explanation of something I have struggled with has made it intuitive. Your method of visualisations (with it's own language) is changing the way that math can be taught - by relying less on formulas and text and more in intuitive, visual ideas that our brains can retain so much more effectively. Thank you for your work Grant!
Getting your calc videos on my recommend is by far the best thing UA-cam algorithm has done for me!
I guess we're assuming that no librarians are farmers.
Isn't that like the intersection part A intersection B
"New evidence does not completely determine your beliefs in a vacuum; it should update prior beliefs." That's the quote i'll take for my life after this video.
Ty Blue, discovered your channel yesterday and been mesmerized by it since then.
This is EXACLY what I've been trying to study and understand for the past week, I even did a ton of exercises this morning. THANK YOU!
This is an amazing video, but I'd like to point out that human speech doesn't occur in a vacuum. More specifically, people give answers that are useful to the addressee more often than answers that are technically true; after all, that's why people communicate (think: 'there's a shovel in the shed if it snows'; does the shovel cease to exist if it doesn't?). In the case of Linda, for example, it is more useful to say that Lind is a bank teller who is involved in the feminist movement (assuming that her description matches being a feminist more than not), given that the addressee seems to know, or at least have assumed, that Linda is a bank-teller already (answering that Linda is not a bank teller is not an option). Again, this video was amazing, but I think it's worth pointing out that a large and useful(!) part of human communication does not hinge on mathematical truth but on interspeaker convenience and we really shouldn't strive to 'correct' human judgments or label them as necessarily wrong.
This is an important point. We think that the text is informative, so the added information must provide some additional effect for the consequences we draw from the text, otherwise the speaker probably would not have provided it. Especially in a task like that where the hole point is to draw consequences. The idea that the pieces of information given in a cooperative conversation should be relevant goes back to the philosopher H Paul Grice, his “Maxime of quantity” and of “relevance”. There is lots of articles written about the Linda fallacy but as far as I know nothing makes this point.
That's kinda the point (that humans are predictably irrational).
Even if mathematicians don't know Grice's maxims, you'd think that psychologists would.
Humans like to embellish their answers with fiction as it gives the impression of knowledge even if it is unsupported or fanciful. . The question was which was the more probable. That is why in courts the lawyers often ask the question and insist on a yes or no answer to cut through the irrelevant waffle!
That's the point. Just because we think like that for most questions doesn't mean it's the way to think in this specific context. And such lack of reevaluation of belief can lead to silly situations at best, big mistakes and their consequences at worse.
I love how you bring in the part about objections to Kahneman & Tversky's research. Gives us a very thorough understanding about context around the topic!
None of these objections are objections against Bayes Theorem used for updating beliefs however. They only propose that in the specific experiment more steps of updating the belief to get a different prior probability would be needed.
Please include in future discussions the relationship between bayesian inference and the scientific method and how all these things are related to deductive and inductive reasoning.
Your content is amazing! Thank you!
And thank YOU, good sir, for inventing calculus. :-)
Weren't you supposed to be dead?
I for one have always thought you as the chosen one, not that pompous brit.
I think the use of that second prompt actually reveals yet another mistake in human cognition: assuming humans are concise rule followers.
85% of people are getting the bank teller question wrong, not because they aren't thinking about the set of sets, but rather because they're inherently correcting for the perceived mistake you've made. They read the question, distilled, as "is she more or less likely to be a part of the feminist movement, than to not be."
The reason for this, is that asking such a question of someone doesn't make any sense, since it's 'intuitively obvious', so they assume you've made an error and correct for it. In your rephrasing of the question, that presumed error goes away, because you're asking the percentage of generic people filling particular categories, and the question actually makes sense to ask, since an rational person can come up with genuinely different answers for each. In the previous example, one cannot answer any differently than a bank teller, which triggers their instinct that you've made a mistake in writing your question.
You can see this very thing at work when people read articles with misspellings, or read texts with words that don't make sense in context. They'll automatically fix the spelling when reading, or find a word close in spelling that does make sense contextually.
The assumption that people are like machines, doing things wholly within the defined ruleset, whether that's the rules of English, of culture, of whatever, is a fallacy. People are intuitive thinkers, they don't follow a prescribed set of rules as defined, they follow what they perceive or believe the rules are intended to be.
That's why we can read the same set of rules and come up with different interpretations, because we have different priors and knowledge before reading and attempting to interpret said rules, despite the words we both read being identical.
Same thing with the librarian bit.
I would be thinking about who wrote the description and would guess that 95% of people would describe a librarian as someone organized and with a farmer they would say something about nature.
This perceived probability strongly over rules any % of farmers and librarians in the population.
It's not that the farmers don't fit the description, many probably would, but it wouldn't be the first and only thing you say about them.
@@dp2404 Another point with the librarian bit is that it reads like "am I, the writer of this question, thinking of a librarian or a farmer when I made up this character Steve"? If it was phrased like "select a random individual from the actual population of the USA, with these traits" then it would naturally lead to thinking about real-world proportions of farmers and librarians.
@@benjiunofficial exactly!
You are more thinking about "why am I being asked this question?"
@@dp2404...but the description includes the fact that "Steve" has "very little interest in the world of reality" -- and that fits precisely 0% of all the farmers in the world. It might fit a non-zero percentage of bankrupt ex-farmers, but working farmers depend on "the world of reality" for everything they do... The video as a whole is excellent, but that one phrase in the description in the beginning really broke my immersion.
@@s_m_we addressed this kind of thing at 3:05. The point is that most people don't even consider the prior probabilities. If you do, and come to the conclusion that this evidence makes the probability of being a farmer extremely low, you've still applied the theorem correctly. But this is not how most people get to their judgement in this question.
This geometrical approach to probability is gold worth. I struggled a lot to conceptualize a certain task we had, because our teacher introduced new variables instead of using the "not" symbol. I understood the steps but i don't not understand it intuitively. THIS, this worked excellent, i just need to add a couple of squares together and i get the result. It made it super easy to understand the task at hand as well, thanks a lot.
Years of bashing my head against the wall trying to build something resembling an intuition on Bayes Theorm, and in 15 minutes everything has now fallen into place. You are amazing, Grant! Thank you so much!!!!!!!
If you want to intuitively understand Bayes theorem, forget about Steve, leave the Bayes Church and adopt the frequency interpretation of probabilities (or at least accept that is exists). And then write down the theorem in a symmetric form. It becomes an almost trivial and intuitive statement about sets.
But, of course, writing it in an unintuitive unsymmetric form and rejecting the interpretation of probabilities as predicted frequencies makes it somehow deep and mysterious.
So you have to choose a side. You cannot have it both. Either you can intuitively understand it, or you can stick to the believe that there is something non-trivial behind it.
You just unlocked a different spectrum of my brain
I'll just add that the explanation is GREAT! A pure example of how if you really understand something you can explain it with simple words to everyone. Thanks
This is a way of getting people to be much more rational in their beliefs. And 3Blue1Brown is a great teacher putting up stuff for free for us all to learn from and if everyone saw this and took the time to understand it we would have a better world. This guy is amazing!
11:12 I believe this too is a problem with the education system. In MCQ type questions, if multiple options are correct, we are expected to choose the "more correct" option.
As an example: Q is a gaseous element that reacts with oxygen to create common water.
1. Q is an element in the periodic table
2. Q is the first element of periodic table
Even though, 2 is a subset of 1, I can say with utmost certainty that the majority of students will answer 2.
but given the prior that the students know it's the first element, the probability for both is equal, so there is no answer more correct than the other. Once you know that Q is hydrogen, which happens before the MCQ, then all you need to do to reach this conclusion is to evaluate the truthfulness probability of the choices by plugging the answer, and this becomes
What is the probability for each of the following statements being true?
Hydrogen is an element in the periodic table
Hydrogen is the first element of the periodic table
They are both true, so none of the answer is more correct than the other, since you already knew the answer before the question being asked.
The result above can also be determined with the formula explained.
Say that we want to test answer 1, which becomes the first tested hypothesis, H1.
The formula, as presented in the video is
P(H|E) = (P(E|H)*P(H))/(P(E|H)*P(H)+P(E|~H)*P(~H))
To calculate P(H1|E) we need all the above terms, but let's start with the easy ones
P(H1) = ? , or in other words, what is the probability that Q is an element? Obviously this depends on how we define our space, but let's use the periodic table as a space, so then
P(H1)=1
P(~H1) = ? , or in other words, what is the probability that Q is NOT an element? Obviously,
P(~H1)=0
P(E|H1) = ? , or in other words, what is the probability that Q is a gaseous element that reacts with oxygen to create common water given that Q is an element in the periodic table? Well, once again, we know there's exactly only hydrogen out of all the elements, so the answer is
P(E|H1) = 1/n , where n is the number of elements in the periodic table. Let's simplify and say that we only discovered the first 100 elements, so n=100
P(E|H1) = 1/100
P(E|~H1) = ? , or in other words, what is the probability that Q is a gaseous element that reacts with oxygen to create a common water given that Q is not an element in the periodic table? Obviously 0, although I suspect the more appropriate answer is undefined.
P(E|~H1) = 0
If we plug all these in, we get
P(H1|E) = ((1/100)*1)/((1/100)*1+0*0 = 1
P(H1|E) = 1
Same about H2
P(H2) = ? , or in other words, what is the probability that Q is the first element of periodic table? Considering the same space of the periodic table, then
P(H2)=1/100
P(~H2) = ? , or in other words, what is the probability that Q is NOT the first element? Obviously
P(~H2)=99/100
P(E|H2) = ? , or in other words, what is the probability that Q is a gaseous element that reacts with oxygen to create common water given that Q is the first element of periodic table? Well, we know there's exactly only hydrogen to be first
P(E|H2) = 1
P(E|~H2) = ? , or in other words, what is the probability that Q is a gaseous element that reacts with oxygen to create a common water given that Q is not the first element of periodic table? Obviously 0
P(E|~H2) = 0
So, we get
P(H2|E) = (1*(1/100))/(1*(1/100)+0*(99/100)) = 1
P(H2|E) = 1
P(H1|E) = P(H2|E) , so both answers should be accepted as being the most correct answers.
This problem is different than the Linda problem in the video. To make it equivalent, assume that you personally know that Linda is a bank teller and that she is active in the feminist movement.
The 2 problems are also equivalent if you eliminate the priors, and then everyone would give the more inclusive answer. Say that you're only asking your question to people who don't know that Q is hydrogen, or that do not see any correlation between being active in the feminist movement and the evidence presented in the question. Then these people would pick the more likely answer, meaning the first, each time.
This means that people are very selective with the priors they use. Moreover, people are often tricked by the fact that hypotheses overlapping, but people guess they are distinct (and sometimes complementary, and sometimes equal). So given H1 and H2, people make the following assumption P(H1|H2)=P(H2|H1)=0 (and sometimes P(H1)+P(H2)=1, and sometimes P(H1)=P(H2)=0.5), which means that the only possible way of testing actual knowledge using MCQ is for choices to hold as manu natural assumptions as possible, but at least the first, P(H1|H2)=P(H2|H1)=0 . So the proper choices for your question should be:
1. Q is an element on an odd position in the periodic table
1. Q is an element on an even position in the periodic table
This way, the student will only perform better than chance if they truly know the exact answer.
Thank you!! Learned a lot!
"Rationality is not about knowing facts, it's about recognizing which facts are relevant". Great quote.
And why Bayesianism is an incomplete philosophy.
I think the discrepancy in the Linda part can be that people see the two options they juxtapose them and intuitively take "a bank teller" to mean "just a bank teller and nothing else". Thinking fast and slow is pretty good, just about finished with it. I'd highly recommend it. Really changes your brain.
11:30 the problem with this question is the answer "banker" implies "Banker but not feminist", making them disjoint. It doesnt mathematically mean that, but in English it does. Consider if you had toast for breakfast, or toast and and egg, the first doesnt explicitly say you didnt have an egg, but it is implied by the second option
Such a soothing voice, killer animations and deep knowledge
"Rationality is not about knowing facts, it’s about recognizing which facts are relevant."
I would like to know if Mr. Sanderson himself wrote this line or someone else. It took me three weeks to fully absorb this. It helped me with my analytical ability, and is now one of the constructive pillars of my discussions.
We use Bayes Theorem when we assess and control risks on a day-to-day basis. It can even be common sense one could say. When we see something unsafe we are wondering about the likely hood and the consequences of continued action or inaction. When something might be unsafe there is a primal need to explore and gather more information in order to make better choices. This is displayed time and time again in nature and society. The snap judgements humans and other animals make seem to suggest an almost innate ability to see probabilities. Love the line in this about rational thinking being the ability to know which information is relevent. Keep up the good work!
It's incredible how logically sound things become when you explain them.
Yes, and the irony is that he is making us understand by using our intuition. So basically he is using intuition to explain things logically.
A year later, watching again. Still good!
This also gives good advice on how to argue with people who hold beliefs that are not backed by evidence. A lot of people target the likelihood, getting bogged down in trying to adjust the person's percentages. We forget to take into account the size of their prior.
Came across this video on a whim and I gotta say, I studied computer science in college with multiple classes touching on this subject and this is by far the best explanation I’ve ever seen. Fantastic teaching
Please make a series on probability like u have done for linear algebra and calculus. They have helped me a lot to visualize the topic but also to appreciate what I'm learning.
Thank u for your work
I think he is doing this right now
The gods of Mathematics have answered our prayers😀. Liking before even watching because I have been waiting.
that username
Can't Thank you enough for the illustrations that make everything clear and easy to recall. Also, the fact that it is not just about teaching the formula but the concept and the notion of it is what we all need. Thanks a million.
Where Y is a subset of X, perhaps asking if she is more likely "an X or an (X and a Y)" is being interpreted as given that she is an X, is she more likely:
A: (X and Y)
B: (X and not Y)
This is the same as swapping out the "or" for an "xor"? The two are used interchangeably, often the wrong way round in plain English! "It's this or that?" usually means "It's this xor that?".
Yeah I'm highly sceptical about the psychological import of these experiments. I feel like it's mostly explained by the vagueness in the word "likely". As soon as you put the problem in context, the incorrect answers disappear. Which totally makes it sound like a communication issue rather than a psychological flaw.
X and not Y is a subset of X. Therefore P(X) = P(X and Y) + P(X and not Y) which implies that P(X and Y) < P(X) which means Lynda is more likely to be bank teller than a bank teller who is part of the feminist movement.
@@Karthik-lq4gn You've misunderstood.
Yes, P(X and Y) < P(X) is always true, but whether P(X and Y) < P(X and not Y) is not known, which is how Alan is saying people are interpreting the question.
@@gorgolyt Yeah, these experiments are no longer considered valid in as far as the original conclusions that were made, but are still important in that they provide good data showing that how a question is phrased can change the way a person interprets a question, and therefore how they will answer it (which is really important in any country where the citizens vote).
I think there's an additional piece of information when considering the Steve problem, which is that being told about the personality traits makes you think that his personality is what's most relevant to the correct answer. Basically it's like when you're given a formula or a hint on a test question, the fact that you're given the formula at all automatically makes you think that you have to use it somewhere in your solution (even if it's totally irrelevant and the examiner is just being a dick)
It's not a dick move to test how well you actually understand the problem. If the irrelevant formula trick works on you, it illustrates that you understand the material so poorly that you have no idea what pieces of information you need to solve the problem.
Likewise, if you're so foolish as to think that completely irrelevant information about Steve is relevant just because somebody mentioned it, that's still an illustration of your irrational tendencies. These tendencies can and likely will be exploited in your day to day life by sociopaths such as in politics. We can't go through our lives expecting examiners to wipe our asses for us.
@@tj12711 Humans are not superrational machines and things like proximity and wording will inevitably affect how they think. This has been scientifically proven hundreds of times and telling people to "be more rational" does not change this. This is how you can get drastically different responses in a poll just by changing the wording around, even if semantically, both polls ask the same thing.
If you're telling yourself that you're so rational that you don't have any cognitive biases, you definitely aren't self-examining close enough. If you're exploiting cognitive biases on exam questions to make the exam harder, you should probably ask yourself whether that's the most effective way to test the students' understanding of the material.
@@tj12711 However, provided that the examiners did not remind the students about the relevant info (like the proportion of librarians to farmers, or the fact that the options were not mutually exclusive), the examiners capability of proving their hypothesis about the students was impaired, since the students probably only used the available context to answer both questions.
So, yeah, in your everyday life, and specially in politics, it is better to consider statistics more than anedoctal evidence, but in a test (specially written ones) it is not feasible to expect students to consider every piece of possible relevant information without proper context, specially in Mathematics with its formulas.
if it is true that certain personality types are more likely to frequent an occupation, then knowing that person's personality type does make it more likely to better place them. being told something does not make it more likely, but the correlation between two similar things can make it likelier.
if we are told steve hates working outside or getting dirty we can assume he is probably less likely to work as a farmer. at the same time that doesn't make it impossible (steve may not have a choice and is desperate for a job)
Brains are trained to compare similarities (which is really useful), not probabilities (which is an artificial concept invented by mathematicians). So when you speak mathematician-gibberish to an average person, they still run the intuitive similarity matching in their head.
This music is very calming. I can't tell if it's distracting, but it's very calming. The content is very good too. Will keep coming back to it until the knowledge sets in a little better.
Understanding even complex maths is fun if we have teachers like you. Excellent work!
I wish every teacher would take students from understanding a mathematical theorem conceptually into reasoning with them to the actual formulas with this order, Truly remarkable!
I had to watch the video about 5 times, bring in some rough paper and pen for some visualization but I learnt a lesson for a lifetime. Thank you for such amazing explanation, its just amazing how new evidence can shape our prior perspectives and affect our mindsets completely. Kudos ❤
This is simply the best explanation of the topic I've come across, very well done and thank you
I feel like the hard problem here is recognizing when you have missed something important like how people missed that the ratio of librarians to farmers, was something they should have taken into consideration. Most people, if given a story problem, will reflexively self limit themselves to only the evidence in the stated problem.
I work at a high-tech company and you have just saved me a lot of pain! Now I can finally quantify my believes, present and update them! Thank you so so much!!!
This is why it's so important in machine learning to have a balanced dataset.
If a model is programmed to reply 'farmer' every time, the model has an accuracy of 95.4%. This sounds both great and horrible at the same time.
Actually it is way more important to have a relevant loss function than a balanced dataset. If your goal is to minimise the number of errors then replying 'farmer' every time would be not so bad at all. Things change drastically when every wrong answer takes $100 from you and right ones give you just $1
@Uładzisłaŭ Astrašab nope, in the second case I care about the money I win or lose
@Uładzisłaŭ Astrašab the loss function is an optimization of the measure of performance
@@dmitrynovikov5844 that wouldn't be enough. With such extremes in the dataset and an overcompensating loss function, it will likely make the distinction between 'farmers' and 'non-farmers' considering it has so many farmers to work with. This means that in practice it would classify atypical farmers that it didn't train on as librarians.
It would see cats and firemen as librarians as well.
In practice you'd want the model to return a low confidence on both librarian and farmer when a cat goes through the model.
But that depends on what you'd want from the model. It might be right in most cases, but it's just not as elegant.
@@deldarel Yes, having a balanced dataset is very important if possible, but in cases where it isn't, there are other ways of evaluating the performance of an algorithm besides accuracy - (number of correct answers) / (number of total examples). Such as: dividing up all the answers into 4 categories: True positives, false positives, true negatives, and false negatives instead of just whether it predicted positive vs negative can help.
Let's go with your example of an algorithm that gets 95.4% accuracy by always predicting "Farmer." If you know your dataset is skewed, and you notice in your test results that it predicts "Farmer" 100% of the time, instead of accuracy you can use a formula like (# of correctly predicted librarians) / (actual # of librarians) to evaluate your algorithm's performance. This formula basically asks the question "out of the number of people we predicted to be librarians (all predicted positives), how many are actually librarians (true positives)?" This alternate way of evaluating can quickly reveal an error, as if your algorithm is predicting "Farmer" every single time it would quickly be revealed that it never predicts "Librarian" because (# of correctly predicted librarians) in the numerator is equal to 0 so the performance would be 0%.
Bae's theorem: The probability that your bae is hungry, provided that she is angry is equal to the probability that your bae is hungry and angry divided by the probability that your bae is angry.
The probability is surely 1.
"Bae's" theorem haha
@@shreerangvaidya9264 Yeah, isn’t it 1 in this case ? The angry bae’s cancel out ?
She's hot and crabby.
@@townley1017 (1 * 1) / 1 is still 1 though
Thank you Grant. You were my weekend movie for 1 year in 2020. And now, you will be my pastime, my hobby and weekend entertainment for 2 years at least.
The problem with Lisa is in my opionion the phrasing. Intuitively, people might assume that "Linda is a bank teller" is more like some prior knowledge. Or, to phrase it differently, the question sounds more like P(Bank Teller, Feminist | Story) vs P(Bank Teller, NOT Feminist | Story) which essentially boils down to P(Feminist | Story) vs P(NOT Feminist | Story), which clearly edges for second answer.
Same with Steve. I think the point was a lot of people jump straight to: "Which group of people is more likely to be shy and withdrawn?" i.e. straight to the likelihood, neglecting the prior.
But that's the whole point of it. The question was entirely unambiguous but people managed to misinterpret it and arrive at a nonsensical conclusion _because_ they're grappling with unfamiliar, abstract mathematical ideas. Re-framing the question in more familiar, concrete terms resulted in a very different outcome.
Another example they cite in the book (Thinking Fast And Slow) , that follows the exact same structure as the Linda example is:
Which alternative is more probable?
Mark has hair.
Mark has blond hair.
by your logic people assume the options are (Mark has hair, excluding blond hair) and (Mark has blond hair) are the actual assumptions people have because these follow the exact same logic structure as the Linda problem.
@@raafaytariq9132 If blond hair was exceedingly common (given a story), then under that interpretation B would be correct
like maybe the setup is that "mark's hair often shines bright in the day / mark's hair is a light tone" or something like that
Given the interpretation that A v A & B implies the question is A & -B v A & B
@@MagicGonads I completely agree that if the given context was one that assumed blond hair was more common, it changes the story. However, it doesn't make sense to say that you can both infer A &-B v A & B because of context of the wording, yet you can't infer context of the question. You can either take it at straight value, and infer no context. Or infer your frame of context for both the logical argument and context of the argument.
I need some time to reprogram my feelings after this video, it was a hell of an exciting ride and i'm not even that interested in probabilities !!
OH MY GOD THIS WAS SO GOOD.
My Intro to Statistical Reasoning prof assigned this video for us to watch and take a quiz on. What a fascinating topic! You are making seemingly complicated math accessible to everyone. Thank you so much!
This is the second time Kurzgesagt and 3b1b uploaded in the same day. I am blessed.
i see you're a gentlemen
As always, another wow moment.
I'm waiting for the day when the intuition behind solving partial differential equation will be explained. Especially about CF and PI and how you interpret them physically on a graph
This is one of the best descriptions (and visuals) of Bayes theorem!
I just wanna take the moment to present my gratitude to you. I really appreciate the work that you put in to make us all understand such important and not so intuitive concepts. Thank You.
13:38 I actually thought farmer was more likely because I share a lot of personality traits with Steve so I approached it thinking "what job would I more likely have ended up in" and since I live in the country, farming was way more likely.
I picked farmer because I assumed they were trying to trick me into picking librarian.
@@abebuckingham8198 that's one way to do it
Thinking "probability" as "proportion" instead of "uncertainty" is mind-blowing. Thank you!
UA-cam: "3blue1brown uploaded two new videos"
Me: Ah! That's the perfect Christmas present! :D
This is so weird...
When presented geometrically like this, I'm having a hard time understanding the context.
But I knew Bayes theorem well enough. It's just my brain can't comprehend it when you present it using geometry like this.
I had a similar issue. I guess I'm just not used to thinking about it that way.
I learned conditional probability using contingency tables and tree diagrams; so, maybe is just a matter of not being used to seeing things this way.
Still, it's cool to see other forms of thinking about Bayes theorem and conditional probability and getting a different perspective. Plus, the explanation was great.
@@rjsr03 I agree, the explanation was great. I wonder if the first exposure I had about the theorem is using this kind of explanations, then maybe the way I think about and use the theorem will also be different too.
The same thing happened with me in Calculus series. I understood differentials and integrals quite well, but then Grant came out with visuals for those, and it seemed kinda unnecessary to my knowledge. However I do think that we should learn to look at one thing through different angles.
channel's author has an unbelievable gift of explaining stuff.
Hey! Why you gotta make fun of some poor guy with the name "Tommy Thompson"? 0:13. That hurt! 😂
It took me like 3 times watching this video in different days to get a slightly idea what he is talking about 🤣.
I do not know how he got all these views
@@vijayakumarm1996 This video was so much better. I guess people just have different preferences for how they like to learn
@@vijayakumarm1996 exactly!!
I think that, because the topic is complex, you have to slow down the video, go back a few times, but eventually you'll get it. I think this is a fantastic video!
Okay! I'm not the only 1 then ☹️
Giving the new evidence I must say that this is one of the best UA-cam math channels .