THANKS MAN! Tomorrow is my official school graduation exam and honestly i didn't ecen know a word about this concept so i was worried and your video popped up! Thanks a bunch for making me understand it! Ill be back to report my marks if i am reminded of this comment! P. S: keep doing this. We love it and will support you through the best of our efforts!
Thank you so much for your comment! It's stories like yours that give me the fuel to make these videos. I wish you luck on your exam! And thank you for the support!
Thanks a lot for this intuitive example. It helped me a lot to understand this mechanism when I understood that as p(cloudy) becomes smaller, p(rain|cloudy) becomes greater, all else being equal. Since p(cloudy) is the numerator. This makes sense intuitively, because in a situation clouds are rare (i.e. p(cloudy) is smaller), but when it rains, there were often clouds in the morning (i.e. p(cloudy|rain) is large), the prediction value of it being cloudy in the morning is high. Reversely, in a climate where it is always cloudy (i.e. p(cloudy) is near one), the fact that it's cloudy in the morning does not tell you much in terms of how much rain you will get.
I think the example in this video is better than what 3B1B gave in his Bayes' theorem video. The starting wasn't good because you just spammed the formula but the example and the way you conveyed it is really good. One can understand the principle through your example. Good work!
Great videos Mark, you inspire us every day with your slogan "You've big dreams, don't let a class get in your way." The likelihood of it being rain while picnic seems low, as it's less than 0.5 / 50%. So I think.i would still go on the picnic.
That was easy and simple to understand. Master it is another thing, but my guess is that you saved me some precious time with this video. Thank you a lot.
I am from Bangladesh, have started to learn machine learning. For which I have to learn probability and statistics. I have clear out all the topics on probability by 11-12th books. But BAYES' THEOREM was seemed to tough to understand. So I came to UA-cam and saw lots of videos which was even approximately half an hour! Although they tried for a long time, they all were gloomy to understand. But your 8 minute video is so effective than all those videos. Thanks. I have subscribed your channel. I will visit again if any other topics I have to understand in future.
God bless you sir for this video. I HAVE went through few videos on UA-cam and this was one of the best where my mind has understood this fully. Now lets see if you have stuff on Binomial distribution. Thanks just subscribed now
This is the best UA-cam video I've ever seen on bayes theorem I don't think I've ever understand bayes theorem as much as this before❤ 0.48 is not enough to make me cancel my picnic I think I will go for the picnic since it's 0.48% chance of rain 🌧️ ☔
THANK YOU! I was trying to learn Bayes' Theorem off the example of "Go For Broke" the gameshow. 🥵 my brain was twisting in on itself. THIS I can understand.
OMG THANK YOUUUUUUUUUUUUUUUUUUUUUUUU!!!!!!!!!!!!!!!!! You just made this SOOOOOOOOO much easier!! 😭 the instructions I had was really making me punch the air. Thank you, thank you, thank you! ♥
Thank you. I'm still having concept issues. As a teaching technique is it possible for you to summarize the meaning of the numerator and what the denominator accomplishes in the equation
Hi, so is conditional probability used with limited information in a question, however Bayes theorem can be used to answer a question that has more information? I'm just struggling with which one to use in an exam question
Thanks so much for your kind words! I'm not sure a probability distribution would help much, but some more Venn diagrams could be helpful! We'll consider this in a follow-up video! Thanks for the feedback!
I think this solution is incorrect actually. We should have calculated P(C) with the rainy days' 0.85 ratio with the formula : P(C)=P(C∣R)×P(R)+P(C∣¬R)×P(¬R) where C is being cloudy and R is rainy. So it would be 0.12 for P(C∣R)×P(R) and 0.215 for P(C∣¬R)×P(¬R) and the calculations made it's 0.36. Can you clarify please
Wow fairly good explanation. i just understand it perfectly today. However, in the process, i found another better way to comprehend this theorem. To those who still do not understand. Read this. First you must understand what p(a/b) is. It is the probability that a happening when we already know that b happened. To find p(a/b) we need to find prob that a and b happening at the same time , and divided it by prob of b happening. To find prob that and b happening at the same time (p (a interect b)), you can find that indirectly from prob that b happening when we already know that a happened multiplied by prob of a happening Ah.... i need a pen and a paper to convey this concept 🙄
So what you’re saying (if I pause the video @6:23) is: If the probability of it being cloudy outside is 4 times greater if it also rains that day, then the probability of it raining on any given day is also 4 times greater if it happens to be a cloudy day?
Given the statement "it’s 4 times more likely to be cloudy outside, given it is also raining," we can express this as: P(Cloudy | Rain) = 4 * P(Cloudy) Substituting this into the Bayes' theorem equation, we get: P(Rain | Cloudy) = (4 * P(Cloudy) * P(Rain)) / P(Cloudy) The P(Cloudy) term cancels out, resulting in: P(Rain | Cloudy) = 4 * P(Rain) So, yes, from a simple Bayesian probability standpoint, if it's 4 times more likely to be cloudy outside given that it's raining, then it's 4 times more likely to rain given that it's cloudy outside.
This series is amazing. Must have been hard to make these beautiful animations. Thank you so much❤. Can you please make two more distributions viz: 1. "Poisson Distribution" and 2. "Exponential Distribution" And explain the intuition behind the mean and standard deviation in these distributions like you did in Uniform distribution video?
if one knows that there are 12 rainy cloudy days (the 80% of 15) in 100 days and 25 cloudy days in total why one can’t just calculate 12/25= 0.48, without all the machinery and the language that the Bayes formula brings along? Is there something wrong in just applying the definition of probability?
Depends on how much you like picnics, how often you want to go on picnics, if you think rain ruins it and maybe you might already think a cloudy day isn't nice for picnics. But before we think about that, let's think about the ethics of holding a picnic and it's core components. We NEED to apply divide and conqure on this problem before we could even start to make a decision. We might even need to apply derivatives to calculate the slope at how long the picnic takes (x) and how much fun it is (y). Then we can decide the optimal time to hold the picnic 🙊
TL;DR - I would postpone the picnic! - not rain is 1.08x more likely to cloudy than rain. - not rain is 24.00x more likely to not cloudy than rain. - cloudy is 12.00x more likely to rain than not cloudy. - not cloudy is 1.85x more likely to not rain than cloudy. - cloudy given rain is 5.23x more likely than cloudy given not rain. - not cloudy given not rain is 4.24x more likely than not cloudy given rain. Let’s say the prevalence or prior probabilities for rain is 15.00% (odds of 0.18x or chances of 100 for every 667), and for not rain is 85.00% (5.67x or 100 for every 118), whether or not cloudy. In a world of rain, 80.00% (4.00x or 100 for every 125) is cloudy, let’s say, and 20.00% (0.25x or 100 for every 500) is not cloudy. In a world of not rain, 15.29% (0.18x or 100 for every 654) is cloudy, let’s say, and 84.71% (5.54x or 100 for every 118) is not cloudy. Thus, rain is 5.23x as likely cloudy as not rain. Also, rain is 0.24x as likely not cloudy as not rain. We know this as the Likelihood Ratio, Risk Ratio, or Bayes Factor. The Relative Risk Increase is 423.22%, and the Absolute Risk Increase is 64.71% (1.83x or 100 for every 155). The prevalence of cloudy, or not cloudy, regardless of rain or not rain, is 25.00% (0.33x or 100 for every 400), and 75.00% (3.00x or 100 for every 133), respectively. Therefore, which is more likely? In a world of cloudy, the posterior probability of rain is 48.01% (0.92x or 100 for every 208), and not rain is 51.99% (1.08x or 100 for every 192). In a world of not cloudy, the posterior probability of rain is 4.00% (0.04x or 100 for every 2500), and not rain is 96.00% (24.00x or 100 for every 104). That's an Attributable Risk or Risk Difference of 44.01% (0.79x or 100 for every 227). The Accuracy Rate (that is, 'true-positive' and 'true-negative') is 84.00% (5.25x or 100 for every 119), and the Inaccuracy Rate (that is, 'false-positive' and 'false-negative') is 16.00% (0.19x or 100 for every 625). The probability of rain, and cloudy is 12.00% (0.14x or 100 for every 833). The probability of rain, and not cloudy is 3.00% (0.03x or 100 for every 3333). The probability of not rain, and cloudy is 13.00% (0.15x or 100 for every 769). The probability of not rain, and not cloudy is 72.00% (2.57x or 100 for every 139). Sensitivity analysis: What would the prevalence or prior probabilities for rain, and not rain, whether or not cloudy, need to be such that in a world where rain given cloudy, and not rain given cloudy, that both these posterior probabilities are equally likely? In other words, we’d be indifferent? The prevalence of rain would need to be 16.05% (0.19x or 100 for every 623), and not rain would need to be 83.95% (5.23x or 100 for every 119), all else being equal. Similarly, what would the prevalence or prior probabilities for rain, and not rain, whether or not not cloudy, need to be such that in a world where rain given not cloudy, and not rain given not cloudy, that both these posterior probabilities are equally likely? In other words, we’d be indifferent? The prevalence of rain would need to be 80.90% (4.24x or 100 for every 124), and not rain would need to be 19.10% (0.24x or 100 for every 524), all else being equal. What would the consequent probabilities or likelihoods for cloudy given rain, and not cloudy given rain, need to be such that in a world where rain given cloudy, and not rain given cloudy, that both these posterior probabilities are equally likely? In other words, we’d be indifferent? The likelihood of cloudy given rain would need to be 86.64% (6.49x or 100 for every 115), and not cloudy given rain would need to be 13.36% (0.15x or 100 for every 749), all else being equal. Similarly, what would the consequent probabilities or likelihoods for cloudy given not rain, and not cloudy given not rain, need to be such that in a world where rain given cloudy, and not rain given cloudy, that both these posterior probabilities are equally likely? In other words, we’d be indifferent? The likelihood of cloudy given not rain would need to be 14.12% (0.16x or 100 for every 708), and not cloudy given not rain would need to be 85.88% (6.08x or 100 for every 116), all else being equal.
What would the consequent probabilities or likelihoods for cloudy given not rain, and not cloudy given not rain, need to be such that in a world where rain given not cloudy, and not rain given not cloudy, that both these posterior probabilities are equally likely? In other words, we’d be indifferent? The likelihood of cloudy given not rain would need to be 96.47% (27.33x or 100 for every 104), and not cloudy given not rain would need to be 3.53% (0.04x or 100 for every 2833), all else being equal. Using the Wald test, the relationship or association between rain or not rain, and cloudy is statistically significant (n=10,000). Odds Ratio (OR) = 22.16, p < .001, 95% Confidence Interval (CI) [19.27, 25.48]. We can say the same between rain or not rain, and not cloudy. OR = 0.05, p < .001, 95% CI [0.04, 0.05]. By changing P(E|H') of 15.29% such that the OR would be 19.27 - 25.48, then P(H|E) = 45.10% - 50.99%. Therefore, I would postpone the picnic!
Bayes’ Rule: P(H|E) = P(H) x P(E|H) / P(H) x P(E|H) + P(H’) x P(E|H’) Contingency Table: ||H|H’|Total| |:-:|-:|-:|-:| |E|1200|1300|2500| |E'|300|7200|7500| |Total|1500|8500|10000|
Great point! This version is a more general formula if there are more than 2 events being considered. In this video, we just used the simplified version of 2 events to make it easier.
I don't know how this is not the most viewed Bayes' theorem video because its the most helpful in youtube
That means so so much to me! Thank you for saying that! I appreciate it!
I am Indian I have no words for saying video but I say few word this is very amazing and very helpfull in all students
Indeed, great vid. I also found this one to be very useful:
ua-cam.com/video/1QulO1jS2Hk/v-deo.htmlfeature=shared
I totally agree
because it's not
The best explanation of Bayes' theorem on youtube, thank you
Saying this video is the best is an understatement. Thank you so much for posting this beyond-amazing video!
THANKS MAN! Tomorrow is my official school graduation exam and honestly i didn't ecen know a word about this concept so i was worried and your video popped up! Thanks a bunch for making me understand it! Ill be back to report my marks if i am reminded of this comment!
P. S: keep doing this. We love it and will support you through the best of our efforts!
Thank you so much for your comment! It's stories like yours that give me the fuel to make these videos. I wish you luck on your exam! And thank you for the support!
This is the most simple explaination I've ever found without any confusion. Thank you
Please keep making more videos. I am an MPH student at Harvard, and you make the concepts extremely understable. Sending you a lot of love
Thank you so much for your kind words! I really appreciate it and will keep work on putting out videos.
@@AceTutors1 we need probability testing statistics if possible
Thanks a lot for this intuitive example. It helped me a lot to understand this mechanism when I understood that as p(cloudy) becomes smaller, p(rain|cloudy) becomes greater, all else being equal. Since p(cloudy) is the numerator.
This makes sense intuitively, because in a situation clouds are rare (i.e. p(cloudy) is smaller), but when it rains, there were often clouds in the morning (i.e. p(cloudy|rain) is large), the prediction value of it being cloudy in the morning is high.
Reversely, in a climate where it is always cloudy (i.e. p(cloudy) is near one), the fact that it's cloudy in the morning does not tell you much in terms of how much rain you will get.
Probably the clearest explanation of Bayes Theorem I have seen so far. Beautifully done. Got to watch all your videos now.
Got an exam in 8 min, this was good help
Finally I understand it!. So P(A|B) is not the probability of "A" being the result of "B", but the probability of "A" when "B" is observed. Thanks!
I think the example in this video is better than what 3B1B gave in his Bayes' theorem video. The starting wasn't good because you just spammed the formula but the example and the way you conveyed it is really good. One can understand the principle through your example. Good work!
I’ve never seen a more clear explanation of how Bayes’ Theorem can be applied. This is extremely helpful! Thank you so much!
Thank you so much for saying that! I really appreciate the support!
Great videos Mark, you inspire us every day with your slogan "You've big dreams, don't let a class get in your way."
The likelihood of it being rain while picnic seems low, as it's less than 0.5 / 50%. So I think.i would still go on the picnic.
Thank you so much for your kind words and support! Ahh, you might have a higher risk tolerance than me! haha :)
At last! I got it! I wish my professor were so clear. Subscribed.
This is so beautiful.
I didn't understand it really at first, but after now I have a pretty great idea of it
That's terrific to hear! That is exactly my goal with my videos!
Loved this, I never really understood until now after watching your tutorial..... Your a genius or just a great teacher, thank you. ❤
Thank you so much. It took 1 video of you understand 3 hours of lecture.
I struggled a lot to understand this and you have clarified this for me so well. Thank you.
That was easy and simple to understand. Master it is another thing, but my guess is that you saved me some precious time with this video. Thank you a lot.
This is really an excellent explanation of Bayes Theorem.
honestly u did better job than many others
Great video! The perfect first step to understanding Bayes' Theorem.
Please make more videos on the probabilities. Thank you so much We appreciate your effort.
English is not my first language and you still made it very easy :D
I am from Bangladesh, have started to learn machine learning. For which I have to learn probability and statistics.
I have clear out all the topics on probability by 11-12th books. But BAYES' THEOREM was seemed to tough to understand.
So I came to UA-cam and saw lots of videos which was even approximately half an hour! Although they tried for a long time, they all were gloomy to understand.
But your 8 minute video is so effective than all those videos. Thanks. I have subscribed your channel. I will visit again if any other topics I have to understand in future.
I have statistics mid semester exams and i'm hoping to get good grades 😊
How did you made your subscribe button glow at 0:25 ??
UA-cam automatically does that, whenever the creator says "subscribe" the button glows in rainbow colour.
This was very helpful am taking statistics class and was so lost. Thanks
God bless you sir for this video. I HAVE went through few videos on UA-cam and this was one of the best where my mind has understood this fully. Now lets see if you have stuff on Binomial distribution. Thanks just subscribed now
Amazing video, thank you for the explanation it finally clicked.
This is the best UA-cam video I've ever seen on bayes theorem I don't think I've ever understand bayes theorem as much as this before❤ 0.48 is not enough to make me cancel my picnic I think I will go for the picnic since it's 0.48% chance of rain 🌧️ ☔
It's 48% chance of rain.
How did u understand anything if you don t even know that 0.48 is 48%
THANK YOU!
I was trying to learn Bayes' Theorem off the example of "Go For Broke" the gameshow. 🥵 my brain was twisting in on itself. THIS I can understand.
OMG THANK YOUUUUUUUUUUUUUUUUUUUUUUUU!!!!!!!!!!!!!!!!!
You just made this SOOOOOOOOO much easier!! 😭 the instructions I had was really making me punch the air.
Thank you, thank you, thank you! ♥
it would make me 48% worried about rain and 48% considerable of postponing the picnic
Good and simple explanation. Thanks.
Which software are you using to make these videos?
Thank you. I'm still having concept issues. As a teaching technique is it possible for you to summarize the meaning of the numerator and what the denominator accomplishes in the equation
Thanks a lot, I appreciate it. Tomorrow I have a quiz on probability theory and needed a quick revision of the topic.
Very clear to understand thank you
Sir You are just awesome....maintaining your work and contributing to students is great...😊 May i know which software you use to edit?
This is an amazing video, thank you !
Well done! Excellent Explanation. And I would definitely cancel the picnic I'm worry wart. LOL
Definitely demystified sir thank you ❤
Thank you. a great explanation for a rusty brain like mine!
Hi, so is conditional probability used with limited information in a question, however Bayes theorem can be used to answer a question that has more information? I'm just struggling with which one to use in an exam question
Thank you soo much for well explaining this concept. I now can say that I understand it better!
That's amazing to hear! Thanks for watching!
i feel like i want to cry
#metoo 😢
Ubhala ksasa i stats 130 ngo 6 e commerce??
Find the probability given that she saw the video 👁️👄👁️
Impossible event 😂@@aditya_kashyap-bathroom-singer
So farthis is the best bayes explanation. Can you explain this thing using a venn diagram and a probability distribution for the cloudy rain example
Thanks so much for your kind words! I'm not sure a probability distribution would help much, but some more Venn diagrams could be helpful! We'll consider this in a follow-up video! Thanks for the feedback!
Thanks for your videos. It helped a lot. Please do something on hypothesis. Thanks
Very precisely explained.Thank you Sir❤
How you will connect prior and posterior terms with this?
I think this solution is incorrect actually. We should have calculated P(C) with the rainy days' 0.85 ratio with the formula : P(C)=P(C∣R)×P(R)+P(C∣¬R)×P(¬R) where C is being cloudy and R is rainy. So it would be 0.12 for P(C∣R)×P(R) and 0.215 for P(C∣¬R)×P(¬R) and the calculations made it's 0.36. Can you clarify please
very helpful great example used
Wow fairly good explanation. i just understand it perfectly today. However, in the process, i found another better way to comprehend this theorem.
To those who still do not understand. Read this.
First you must understand what p(a/b) is. It is the probability that a happening when we already know that b happened.
To find p(a/b) we need to find prob that a and b happening at the same time , and divided it by prob of b happening.
To find prob that and b happening at the same time (p (a interect b)), you can find that indirectly from prob that b happening when we already know that a happened multiplied by prob of a happening
Ah.... i need a pen and a paper to convey this concept 🙄
I think the most difficult part overall regarding to probability problems... are the wordings. They seem to be confusing
Thanks bro
you earned a sub
superb video. How easily the theorem is explained with the help an excellent example...thanks.
Can you please make a video on linear regression and correlation
Thank you very, very much😊
Mesmerizing awesome for beginners
love your work man, keep up the good work!
Thank you so much for the support!
You did great. 👍🏻
Great videos man they help a lot
Thank you so much! I appreciate it!
Super helpful and straightforward. Thank you sir please keep posting
It indeed helped ! appreciate it gentleman
Great, I'm so glad! Thanks for your support!
Thank you for creating this theorem, Bae 😍
Thank you so much for this vid man, your method of explanation was impressive
absolutely AMAZING
Very nicely explained
So what you’re saying (if I pause the video @6:23) is:
If the probability of it being cloudy outside is 4 times greater if it also rains that day, then the probability of it raining on any given day is also 4 times greater if it happens to be a cloudy day?
I essentially just pretended that P(cloudy) was equal to 0.2 instead of 0.25, and then I just isolated the ratio of: P(cloudy/rain)/P(cloudy)
Given the statement "it’s 4 times more likely to be cloudy outside, given it is also raining," we can express this as:
P(Cloudy | Rain) = 4 * P(Cloudy)
Substituting this into the Bayes' theorem equation, we get:
P(Rain | Cloudy) = (4 * P(Cloudy) * P(Rain)) / P(Cloudy)
The P(Cloudy) term cancels out, resulting in:
P(Rain | Cloudy) = 4 * P(Rain)
So, yes, from a simple Bayesian probability standpoint, if it's 4 times more likely to be cloudy outside given that it's raining, then it's 4 times more likely to rain given that it's cloudy outside.
Thank u for getting me
good sleep
Can we solve the same example by conditional probability?
great, great and best explanation
Good explanation.Thank you
Omg thank you sir thank you so much ❤ the way u explained it,, cleared my all doubts regarding this topic ❤
That's really awesome to hear! Thanks for the kind words!
I'm still puzzeld on which data is A and which is B - and why. Swapping things around changes the outcome of the formula, doesn't it?
This was super helpful thank you!
this was great thnx for the example
Dude, when the guy says hit the subscribe button, the button lights up. I noticed it just now
This series is amazing. Must have been hard to make these beautiful animations. Thank you so much❤.
Can you please make two more distributions viz:
1. "Poisson Distribution" and
2. "Exponential Distribution"
And explain the intuition behind the mean and standard deviation in these distributions like you did in Uniform distribution video?
Super explanation. Thanks Sir
thank you for this sharing
Thank you NRI ❤
Thanks for the concept ☺️ I think I will try to reschedule 😅❤
That's so helpful ❤🤝
hi I want to ask so in this case what the addictional knowledge? the probability of beibg cloud?
Ek thanks toh banta hein!😛❤
Super helpful ,Thank you 🌹
Great explaination sir
if one knows that there are 12 rainy cloudy days (the 80% of 15) in 100 days and 25 cloudy days in total why one can’t just calculate 12/25= 0.48, without all the machinery and the language that the Bayes formula brings along? Is there something wrong in just applying the definition of probability?
Great job
Good explanation 🎉
Depends on how much you like picnics, how often you want to go on picnics, if you think rain ruins it and maybe you might already think a cloudy day isn't nice for picnics.
But before we think about that, let's think about the ethics of holding a picnic and it's core components. We NEED to apply divide and conqure on this problem before we could even start to make a decision.
We might even need to apply derivatives to calculate the slope at how long the picnic takes (x) and how much fun it is (y). Then we can decide the optimal time to hold the picnic 🙊
I would have liked if you incorporated dark clouds vs white clouds in the calculation.
Do you use the Python library called "manim" to create these beautiful animations for your great videos.
That's great when the example gives what P A|B is
TL;DR - I would postpone the picnic!
- not rain is 1.08x more likely to cloudy than rain.
- not rain is 24.00x more likely to not cloudy than rain.
- cloudy is 12.00x more likely to rain than not cloudy.
- not cloudy is 1.85x more likely to not rain than cloudy.
- cloudy given rain is 5.23x more likely than cloudy given not rain.
- not cloudy given not rain is 4.24x more likely than not cloudy given rain.
Let’s say the prevalence or prior probabilities for rain is 15.00% (odds of 0.18x or chances of 100 for every 667), and for not rain is 85.00% (5.67x or 100 for every 118), whether or not cloudy. In a world of rain, 80.00% (4.00x or 100 for every 125) is cloudy, let’s say, and 20.00% (0.25x or 100 for every 500) is not cloudy. In a world of not rain, 15.29% (0.18x or 100 for every 654) is cloudy, let’s say, and 84.71% (5.54x or 100 for every 118) is not cloudy. Thus, rain is 5.23x as likely cloudy as not rain. Also, rain is 0.24x as likely not cloudy as not rain. We know this as the Likelihood Ratio, Risk Ratio, or Bayes Factor.
The Relative Risk Increase is 423.22%, and the Absolute Risk Increase is 64.71% (1.83x or 100 for every 155). The prevalence of cloudy, or not cloudy, regardless of rain or not rain, is 25.00% (0.33x or 100 for every 400), and 75.00% (3.00x or 100 for every 133), respectively.
Therefore, which is more likely? In a world of cloudy, the posterior probability of rain is 48.01% (0.92x or 100 for every 208), and not rain is 51.99% (1.08x or 100 for every 192). In a world of not cloudy, the posterior probability of rain is 4.00% (0.04x or 100 for every 2500), and not rain is 96.00% (24.00x or 100 for every 104).
That's an Attributable Risk or Risk Difference of 44.01% (0.79x or 100 for every 227). The Accuracy Rate (that is, 'true-positive' and 'true-negative') is 84.00% (5.25x or 100 for every 119), and the Inaccuracy Rate (that is, 'false-positive' and 'false-negative') is 16.00% (0.19x or 100 for every 625). The probability of rain, and cloudy is 12.00% (0.14x or 100 for every 833). The probability of rain, and not cloudy is 3.00% (0.03x or 100 for every 3333). The probability of not rain, and cloudy is 13.00% (0.15x or 100 for every 769). The probability of not rain, and not cloudy is 72.00% (2.57x or 100 for every 139).
Sensitivity analysis:
What would the prevalence or prior probabilities for rain, and not rain, whether or not cloudy, need to be such that in a world where rain given cloudy, and not rain given cloudy, that both these posterior probabilities are equally likely? In other words, we’d be indifferent? The prevalence of rain would need to be 16.05% (0.19x or 100 for every 623), and not rain would need to be 83.95% (5.23x or 100 for every 119), all else being equal.
Similarly, what would the prevalence or prior probabilities for rain, and not rain, whether or not not cloudy, need to be such that in a world where rain given not cloudy, and not rain given not cloudy, that both these posterior probabilities are equally likely? In other words, we’d be indifferent? The prevalence of rain would need to be 80.90% (4.24x or 100 for every 124), and not rain would need to be 19.10% (0.24x or 100 for every 524), all else being equal.
What would the consequent probabilities or likelihoods for cloudy given rain, and not cloudy given rain, need to be such that in a world where rain given cloudy, and not rain given cloudy, that both these posterior probabilities are equally likely? In other words, we’d be indifferent? The likelihood of cloudy given rain would need to be 86.64% (6.49x or 100 for every 115), and not cloudy given rain would need to be 13.36% (0.15x or 100 for every 749), all else being equal.
Similarly, what would the consequent probabilities or likelihoods for cloudy given not rain, and not cloudy given not rain, need to be such that in a world where rain given cloudy, and not rain given cloudy, that both these posterior probabilities are equally likely? In other words, we’d be indifferent? The likelihood of cloudy given not rain would need to be 14.12% (0.16x or 100 for every 708), and not cloudy given not rain would need to be 85.88% (6.08x or 100 for every 116), all else being equal.
What would the consequent probabilities or likelihoods for cloudy given not rain, and not cloudy given not rain, need to be such that in a world where rain given not cloudy, and not rain given not cloudy, that both these posterior probabilities are equally likely? In other words, we’d be indifferent? The likelihood of cloudy given not rain would need to be 96.47% (27.33x or 100 for every 104), and not cloudy given not rain would need to be 3.53% (0.04x or 100 for every 2833), all else being equal.
Using the Wald test, the relationship or association between rain or not rain, and cloudy is statistically significant (n=10,000). Odds Ratio (OR) = 22.16, p < .001, 95% Confidence Interval (CI) [19.27, 25.48]. We can say the same between rain or not rain, and not cloudy. OR = 0.05, p < .001, 95% CI [0.04, 0.05]. By changing P(E|H') of 15.29% such that the OR would be 19.27 - 25.48, then P(H|E) = 45.10% - 50.99%. Therefore, I would postpone the picnic!
Bayes’ Rule: P(H|E) = P(H) x P(E|H) / P(H) x P(E|H) + P(H’) x P(E|H’)
Contingency Table:
||H|H’|Total|
|:-:|-:|-:|-:|
|E|1200|1300|2500|
|E'|300|7200|7500|
|Total|1500|8500|10000|
clearly understandable
Ive seen bayes theorem be written as P(Ei|A) = P(Ei)P(A|Ei) / ∑ P(Ek)P(A|Ek)
can you explain this version?
Great point! This version is a more general formula if there are more than 2 events being considered. In this video, we just used the simplified version of 2 events to make it easier.
Thank you so very much!