Thanks for watching this, my 4th video, looking at one way in which negative “probabilities” can (maybe) appear in quantum theory. It was not in my original list of video ideas, but was suggested in response to some recent comments on Twitter. Any suggestions for future topics, please let me know in the comments. Thanks!
Taleb is trying to make the same point about probabilities that you are making about the wave function. They only make *physical* sense when they are applied as the kernel of an integral. You can mathematically make sense of them outside of that context, and it's often useful to use this thinking to solve problems. But attempting to make physical sense of them as independent objects just leads to confusion and even apparent paradoxes.
Yes, I kind of think that's what he is saying, although that would be a probability *density*. Also, I think he is referring to the practice of inferring a probability density from option prices, which can be interpreted as a probability but is really just a tool used for option pricing. If this goes negative, it suggests the existence of arbitrage, so no philosophical problems. However, he backed up the mention of negative probabilities by saying that they are used quantum mechanics, which I followed up on here. A bit more interesting than the option pricing example (although maybe option pricing is a subject of a future video)
I guess my first instinct when I see probabilities that add to less than 1 is to say you're not counting everything you should or you've got the wrong probabilities assigned or your system is wrong.
Hej I've been reading your blog for like a year now I didn't even know you have a youtube channel :) Love your posts about stochastic analysis! Edit: Great video, quite easy to follow, I like that you didn't go into that much "unnecessary" detail and instead focused on why we want model negative probabilities through real world examples.
Nice video! The QM approach to probability (squared norms) being different from everything else really does baffle me. I'd love to see a video on topics like probability via expectation, free probability, etc. Cheers.
Thanks for the comment. Probability via expectation and free probability are both interesting topics. I expect I’ll do something on those a bit further down the line
Hey George ! Huge fan of your blog and now of your UA-cam channel ! I remember suggesting, on X, a book collection tour as a video if you would like to share your book collection. Otherwise, Can you please suggest your best readings on probability theory and stochastics processes and calculus !
I will think about that. My book collection isn’t that extensive, not sure it is enough for a video on that alone. The Williams probability with martingales book, Rogers & Williams, Protter, and Kallenberg, are main ones I’ve used. They’re all 20 years old or more by now though, maybe there’s a good recent book at add to this?
@@almostsure I am not sure ! But the one you've mentioned are still the main references. Not sure if there's new textbooks on the subject Thanks George ! Keep up the good work !
it's ok to have signed measures but how is it useful to think of them as probabilities? Alternative interpretations of probability can be fine (e.g. allowing infinitesimals) but it needs to genuinely be a useful idea.
Yes, signed measures (and complex or vector valued) measures exist as mathematical objects. As discussed in my video, negative probabilities do appear when you try extending actual probabilities from observable events to unobservable ones. Whether they should be called probabilities, and how useful it is, is up for debate, although they can be used to try and bridge the quantum-classical divide. More-so with Wigner quasiprobabilities than the single qbit discussed here
There's a wikipedia page about negative probabilities. I've seen negative probabilities pop up in a few places in queueing theory and finance, such as: Cox, 1955: Complex probabilities; Nojo and Watanabe, 1987: Negative branching Probability (NP) distrib.; Graham, Knuth, Patashnik 1994 pg 403; Tijms H, Staats K. NEGATIVE PROBABILITIES AT WORK IN THE M/D/1 QUEUE. Probability in the Engineering and Informational Sciences. 2007;21(1):67-76. doi:10.1017/S0269964807070040
Thanks for those references! I could clearly have gone in various different ways with the negative probability concept, but just picked one simple way from QM
Wouldn't it make more sense to think of a polarizer as an array of prisms, rather than a filter? Aren't they just made of a bunch of really tiny dichroic crystals all lined up with each other?
The ones I used in the video are Polyethylene terephthalate (PET). Don’t think it consists of crystals. In any case, the point in the video is that they let light of one polarisation through while blocking the opposite polarisation, the underlying mechanism not really being relevant to the discussion there
Well, if the probability for you marrying is 20%. The probability that a marriage is arranged by your parents is 30%, then for sure the probability for you to find the girlfriend yourself is -10%
Where negative money or negative apples would make sense: Lets say I own a market stall that sells apples. In the morning, I buy some stock for the day, that is negative money (money going out, and positive apples, apples coming in). When a customer visits, that is positive money and negative apples. A loan would also be negative money, as would selling a futures contract for apples (I sell apples I don't have, and have an obligation to deliver apples at a future date). An apple farmer might do this at the beginning of the season with a delivery date after harvest, or commodity traders might do it to gamble on apple prices.
Agreed. Negative money makes sense. But how would you even define negative probability? You can't owe anyone a probability. If you rolled 9 straight heads, the next coin still has a 50% chance. Moreover a probability of 1 means it has a 100% chance, a probability of 0 means it's never gonna happen, so how would one describe negative probability?
Probability amplitudes can be negative. Probability densities are always positive -- the Born rule. Amplitudes are dual to densities -- probability or AC/DC. "Negation of the negation gives a positive" -- Hegel. Negative probability can be associated with negative frequencies -- the Fourier transform. The time domain is dual to the frequency domain -- Fourier analysis. Positive (clockwise) is dual to negative (anti-clockwise) -- numbers, frequencies, electric charge, curvature. Negative probability would imply negative information. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual -- photons are dual or pure energy is dual. "Always two there are" -- Yoda. Exponentials (probability waves) are dual to logarithms (information). Lie groups are dual to Lie Algebra.
i think if you're going to present something, you should really become familiar with the topic. This presentation was all over the place, and the made the topic even more confusing and self-contradicting.
@@almostsure I clicked on the video because I'm interested in maths and probability. But I'm not an expert on that nor physics, and your explanation became confusing and rambling past the halfway point. I can only speak for myself, but I think you should improve your explanations. Maybe run the script by a layperson first.
@@hoagie911 I'm no expert in Physics and I followed it pretty easily. Trig and probability knowledge should take you through it all. The demonstrations of filters gave a clear, real-world example of how it applies.
Thanks for watching this, my 4th video, looking at one way in which negative “probabilities” can (maybe) appear in quantum theory. It was not in my original list of video ideas, but was suggested in response to some recent comments on Twitter.
Any suggestions for future topics, please let me know in the comments. Thanks!
Taleb is trying to make the same point about probabilities that you are making about the wave function. They only make *physical* sense when they are applied as the kernel of an integral. You can mathematically make sense of them outside of that context, and it's often useful to use this thinking to solve problems. But attempting to make physical sense of them as independent objects just leads to confusion and even apparent paradoxes.
Yes, I kind of think that's what he is saying, although that would be a probability *density*.
Also, I think he is referring to the practice of inferring a probability density from option prices, which can be interpreted as a probability but is really just a tool used for option pricing. If this goes negative, it suggests the existence of arbitrage, so no philosophical problems.
However, he backed up the mention of negative probabilities by saying that they are used quantum mechanics, which I followed up on here. A bit more interesting than the option pricing example (although maybe option pricing is a subject of a future video)
Neat video. I like how it wasn't just a talky video, but you did demonstrations and showed images.
I guess my first instinct when I see probabilities that add to less than 1 is to say you're not counting everything you should or you've got the wrong probabilities assigned or your system is wrong.
Or that the samples aren't independent.
Not quite
Hej I've been reading your blog for like a year now I didn't even know you have a youtube channel :)
Love your posts about stochastic analysis!
Edit: Great video, quite easy to follow, I like that you didn't go into that much "unnecessary" detail and instead focused on why we want model negative probabilities through real world examples.
Thanks!
Its the interaction of probabilities that shows the necessity for negative values.
Nice video! The QM approach to probability (squared norms) being different from everything else really does baffle me. I'd love to see a video on topics like probability via expectation, free probability, etc. Cheers.
Thanks for the comment. Probability via expectation and free probability are both interesting topics. I expect I’ll do something on those a bit further down the line
Hey George !
Huge fan of your blog and now of your UA-cam channel !
I remember suggesting, on X, a book collection tour as a video if you would like to share your book collection. Otherwise, Can you please suggest your best readings on probability theory and stochastics processes and calculus !
I will think about that. My book collection isn’t that extensive, not sure it is enough for a video on that alone.
The Williams probability with martingales book, Rogers & Williams, Protter, and Kallenberg, are main ones I’ve used. They’re all 20 years old or more by now though, maybe there’s a good recent book at add to this?
@@almostsure I am not sure ! But the one you've mentioned are still the main references. Not sure if there's new textbooks on the subject
Thanks George ! Keep up the good work !
it's ok to have signed measures but how is it useful to think of them as probabilities? Alternative interpretations of probability can be fine (e.g. allowing infinitesimals) but it needs to genuinely be a useful idea.
Yes, signed measures (and complex or vector valued) measures exist as mathematical objects. As discussed in my video, negative probabilities do appear when you try extending actual probabilities from observable events to unobservable ones. Whether they should be called probabilities, and how useful it is, is up for debate, although they can be used to try and bridge the quantum-classical divide. More-so with Wigner quasiprobabilities than the single qbit discussed here
Quasi, or Pseudo, or Semi. Or Almost.
Welcome to quasisure!
There's a wikipedia page about negative probabilities. I've seen negative probabilities pop up in a few places in queueing theory and finance, such as: Cox, 1955: Complex probabilities; Nojo and Watanabe, 1987: Negative branching Probability (NP) distrib.; Graham, Knuth, Patashnik 1994 pg 403; Tijms H, Staats K. NEGATIVE PROBABILITIES AT WORK IN THE M/D/1 QUEUE. Probability in the Engineering and Informational Sciences. 2007;21(1):67-76. doi:10.1017/S0269964807070040
Thanks for those references! I could clearly have gone in various different ways with the negative probability concept, but just picked one simple way from QM
Wouldn't it make more sense to think of a polarizer as an array of prisms, rather than a filter? Aren't they just made of a bunch of really tiny dichroic crystals all lined up with each other?
The ones I used in the video are Polyethylene terephthalate (PET). Don’t think it consists of crystals. In any case, the point in the video is that they let light of one polarisation through while blocking the opposite polarisation, the underlying mechanism not really being relevant to the discussion there
The probability of me finding a girlfriend is for sure negative
So she stalking you and you running away. Might probably find that negative.
Don't worry. You still have the probability of BECOMING a girlfriend. ♥
bro same...lol
Well, if the probability for you marrying is 20%. The probability that a marriage is arranged by your parents is 30%, then for sure the probability for you to find the girlfriend yourself is -10%
@@steffenbendel6031 Probability that you find the girlfriend yourself is 14% surely? (70% of 20%).
Thank you!
Where negative money or negative apples would make sense:
Lets say I own a market stall that sells apples. In the morning, I buy some stock for the day, that is negative money (money going out, and positive apples, apples coming in). When a customer visits, that is positive money and negative apples.
A loan would also be negative money, as would selling a futures contract for apples (I sell apples I don't have, and have an obligation to deliver apples at a future date). An apple farmer might do this at the beginning of the season with a delivery date after harvest, or commodity traders might do it to gamble on apple prices.
Agreed. Negative money makes sense. But how would you even define negative probability? You can't owe anyone a probability. If you rolled 9 straight heads, the next coin still has a 50% chance. Moreover a probability of 1 means it has a 100% chance, a probability of 0 means it's never gonna happen, so how would one describe negative probability?
Maybe ist can in quantum theory
Before watching.
Usually when people say negative probability they're talking about the square root of the probability for quantum/wave stuff
Probability amplitudes can be negative.
Probability densities are always positive -- the Born rule.
Amplitudes are dual to densities -- probability or AC/DC.
"Negation of the negation gives a positive" -- Hegel.
Negative probability can be associated with negative frequencies -- the Fourier transform.
The time domain is dual to the frequency domain -- Fourier analysis.
Positive (clockwise) is dual to negative (anti-clockwise) -- numbers, frequencies, electric charge, curvature.
Negative probability would imply negative information.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Real is dual to imaginary -- complex numbers are dual -- photons are dual or pure energy is dual.
"Always two there are" -- Yoda.
Exponentials (probability waves) are dual to logarithms (information).
Lie groups are dual to Lie Algebra.
Cough
i think if you're going to present something, you should really become familiar with the topic. This presentation was all over the place, and the made the topic even more confusing and self-contradicting.
Almost impossible to follow. Disliked.
Oh well, I guess a channel going into mathematical ideas isn’t for everyone!
@@almostsure I clicked on the video because I'm interested in maths and probability. But I'm not an expert on that nor physics, and your explanation became confusing and rambling past the halfway point. I can only speak for myself, but I think you should improve your explanations. Maybe run the script by a layperson first.
Kid hittin wall and blaming wall for pain, maybe stop being imbecile better choise?
@@hoagie911 I'm no expert in Physics and I followed it pretty easily. Trig and probability knowledge should take you through it all. The demonstrations of filters gave a clear, real-world example of how it applies.