8 ms·

Nobody Understands Probability

84 points by kmod 16y ago

frossie 16y ago
For those who didn't make it all the way down: People often intuitively think of probabilities as a fact about the world, when in reality probabilities are a fact about our model of the world.
- tome 16y ago
  There are at least two interpretations of probability. Firstly the epistemological, as you say, represents our lack of knowledge about the world. Secondly the aleatory truly represents the phenomenon of chance in the world. The best interpretation of quantum theory for example (as I understand it) takes the latter view that randomness is genuinely physically manifested, and does not simply represent our inability to model reality.
  - Confusion 16y ago
    The best interpretation of quantum theory [..] takes the [..] view that randomness is genuinely physically manifested We could argue endlessly about whether that is 'the best' interpretation (which ethical assumptions does your 'the best' presuppose?) but fortunately it doesn't matter. As Mermin (from the famous Ashcroft and Mermin book on Solid State Physics) famously quipped: you can 'shut up and calculate'. The usefulness of the model does not depend on its interpretations (although the interpretations are certainly important with respect to scientific progress).
    - tome 16y ago
      Well indeed, and this doesn't just apply to quantum mechanics but to the OP's quotation from the article too: People often intuitively think of probabilities as a fact about the world, when in reality probabilities are a fact about our model of the world.
  - scott_s 16y ago
    I think you're confusing concepts. The original point is about the difference between the map and the territory. And, because I'm (finally) systematically going through Eliezer's sequences: http://wiki.lesswrong.com/wiki/Map_and_Territory_(sequence) http://wiki.lesswrong.com/wiki/Map_and_Territory_(sequence) Your second paragraph is about a particular map: quantum theory. Quantum theory has probabilities in it. The dominant interpretation of quantum theory is that the probabilities accurately represent what happens in the universe; they are not artifacts for us to correct. But there is still a difference between our map (quantum theory) and the territory (the universe itself). Put another way: quantum theory is a map with uncertainty baked into it. But this uncertainty has been accurately mapped.
    - tome 16y ago
      No I don't think so. Unlike statistical physics, where probabilities are simply a mathematical technique for dealing with uncertainty, quantum mechanics actually postulates that randomness is inherent to the universe. If you disagree with me, please describe how your concept of "maps and territories" applies to the StatPhys/QM distinction.
      - scott_s 16y ago
        Suppose I accurately map the coastline, and every relevant part of the coastline is depicted in my map. But the map is not the same as the coastline itself. If my coastline has some feature that blips in and out of existence in a predictable way, I can integrate that into my map. My map then has uncertainty in it. That uncertainty is an accurate reflection of the coastline itself - but there is still a distinction between the map and the coastline. I don't disagree with your second sentence. But there is still a difference between our theory of quantum mechanics and the universe itself.
- tel 16y ago
  Obligatory quote: "Essentially, all models are wrong, but some are useful" - George Box
- Eliezer 16y ago
  Not an exact quote, but E. T. Jaynes: "If I am ignorant about a phenomenon, that is a fact about my state of mind, not a fact about the phenomenon."
  - deleted 16y ago
    [deleted]
scottw 16y ago
The odds of nobody understanding probability is near-zero...
- westbywest 16y ago
  This may arguably be one of the few probabilities in the natural world which does == 0.
- JacobAldridge 16y ago
  I figured there was a good chance someone would say that.
equark 16y ago
This article is not particularly clear. It doesn't have a clear discussion of Bayesian versus frequentist interpretations of probability or inferential statements that are conditioned on the unobserved true parameter versus the observed data. It's hard to understand the subtlety of probability without understanding p(theta), p(x), p(theta|x) and p(x|theta).
- aphyr 16y ago
  I've read several pieces on Bayesian stats, and I've done some nontrivial statistics before. It still confuses me that p(data) != 1. I kinda wish the author had gone into detail about how to calculate the probability of an already-observed event.
  - tel 16y ago
    You're confusing p(data) with p(data|data) which is, trivially, equal to 1. p(data) is better formulated as p(data|F) where F codifies your assumptions about the possible generative probability models that you're building your likelihood function from. Or, similarly, F codifies your understanding of the world and the possible things that could occur within it. This makes p(data|F) a perfect normalizing constant for the numerator of Bayes' Theorem since the numerator implies a choice of a specific model in the family F, but p(data|F) averages over all possible models/worlds/parameter choices (contained in F).
- Helianthus16 16y ago
  It sounds like he didn't get to it before running out of steam. Which is unfortunate, because that's really the reason I kept reading.
deleted 16y ago
[deleted]
RickHull 16y ago
> However, the answer is not, in fact, 1/3. Why is this? This seems like a canard to me. Here is my defense of 1/3 as a correct answer: http://gist.github.com/578386 http://gist.github.com/578386 > Is Bayes’ theorem wrong? > No, the answer comes from an unfortunate namespace collision in the word “given”. The man “gave” us the information that he has at least one male child. By this we mean that he asserted the statement “I have at least one male child.” Now our issue is when we confuse this with being “given” that the man has at least one male child, in the sense that we should restrict to the set of universes in which the man has at least one male child. This is a very different statement than the previous one. For instance, it rules out universes where the man has two girls, but is lying to us. No, we are assuming that the givens are facts that are true. > Even if we decide to ignore the possibility that the man is lying, we should note that most universes where the man has at least one son don’t even involve him informing us of this fact, and so it may be the case that proportionally more universes where the man has two boys involve him telling us “I have at least one male child”, relative to the proportion of such universes where the man has one boy and one girl. In this case the probability that he has two boys would end up being greater than 1/3. No, we don't have to consider universes where the man has at least one male child but does not inform of us of this fact. We have a set of givens that are assumed to be true, and based on those givens and the rules of logic, we can make justifiable statements of probabilities.
- aphyr 16y ago
  It's a canard, but still informative. I do wish he hadn't claimed that 1/3rd is wrong by Bayesian statistics because the frequentist approach, with the same interpretation of the problem, yields exactly the same results. It's still a valuable example of how to represent unreliable measurement processes in your model, and the importance of doing so.
- thwarted 16y ago
  1/3 isn't correct, and I find the OP's explanation to be overly complex. The set of possibilities for two genders of two children is GG, GB, and BB. In your possibilities, GB and BG are exactly the same set (order doesn't matter in a set, only membership), so you don't have 4 possibilities, you have 3 total. Since the guy asserted that one of them is a boy, you can rule out the GG possibility. This leaves only GB and BB as possible results, both of which have a 1/2 chance of being the correct one. The guy never makes a claim that the first child or the second child is the boy (but, this doesn't change the possibility that he has two boys, it just changes which one you remove from the possibilities based on the provided information). I'm not sure that it's that people don't understand statistics (although I'm not in a position to confirm or deny that), it's that people don't understand set theory. At least if you're going to use this "genders of two children" as example.
iliketosleep 16y ago
'Let’s consider an example. Suppose that a man comes up to you and says "I have two children. At least one of them is a boy." What is the probability that they are both boys?'. Am I missing something or in his attempt to solve the problem, does he implicitly assume statistical dependence? If statistical independence is assumed, with P(Boy) = P(Girl) = 1/2, then the answer to the problem is very simple. P(Boy | Boy) = P(Boy) = 1/2. Maybe I just don't understand probability :(
- tel 16y ago
  The basic formation (which the author argues is not subtle enough to be true) is better thought of step by step. Suppose a man comes up to you and says "I have two children" At this point you build a set of possible realities, your model. There are four possibilities: {BB, BG, GB, GG}. This space fully describes a model whereupon there are two distinct, children with genders. Additionally, via assumption of independence and equal likelihood, you can assign probabilities to each observation, {BB:1/4, BG:1/4, GB:1/4, GG:1/4}. "At least one of them is a boy." At this point, you update your realities by removing the one firmly contradicted by the new evidence. Your new space is {BB, BG, GB} and when you renormalize the probabilities you get {BB:1/3, BG:1/3, GB:1/3} which leads to the idea that the probability at this point that the man has two boys is 1/3rd. The author suggests however that during that second step, you should also take into account the possibility that this guy is lying or that the fact that he's proffering this information actually changes the likelihoods of those four scenarios in a way different from just multiplying one of them by 0. So perhaps the likelihood of hearing "At least one of them is a boy" is reflected like this: {BB:0.35, BG:0.32, GB:0.32, GG:0.01} And your new belief in each of these realities reflects that like so (renormalized) {BB: 0.35, BG:0.32, GB:0.32, 0.01} So now I feel even more confident that he has two boys.
  - iliketosleep 16y ago
    Thanks for your explanation. I think his first renormalization process is wrong. Because once we know there is a boy, the problem space is reduced do "What's the probability of a boy?" which is 1/2. It has nothing to do with probabilities involving the known child.
signa11 16y ago
since we are talking about probability-theory, thought folks here might find gnedenko pretty interesting: [ http://www-history.mcs.st-andrews.ac.uk/Biographies/Gnedenko.html http://www-history.mcs.st-andrews.ac.uk/Biographies/Gnedenko... ]
jmtulloss 16y ago
If you're lucky enough to attend UIUC, I highly recommend taking ECE 413 to get a thorough introduction to these concepts. It's unfortunate that none of the class materials are online since it goes well above and beyond what is taught in most undergraduate CS courses on statistics. Taking it was hard, but it made me a much better engineer. Edit: I suppose this applies to anybody in college. Take the hard statistics course that goes over this stuff. It's really valuable, and pretty hard to pick up on your own.
adolph 16y ago
This essay by Yudkowsky is also helpful. http://yudkowsky.net/rational/bayes http://yudkowsky.net/rational/bayes
rmathew 16y ago
A more thorough introduction to this topic is "Probability Theory: The Logic of Science" by E. T. Jaynes (http://www-biba.inrialpes.fr/Jaynes/prob.html http://www-biba.inrialpes.fr/Jaynes/prob.html).
AmberShah 16y ago
"Nobody"? That's not likely...
drakep 16y ago
Never use absolutes?

84 points by kmod 16y ago

For those who didn't make it all the way down: People often intuitively think of probabilities as a fact about the world, when in reality probabilities are a fact about our model of the world.

tome 16y ago
There are at least two interpretations of probability. Firstly the epistemological, as you say, represents our lack of knowledge about the world. Secondly the aleatory truly represents the phenomenon of chance in the world. The best interpretation of quantum theory for example (as I understand it) takes the latter view that randomness is genuinely physically manifested, and does not simply represent our inability to model reality.
- Confusion 16y ago
  The best interpretation of quantum theory [..] takes the [..] view that randomness is genuinely physically manifested We could argue endlessly about whether that is 'the best' interpretation (which ethical assumptions does your 'the best' presuppose?) but fortunately it doesn't matter. As Mermin (from the famous Ashcroft and Mermin book on Solid State Physics) famously quipped: you can 'shut up and calculate'. The usefulness of the model does not depend on its interpretations (although the interpretations are certainly important with respect to scientific progress).
  - tome 16y ago
    Well indeed, and this doesn't just apply to quantum mechanics but to the OP's quotation from the article too: People often intuitively think of probabilities as a fact about the world, when in reality probabilities are a fact about our model of the world.
- scott_s 16y ago
  I think you're confusing concepts. The original point is about the difference between the map and the territory. And, because I'm (finally) systematically going through Eliezer's sequences: http://wiki.lesswrong.com/wiki/Map_and_Territory_(sequence) http://wiki.lesswrong.com/wiki/Map_and_Territory_(sequence) Your second paragraph is about a particular map: quantum theory. Quantum theory has probabilities in it. The dominant interpretation of quantum theory is that the probabilities accurately represent what happens in the universe; they are not artifacts for us to correct. But there is still a difference between our map (quantum theory) and the territory (the universe itself). Put another way: quantum theory is a map with uncertainty baked into it. But this uncertainty has been accurately mapped.
  - tome 16y ago
    No I don't think so. Unlike statistical physics, where probabilities are simply a mathematical technique for dealing with uncertainty, quantum mechanics actually postulates that randomness is inherent to the universe. If you disagree with me, please describe how your concept of "maps and territories" applies to the StatPhys/QM distinction.
    - scott_s 16y ago
      Suppose I accurately map the coastline, and every relevant part of the coastline is depicted in my map. But the map is not the same as the coastline itself. If my coastline has some feature that blips in and out of existence in a predictable way, I can integrate that into my map. My map then has uncertainty in it. That uncertainty is an accurate reflection of the coastline itself - but there is still a distinction between the map and the coastline. I don't disagree with your second sentence. But there is still a difference between our theory of quantum mechanics and the universe itself.
tel 16y ago
Obligatory quote: "Essentially, all models are wrong, but some are useful" - George Box
Eliezer 16y ago
Not an exact quote, but E. T. Jaynes: "If I am ignorant about a phenomenon, that is a fact about my state of mind, not a fact about the phenomenon."
- deleted 16y ago
  [deleted]

scottw 16y ago

The odds of nobody understanding probability is near-zero...

westbywest 16y ago
This may arguably be one of the few probabilities in the natural world which does == 0.
JacobAldridge 16y ago
I figured there was a good chance someone would say that.

equark 16y ago

This article is not particularly clear. It doesn't have a clear discussion of Bayesian versus frequentist interpretations of probability or inferential statements that are conditioned on the unobserved true parameter versus the observed data. It's hard to understand the subtlety of probability without understanding p(theta), p(x), p(theta|x) and p(x|theta).

aphyr 16y ago
I've read several pieces on Bayesian stats, and I've done some nontrivial statistics before. It still confuses me that p(data) != 1. I kinda wish the author had gone into detail about how to calculate the probability of an already-observed event.
- tel 16y ago
  You're confusing p(data) with p(data|data) which is, trivially, equal to 1. p(data) is better formulated as p(data|F) where F codifies your assumptions about the possible generative probability models that you're building your likelihood function from. Or, similarly, F codifies your understanding of the world and the possible things that could occur within it. This makes p(data|F) a perfect normalizing constant for the numerator of Bayes' Theorem since the numerator implies a choice of a specific model in the family F, but p(data|F) averages over all possible models/worlds/parameter choices (contained in F).
Helianthus16 16y ago
It sounds like he didn't get to it before running out of steam. Which is unfortunate, because that's really the reason I kept reading.

deleted 16y ago

[deleted]

RickHull 16y ago

> However, the answer is not, in fact, 1/3. Why is this? This seems like a canard to me. Here is my defense of 1/3 as a correct answer: http://gist.github.com/578386 http://gist.github.com/578386 > Is Bayes’ theorem wrong? > No, the answer comes from an unfortunate namespace collision in the word “given”. The man “gave” us the information that he has at least one male child. By this we mean that he asserted the statement “I have at least one male child.” Now our issue is when we confuse this with being “given” that the man has at least one male child, in the sense that we should restrict to the set of universes in which the man has at least one male child. This is a very different statement than the previous one. For instance, it rules out universes where the man has two girls, but is lying to us. No, we are assuming that the givens are facts that are true. > Even if we decide to ignore the possibility that the man is lying, we should note that most universes where the man has at least one son don’t even involve him informing us of this fact, and so it may be the case that proportionally more universes where the man has two boys involve him telling us “I have at least one male child”, relative to the proportion of such universes where the man has one boy and one girl. In this case the probability that he has two boys would end up being greater than 1/3. No, we don't have to consider universes where the man has at least one male child but does not inform of us of this fact. We have a set of givens that are assumed to be true, and based on those givens and the rules of logic, we can make justifiable statements of probabilities.

aphyr 16y ago
It's a canard, but still informative. I do wish he hadn't claimed that 1/3rd is wrong by Bayesian statistics because the frequentist approach, with the same interpretation of the problem, yields exactly the same results. It's still a valuable example of how to represent unreliable measurement processes in your model, and the importance of doing so.
thwarted 16y ago
1/3 isn't correct, and I find the OP's explanation to be overly complex. The set of possibilities for two genders of two children is GG, GB, and BB. In your possibilities, GB and BG are exactly the same set (order doesn't matter in a set, only membership), so you don't have 4 possibilities, you have 3 total. Since the guy asserted that one of them is a boy, you can rule out the GG possibility. This leaves only GB and BB as possible results, both of which have a 1/2 chance of being the correct one. The guy never makes a claim that the first child or the second child is the boy (but, this doesn't change the possibility that he has two boys, it just changes which one you remove from the possibilities based on the provided information). I'm not sure that it's that people don't understand statistics (although I'm not in a position to confirm or deny that), it's that people don't understand set theory. At least if you're going to use this "genders of two children" as example.

iliketosleep 16y ago

'Let’s consider an example. Suppose that a man comes up to you and says "I have two children. At least one of them is a boy." What is the probability that they are both boys?'. Am I missing something or in his attempt to solve the problem, does he implicitly assume statistical dependence? If statistical independence is assumed, with P(Boy) = P(Girl) = 1/2, then the answer to the problem is very simple. P(Boy | Boy) = P(Boy) = 1/2. Maybe I just don't understand probability :(

tel 16y ago
The basic formation (which the author argues is not subtle enough to be true) is better thought of step by step. Suppose a man comes up to you and says "I have two children" At this point you build a set of possible realities, your model. There are four possibilities: {BB, BG, GB, GG}. This space fully describes a model whereupon there are two distinct, children with genders. Additionally, via assumption of independence and equal likelihood, you can assign probabilities to each observation, {BB:1/4, BG:1/4, GB:1/4, GG:1/4}. "At least one of them is a boy." At this point, you update your realities by removing the one firmly contradicted by the new evidence. Your new space is {BB, BG, GB} and when you renormalize the probabilities you get {BB:1/3, BG:1/3, GB:1/3} which leads to the idea that the probability at this point that the man has two boys is 1/3rd. The author suggests however that during that second step, you should also take into account the possibility that this guy is lying or that the fact that he's proffering this information actually changes the likelihoods of those four scenarios in a way different from just multiplying one of them by 0. So perhaps the likelihood of hearing "At least one of them is a boy" is reflected like this: {BB:0.35, BG:0.32, GB:0.32, GG:0.01} And your new belief in each of these realities reflects that like so (renormalized) {BB: 0.35, BG:0.32, GB:0.32, 0.01} So now I feel even more confident that he has two boys.
- iliketosleep 16y ago
  Thanks for your explanation. I think his first renormalization process is wrong. Because once we know there is a boy, the problem space is reduced do "What's the probability of a boy?" which is 1/2. It has nothing to do with probabilities involving the known child.

signa11 16y ago

since we are talking about probability-theory, thought folks here might find gnedenko pretty interesting: [ http://www-history.mcs.st-andrews.ac.uk/Biographies/Gnedenko.html http://www-history.mcs.st-andrews.ac.uk/Biographies/Gnedenko... ]

jmtulloss 16y ago

If you're lucky enough to attend UIUC, I highly recommend taking ECE 413 to get a thorough introduction to these concepts. It's unfortunate that none of the class materials are online since it goes well above and beyond what is taught in most undergraduate CS courses on statistics. Taking it was hard, but it made me a much better engineer. Edit: I suppose this applies to anybody in college. Take the hard statistics course that goes over this stuff. It's really valuable, and pretty hard to pick up on your own.

adolph 16y ago

This essay by Yudkowsky is also helpful. http://yudkowsky.net/rational/bayes http://yudkowsky.net/rational/bayes

rmathew 16y ago

A more thorough introduction to this topic is "Probability Theory: The Logic of Science" by E. T. Jaynes (http://www-biba.inrialpes.fr/Jaynes/prob.html http://www-biba.inrialpes.fr/Jaynes/prob.html).

AmberShah 16y ago

"Nobody"? That's not likely...

drakep 16y ago

Never use absolutes?