… in order to triangulate closer to whether Platonism is “hard-wired”, do you find it possible to be non-Platonic about mathematical truths? Can someone who is non-Platonic think about them Platonically—is it a choice?
Most of the time I think about math, I do not worry about if it is platonic or not. It was really only in the context of considering my epistemic uncertainty that 2+2=4 that I needed consider the nature of the territory I was mapping, and in this context it did not make sense for the territory to be the physical universe.
In contrast, it is more natural to me to think of the “fundamental truth” as being what the probability of heads is, as a property of the coin and the flip, since the outcome isn’t determined yet.
You mean, the outcome has not been determined by you, since you have not observed all the physical properties of coin, the person flipping it, and the environment, and calculated out all the physics that would tell you whether it would land heads or tails. Attaching a probability to the coin is just our way of dealing with the ignorance and lack of computing power that prevents us from finding the exact answer.
What is your point? You iterate the Bayesian perspective, but do you agree that frequentists and Bayesians have different perspectives about this?
I think it boils down to this: you are a frequentist (and I’ve been using the term Platonist) if you see the 50% probability as a property of the coin and the flip, and you are a Bayesian if you see the 50% probability as just a way of measuring the uncertainty.
(Given your rationale for being Platonic about mathematics, I don’t know if you are really a Platonist (in the hard-wired sense).)
My point is that the view that 50% probability is a fundamental property of the coin is wrong. It is an example of the Mind Projection Fallacy, thinking that because you don’t know the result, somehow the universe doesn’t either. It is certainly not the case that when asked about the result of a single coin flip, that giving a 50% probability for heads is the best possible answer. One could, in principle, do more investigation, and find that under the current conditions, the coin will come up heads (or tails) with 99% probability, and actually be right 99 times out of a hundred.
I don’t like to call this view of the probability as a fundamental property of the coin the frequentist view. It makes more sense to describe their perspective as a the probability being a combined property of the coin and a distribution of conditions in which it could be flipped. From this perspective, the mistake of attaching the probability to the coin is that miss the fact that you are flipping the coin in one particular condition, which will have a definite outcome. The probability comes from uncertainty of which condition from the distribution applies in this case, and of course, limits on computational power.
Are you saying that frequentists are wrong, or just me?
If the former, how can you say that and consider the case closed when frequentists arrive at correct conclusions? What I’m suggesting is that Bayesians are committing the mind projection fallacy when they assert that frequentists are “wrong”.
I am saying that you are wrong, and I am not sure there isn’t more to the frequentist view than you are saying, so I am not prepared to figure out if it is right or wrong until I know more about what it is saying.
If the former, how can you say that and consider the case closed when frequentists arrive at correct conclusions?
Like in the Monty Hall problem, where the frequentists will agree to the correct answer after you beat them over the head with a computer simulation?
What I’m suggesting is that Bayesians are committing the mind projection fallacy when they assert that frequentists are “wrong”.
Huh? What property of our minds do you think we are projecting onto the territory?
In the Monty Hall problem, intuiton tends to insist on the wrong answer, not valid application of frequentist theory.
Just curious—is the monty hall solution intuitively obvious to a “Bayesian”, or do they also need to work through the (Bayesian) math in order to be convinced?
Huh? What property of our minds do you think we are projecting onto the territory?
Just curious—is the monty hall solution intuitively obvious to a “Bayesian”, or do they also need to work through the (Bayesian) math in order to be convinced?
For me at least, it is not so much that the solution is intuitively obvious as that setting up the Bayesian math forces me to ask the important questions.
I meant the typical mind fallacy.
Then how do you think we are assuming that others think like us? It seems to me that we notice that others are not thinking like us, and that in this case, the different thinking is an error. I believe that 2+2=4, and if I said that someone was wrong for claiming that 2+2=3, that would not be a typical mind fallacy.
If the conclusions about reality were different, then the 2+2=4 verses 2+2=3 analogy would hold. Instead, you are objecting to the way frequentists approach the problem. (Sometimes, the difference seems to be as subtle as just the way they describe their approach.) Unless you show that they do not as consistently arrive at the correct answer, I think that objecting to their methods is the typical mind fallacy.
Asserting that frequentists are wrong is actually very non-Bayesian, because you have no evidence that the frequentist view is illogical. Only your intuition and logic guides you here. So finally, as two rationalists, we may observe a bona fide difference in what we consider intuitive, natural or logical.
I’m curious about the frequency of “natural” Bayesians and frequentists in the population, and wonder about their co-evolution. I also wonder about their lack of mutual understanding.
You have a coin.
The coin is biased.
You don’t know which way it’s biased or how much it’s biased. Someone just told you, “The coin is biased” and that’s all they said.
This is all the information you have, and the only information you have.
You draw the coin forth, flip it, and slap it down.
Now—before you remove your hand and look at the result—are you willing to say that you assign a 0.5 probability to the coin having come up heads?
The frequentist says, “No. Saying ‘probability 0.5’ means that the coin has an inherent propensity to come up heads as often as tails, so that if we flipped the coin infinitely many times, the ratio of heads to tails would approach 1:1. But we know that the coin is biased, so it can have any probability of coming up heads except 0.5.”
The frequentists get this exactly wrong, ruling out the only the correct answer given their knowledge of the situation.
The article goes on to describe scenarios in which having different partial knowledge to the situation leads to different probabilities. The frequentist perspective doesn’t merely lead to the wrong answer for these scenarios, it fails to even produce a coherent analysis. Because there is no single probability attached to the event itself. The probability really is a property of the mind analyzing that event, to the extent that it is sensitive to the partial knowledge of that mind.
The competent frequentist would presumably not be befuddled by these supposed paradoxes. Since he would not be befuddled (or so I am fairly certain), the “paradoxes” fail to prove the superiority of the Bayesian approach.
Eliezer responded with:
Not the last two paradoxes, no. But the first case given, the biased coin whose bias is not known, is indeed a classic example of the difference between Bayesians and frequentists.
and in the post he wrote
The frequentist perspective doesn’t merely lead to the wrong answer for these scenarios, it fails to even produce a coherent analysis.
But the frequentist does have a coherent analysis for solving this problem. Because we’re not actually interested in the long-term probability of flipping heads (of which all anyone can say is that it is not .5) but the expected outcome of a single flip of a biased coin. This is an expected value calculation, and I’ll even apply your idea about events with symmetric alternatives. (So I do not have to make any assumptions about the shape of the distribution of possible biases.)
I will calculate my expected value using that the coin is biased towards heads or it is biased towards tails with equal probability. Let p be the probability that the coin flips to the biased orientation (i.e., p>.5).
The probability of heads is p with probability of 0.5. The probability of tails in this case is (1-p)*0.5.
The probability of heads is (1-p) with probability of 0.5. The probability of tails in this case is (p)*0.5.
Thus, the expected value of heads is p .5+(1-p) 0.5 = 0.5.
So there’s no befuddlement, only a change in random variables from the long-term expectation of the outcome of many flips to the long-term expectation of whether heads or tails is preferred and a single flip. Which we should expect, since the random variable we are really being asked about has changed with the different contexts.
You just pushed aside your notion of an objective probability and calculated a subjective probability reflecting your partial information. Congratulations, you are a Bayesian.
I applied completely orthodox frequentist probability.
I had predicted your objection would be that expected value is an application of Bayes’ theorem, but I was prepared to argue that orthodox probability does include Bayes’ theorem. It is one of the pillars of any introductory probability textbook.
A problem isn’t “Bayesian” or “frequentist”. The approach is. Frequentists take the priors as given assumptions. The assumptions are incorporated at the beginning as part of the context of the problem, and we know the objective solution depends upon (and is defined within) a given context. A Bayesian in contrast, has a different perspective and doesn’t require formalizing the priors as given assumptions. Apparently they are comfortable with asserting that the priors are “subjective”. As a frequentist, I would have to say that the problem is ill-posed (or under-determined) to the extent that the priors/assumptions are really subjective.
Suppose that I tell you I am going to pick up a card randomly and will ask you the probability of whether it is the ace of hearts. Your correct answer would be 1⁄52, even if I look at the card myself and know with probability 0 or 1 that the card is the ace of hearts. Frequentists have no problem with this “subjectivity”, they understand it as different probabilities for different contexts. This is mainly a response to this comment, but is relevant here.
Yet again, the misunderstanding has arisen because of not understanding what is meant by the probability is “in” the cards. In this way, Bayesian’s interpret the frequentist’s language too literally. But what does a frequentist actually mean? Just that the probability is objective? But the objectivity results from the preferred way of framing the problem … I’m willing to consider and have suggested the possibility that this “Platonic probability” is an artifact of a thought process that the frequentist experiences empirically (but mentally).
Most of the time I think about math, I do not worry about if it is platonic or not. It was really only in the context of considering my epistemic uncertainty that 2+2=4 that I needed consider the nature of the territory I was mapping, and in this context it did not make sense for the territory to be the physical universe.
You mean, the outcome has not been determined by you, since you have not observed all the physical properties of coin, the person flipping it, and the environment, and calculated out all the physics that would tell you whether it would land heads or tails. Attaching a probability to the coin is just our way of dealing with the ignorance and lack of computing power that prevents us from finding the exact answer.
What is your point? You iterate the Bayesian perspective, but do you agree that frequentists and Bayesians have different perspectives about this?
I think it boils down to this: you are a frequentist (and I’ve been using the term Platonist) if you see the 50% probability as a property of the coin and the flip, and you are a Bayesian if you see the 50% probability as just a way of measuring the uncertainty.
(Given your rationale for being Platonic about mathematics, I don’t know if you are really a Platonist (in the hard-wired sense).)
My point is that the view that 50% probability is a fundamental property of the coin is wrong. It is an example of the Mind Projection Fallacy, thinking that because you don’t know the result, somehow the universe doesn’t either. It is certainly not the case that when asked about the result of a single coin flip, that giving a 50% probability for heads is the best possible answer. One could, in principle, do more investigation, and find that under the current conditions, the coin will come up heads (or tails) with 99% probability, and actually be right 99 times out of a hundred.
I don’t like to call this view of the probability as a fundamental property of the coin the frequentist view. It makes more sense to describe their perspective as a the probability being a combined property of the coin and a distribution of conditions in which it could be flipped. From this perspective, the mistake of attaching the probability to the coin is that miss the fact that you are flipping the coin in one particular condition, which will have a definite outcome. The probability comes from uncertainty of which condition from the distribution applies in this case, and of course, limits on computational power.
Are you saying that frequentists are wrong, or just me?
If the former, how can you say that and consider the case closed when frequentists arrive at correct conclusions? What I’m suggesting is that Bayesians are committing the mind projection fallacy when they assert that frequentists are “wrong”.
I am saying that you are wrong, and I am not sure there isn’t more to the frequentist view than you are saying, so I am not prepared to figure out if it is right or wrong until I know more about what it is saying.
Like in the Monty Hall problem, where the frequentists will agree to the correct answer after you beat them over the head with a computer simulation?
Huh? What property of our minds do you think we are projecting onto the territory?
In the Monty Hall problem, intuiton tends to insist on the wrong answer, not valid application of frequentist theory.
Just curious—is the monty hall solution intuitively obvious to a “Bayesian”, or do they also need to work through the (Bayesian) math in order to be convinced?
Oops. I meant the typical mind fallacy.
For me at least, it is not so much that the solution is intuitively obvious as that setting up the Bayesian math forces me to ask the important questions.
Then how do you think we are assuming that others think like us? It seems to me that we notice that others are not thinking like us, and that in this case, the different thinking is an error. I believe that 2+2=4, and if I said that someone was wrong for claiming that 2+2=3, that would not be a typical mind fallacy.
If the conclusions about reality were different, then the 2+2=4 verses 2+2=3 analogy would hold. Instead, you are objecting to the way frequentists approach the problem. (Sometimes, the difference seems to be as subtle as just the way they describe their approach.) Unless you show that they do not as consistently arrive at the correct answer, I think that objecting to their methods is the typical mind fallacy.
Asserting that frequentists are wrong is actually very non-Bayesian, because you have no evidence that the frequentist view is illogical. Only your intuition and logic guides you here. So finally, as two rationalists, we may observe a bona fide difference in what we consider intuitive, natural or logical.
I’m curious about the frequency of “natural” Bayesians and frequentists in the population, and wonder about their co-evolution. I also wonder about their lack of mutual understanding.
From Probability is in the Mind:
The frequentists get this exactly wrong, ruling out the only the correct answer given their knowledge of the situation.
The article goes on to describe scenarios in which having different partial knowledge to the situation leads to different probabilities. The frequentist perspective doesn’t merely lead to the wrong answer for these scenarios, it fails to even produce a coherent analysis. Because there is no single probability attached to the event itself. The probability really is a property of the mind analyzing that event, to the extent that it is sensitive to the partial knowledge of that mind.
I like the response of Constant2:
Eliezer responded with:
and in the post he wrote
But the frequentist does have a coherent analysis for solving this problem. Because we’re not actually interested in the long-term probability of flipping heads (of which all anyone can say is that it is not .5) but the expected outcome of a single flip of a biased coin. This is an expected value calculation, and I’ll even apply your idea about events with symmetric alternatives. (So I do not have to make any assumptions about the shape of the distribution of possible biases.)
I will calculate my expected value using that the coin is biased towards heads or it is biased towards tails with equal probability. Let p be the probability that the coin flips to the biased orientation (i.e., p>.5).
The probability of heads is p with probability of 0.5. The probability of tails in this case is (1-p)*0.5.
The probability of heads is (1-p) with probability of 0.5. The probability of tails in this case is (p)*0.5.
Thus, the expected value of heads is p .5+(1-p) 0.5 = 0.5.
So there’s no befuddlement, only a change in random variables from the long-term expectation of the outcome of many flips to the long-term expectation of whether heads or tails is preferred and a single flip. Which we should expect, since the random variable we are really being asked about has changed with the different contexts.
You just pushed aside your notion of an objective probability and calculated a subjective probability reflecting your partial information. Congratulations, you are a Bayesian.
I applied completely orthodox frequentist probability.
I had predicted your objection would be that expected value is an application of Bayes’ theorem, but I was prepared to argue that orthodox probability does include Bayes’ theorem. It is one of the pillars of any introductory probability textbook.
A problem isn’t “Bayesian” or “frequentist”. The approach is. Frequentists take the priors as given assumptions. The assumptions are incorporated at the beginning as part of the context of the problem, and we know the objective solution depends upon (and is defined within) a given context. A Bayesian in contrast, has a different perspective and doesn’t require formalizing the priors as given assumptions. Apparently they are comfortable with asserting that the priors are “subjective”. As a frequentist, I would have to say that the problem is ill-posed (or under-determined) to the extent that the priors/assumptions are really subjective.
Suppose that I tell you I am going to pick up a card randomly and will ask you the probability of whether it is the ace of hearts. Your correct answer would be 1⁄52, even if I look at the card myself and know with probability 0 or 1 that the card is the ace of hearts. Frequentists have no problem with this “subjectivity”, they understand it as different probabilities for different contexts. This is mainly a response to this comment, but is relevant here.
Yet again, the misunderstanding has arisen because of not understanding what is meant by the probability is “in” the cards. In this way, Bayesian’s interpret the frequentist’s language too literally. But what does a frequentist actually mean? Just that the probability is objective? But the objectivity results from the preferred way of framing the problem … I’m willing to consider and have suggested the possibility that this “Platonic probability” is an artifact of a thought process that the frequentist experiences empirically (but mentally).