= 27d567bc2d48d9f40e9ccc20ea370b15)
komponisto comments on Take heed, for it is a trap - Less Wrong
You are viewing a comment permalink. View the original post to see all comments and the full post content.
= 783df68a0f980790206b9ea87794c5b6)
Loading…= b715d02e85f7b3b4ae65ca3d3e450868)
Subscribe to RSS Feed
Powered by Reddit
= f037147d6e6c911a85753b9abdedda8d)
You are viewing a comment permalink. View the original post to see all comments and the full post content.
Comments (187)
Let n be an integer. Knowing nothing else about n, would you assign 50% probability to n being odd? To n being positive? To n being greater than 3? You see how fast you get into trouble.
You need a prior distribution on n. Without a prior, these probabilities are not 50%. They are undefined.
The particular mathematical problem is that you can't define a uniform distribution over an unbounded domain. This doesn't apply to the biased coin: in that case, you know the bias is somewhere between 0 and 1, and for every distribution that favors heads, there's one that favors tails, so you can actually perform the integration.
Finally, on an empirical level, it seems like there are more false n-bit statements than true n-bit statements. Like, if you took the first N Godel numbers, I'd expect more falsehoods than truths. Similarly for statements like "Obama is the 44th president": so many ways to go wrong, just a few ways to go right.
Edit: that last paragraph isn't right. For every true proposition, there's a false one of equal complexity.
Who said anything about not having a prior distribution? "Let n be a [randomly selected] integer" isn't even a meaningful statement without one!
What gave you the impression that I thought probabilities could be assigned to non-hypotheses?
This is irrelevant: once you have made an observation like this, you are no longer in a state of total ignorance.
We agree that we can't assign a probability to a property of a number without a prior distribution. And yet it seems like you're saying that it is nonetheless correct to assign a probability of truth to a statement without a prior distribution, and that the probability is 50% true, 50% false.
Doesn't the second statement follow from the first? Something like this:
Integers and statements are isomorphic. If you're saying that you can assign a probability to a statement without knowing anything about the statement, then you're saying that you can assign a probability to a property of a number without knowing anything about the number.
That is not what I claim. I take it for granted that all probability statements require a prior distribution. What I claim is that if the prior probability of a hypothesis evaluates to something other than 50%, then the prior distribution cannot be said to represent "total ignorance" of whether the hypothesis is true.
This is only important at the meta-level, where one is regarding the probability function as a variable -- such as in the context of modeling logical uncertainty, for example. It allows one to regard "calculating the prior probability" as a special case of "updating on evidence".
I think I see what you're saying. You're saying that if you do the math out, Pr(S) comes out to 0.5, just like 0! = 1 or a^0 = 1, even though the situation is rare where you'd actually want to calculate those things (permutations of zero elements or the empty product, respectively). Do I understand you, at least?
I expect Pr(S) to come out to be undefined, but I'll work through it and see. Anyway, I'm not getting any karma for these comments, so I guess nobody wants to see them. I won't fill the channel with any more noise.
[ replied to the wrong person ]