Watercressed

Foundations of Probability

How do we assign zero probability to 0=1 when we can't prove our logic consistent?

Open Thread for January 8 - 16 2014

Concern trolling in the false flag political operation sense is a thing that happened

An example of this occurred in 2006 when Tad Furtado, a staffer for then-Congressman Charles Bass (R-NH), was caught posing as a "concerned" supporter of Bass' opponent, Democrat Paul Hodes, on several liberal New Hampshire blogs, using the pseudonyms "IndieNH" or "IndyNH". "IndyNH" expressed concern that Democrats might just be wasting their time or money on Hodes, because Bass was unbeatable.[37][38] Hodes eventually won the election.

[LINK] Why I'm not on the Rationalist Masterlist

Does fundamentalist Christianity indicate that the believer would be irrational about issues other than religion?

If yes, what's the difference?

A proposed inefficiency in the Bitcoin markets

You could spend the tax-evaded income on the black market, since you're hiding contraband from the police anyway.

Doubt, Science, and Magical Creatures - a Child's Perspective

Since when did epistemic rationality demand making the truth common knowledge? It just means you should know what's true yourself.

The Statistician's Fallacy

Different information about part of nature is not sufficient to change an inference--the probabilities could be independent of the researcher's intentions.

Probability and radical uncertainty

It depends on your priors

A Limited But Better Than Nothing Way To Assign Probabilities to Statements of Logic, Arithmetic, etc.

One of his "desiderata", his principles of construction, is that the robot gives equal plausibility assignments to logically equivalent statements

I don't see this desiderata. The consistency requirement is that if there are multiple ways of calculating something, then all of them yield the same result. A few minutes of thought didn't lead to any way of leveraging a non 1 or zero probability for Prime(53) into two different results.

If I try to do anything with P(Prime(53)|PA), I get stuff like P(PA|Prime(53)), and I don't have any idea how to interpret that. Since PA is a set of axioms, it doesn't really have a truth value that we can do probability with. Technically speaking, Prime(N) means that the PA axioms imply that 53 has two factors. Since the axioms are in the predicate, any mechanism that forces P(Prime(53)) to be one must do so for all priors.

One final thing: Isn't it wrong to assign a probability of zero to Prime(4), i.e. PA implies that 4 has two factors, since PA could be inconsistent and imply everything?

A Limited But Better Than Nothing Way To Assign Probabilities to Statements of Logic, Arithmetic, etc.

You can skip this pararaph and the next if you're familiar with the problem. But if you're not, here's an illustration. Suppose your friend has some pennies that she would like to arrange into a rectangle, which of course is impossible if the number of pennies is prime. Let's call the number of pennies N. Your friend would like to use probability theory to guess whether it's worth trying; if there's a 50% chance that Prime(N), she won't bother trying to make the rectangle. You might imagine that if she counts them and finds that there's an odd number, this is evidence of Prime(N); if she furthermore notices that the digits don't sum to a multiple of three, this is further evidence of Prime(N). In general, each test of compositeness that she knows should, if it fails, raise the probability of Prime(N).

But what happens instead is this. Suppose you both count them, and find that N=53. Being a LessWrong reader, you of course recognize from recently posted articles that N=53 implies Prime(N), though she does not. But this means that P(N=53) <= P(Prime(N)). If you're quite sure of N=53—that is, P(N=53) is near 1—then P(Prime(N)) is also near 1. There's no way for her to get a gradient of uncertainty from simple tests of compositeness. The probability is just some number near 1.

I don't understand why this is a problem. You and your friend have different states of knowledge, so you assign different probabilities.

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