Wednesday, July 27, 2022

The accuracy argument for probabilism

A standard scoring rule argument for probabilism—the doctrine that credence assignments should satisfy the axioms of probability—goes as follows. If s is a scoring rule on a finite probability space Ω, so that s(c)(ω) is the epistemic utility of credence assignment c at ω in Ω, and (a) s is strictly proper and (b) s is continuous, then for any credence c that does not satisfy the axioms of probability, there is a credence p that does satisfy them such that s(p)(ω) is better than s(c)(ω) for all ω. This means that it’s stupid to have a non-probabilistic credence c, since you could instead replace it with p, and do better, no matter what.

Here is a problem with the dialectics behind this argument. Let P be the set of all credence assignments that satisfy the axioms of probability. But suppose that I think that there is some nonempty set M of credence assignments that do not satisfy the axioms of probability but are rationally just as good as those in P. Then I will think there is some way of making decisions using credences in M, just as good as the way of making decisions using credences in P. The best candidate in the literature for this is to use a level set integral, which allows one to assign an expected value EcU to any utility assignment U even if c is not a probability. Note that EpU is the standard mathematical expectation with respect to p if p is a probability.

The argument for probabilism assumed two things about the scoring rule: strict propriety and continuity. Strict propriety is the claim that:

  1. Eps(p) > Eps(c) whenever c is a credence other than p

for any probability p. In words, by the lights of a probability p, then we get the best expected epistemic utility if we make p be our credence.

Now, if I am not convinced by the argument that (1) should hold for any probability p and any credence c other than p, then I will be unmoved by the scoring rule argument for probabilism. So suppose that I am convinced. But recall that I think that credences in M are just as rationally good as the probabilities in P. Because of this, if I find (1) convincing for all probabilities p, I will also find it convincing for all credences p in M, where Ep is my preferred way of calculating expected utilities—say, a level set integral.

Thus, if I am convinced by the argument for strict propriety, I will just as much accept (1) for p in M as for p in P. But now we have:

Theorem 1. If Ep is strongly monotonic for all p ∈ P ∪ M and coincides with mathematical expectation for p ∈ P, and (1) holds for all p in P ∪ M, where M is non-empty, then s is not continuous on P.

(Strong monotonicity means that if U < V everywhere then EpU < EpV. The Theorem follows immediately from the Pettigrew-Nielsen-Pruss domination theorem.)

Suppose then that I am convinced that a scoring rule s should be continuous (either on P or on all of P ∪ M). Then the conclusion I am apt to draw is that there just is no scoring rule that satisfies all the desiderata I want: continuity as well as (1) holding for all p ∈ P ∪ M.

In other words, the only way the argument for probabilism will be convincing to me is if my reason to think (1) is true for all p in P is significantly stronger than my reason to think (1) is true for all p in M, and I have a sufficiently strong reason to think that there is a scoring rule that satisfies all the true rational desiderata on a scoring rule to conclude that (1) holding for all p in M is not among the true rational desiderata even though its holding for all p in P is.

And once I additionally learn about the difficulties in defining sensible scoring rules on infinite spaces, I will be less confident in thinking there is a scoring rule that satisfies all the true rational desiderata on a scoring rule.


Nagy Zsolt said...

Drake doesn't like: Philosophical mess of scoring rules and Dutch book arguments or approaches to probabilities.
Drake likes: The Principle of Maximum Entropy
"Principle of maximum entropy" from wiki
"The Principle of Maximum Entropy" by Mutual Information

Yeah. This is the REALLY good stuff.

Zsolt Nagy said...

"And once I additionally learn about the difficulties in defining sensible scoring rules on infinite spaces, I will be less confident in thinking there is a scoring rule that satisfies all the true rational desiderata on a scoring rule."

I used to be an adventurer like you and then I took an arrow to the knee.
- once I additionally learned about the principle of maximum entropy, I am much more confident in thinking, that there is actually a scoring rule, that satisfies all the true rational desiderata on scoring on a scoring rule, at least for some cases.
Just sayin.

Nagy Zsolt said...

A coin is tossed on sunday.
Question to sleeping beauty: "What is your credence, that the coin (tossed on sunday) came up heads?"
Answer from sleeping beauty: "Since no further relevant information is given about that coin, like the average of that coin toss with that coin being heads or tails, therefore my credence of that coin toss resulting in heads is 0.5 (or 50%) given the principle of maximum entropy as an appropriate scoring rule for such cases or this particular case."
We simply maximise the entropy S (-average uncertainty), which is appropriate for maximising our scoring rule s (- epistemic utility of credence assignment) here.
s(c)(ω):= S(c):=-k·(c·ln(c)+(1-c)·ln(1-c))
with k>0: some irrelevant positive konstante here,
c∈[0;1]: credence of the coin toss resulting in heads
(1-c)∈[0;1]: credence of the coin resulting not in heads.
⇒ ds/dc=dS/dc=-k·(ln(p)-ln(1-p))=0 for an extremum of s/S at c=p.
⇒ p=1/2=0.5=50%.
(⇒ d²s/(dc)²=d²S/(dc)²=-k·(1/c+1/(1-c))=-4k<0 with c=p=1/2 ⇒ maximum of s/S at c=p=1/2.)

I wonder: Why isn't this the standard solution to the sleeping beauty problem?
I just simply don't understand those Dutch book arguments for the credence better being supposedly 1/3.
I would simply just always simultaneously bet on that coin toss resulting in tails at the same time and get with that my money back. Or even gain money on average by simply never betting on heads in that particular case, since of course that coin toss results in 50% of the times in heads and in 50% of the times in tails and on average making more bets, when the coin toss has resulted in tails:
Simulation of the Sleeping Beauty Scenario and the Duch Book Bets

I guess, only philosophers can make out of such small problems with trivial solutions and a mosquito a big problem and an elephant.
Well, it’s your time to waste and not mine. But please at least try to spare others from non-sense.

Alexander R Pruss said...

Here's an argument for thirding:

Zsolt Nagy said...

P1) If any one is genuinely interested in truth and wisdom, then a particular some one is not just providing another argument for his or hers position, but also that person is giving a genuine response to a previously provided objection at the same time.
P2) You, Alexander, are not giving any genuine response to a previously provided objection, but you are as "a particular some one" providing only another argument for your proposition apparently and currently.
C) Therefore, it's not that any one is genuinely interested in truth and wisdom, but also you as a particular some one are not genuinely interested in truth and wisdom apparently and currently.
(Did I do the quantifier shift here correctly? I think, that I just did that here.)

It's the second time, that you, Alexander, didn't even bother to say a single word about my argument or view on the considered or suggested subject matter. (Yes, I count my antithesis to your so called "theorem" about there being no such infinite graphs with two ends as being the first incident or demonstration of that ignorance of yours.)
This ignorance and approach of yours to valid objections is peculiar given this suggestion and proposition for your colleague here: "The Dos and Donts of Objecting to Arguments (bonus video)" by (Dr.) Liz Jackson - suggesting a good objection to arguments being addressing a specific and particular premise of a previously provided argument and just giving another argument as an objection (,even when that "another" given argument/objection is a valid argument/objection).
Well, I'm more of "Socratic Method"-guy myself: "How to Argue - Induction and Abduction: Crash Course Philosophy #3" by CrashCourse.
Well, I guess, that I should be thankful and grateful, that those comments of mine are not deleted anymore and also the first amendment being not violated by such deletions. And yet I wonder, why is there such an ignorance of yours regarding legitimate objections or what is the specific reason for you to be so ignorant about such legitimate objections?
At least William Lane Craig has a somewhat of an excuse for being not "interested" in responding to objections of people's without some kind of legitimate degrees (even though that is not an essential condition or essential metric for the provided objections to be legitimate objections as most of the by William Lane Craig ignored objections given by layman's are such legitimate objections to his arguments or premises and assumptions).
There is also the alternitive typical response given by William Lane Craig, that his supposedly appropriate response to an objection has already been given by him in one of his books. "So simply look that there and there up." How convenient, isn't that?
Don't get me wrong here: Those are all bad responses, but it's still better than nothing or no responses - at least somewhat responding to something previously mentioned even though those kind of responses are no appropriate responses, but also those are not "not at all"/"totally ignorant" responses.
So what is your "reason"/"excuse" for not properly responding to a provided objection or counterargument, Alexander?

Alexander R Pruss said...

One can't respond to everything due to resource constraints.

Zsolt Nagy said...

But one can give some relevant little remarks here and there like:
"The Principle of Maximum Entropy might imply, that there is a way finding a strict optimal credence satisfying the axioms of probability plus satisfying of the criteria of maximizing the Entropy. But this doesn't imply the impossibility of there to be a credence of equal or better quality than given by the Principle of Maximum Entropy for some unknown specific and particular criteria or conditions."
This is a valid point stated in two sentences.

But then again I might response with:
"Yeah, but then again the Principle of Maximum Entropy appears to be derivable or is actually derived from some very general or reasonable Axioms:
"Axiomatic Derivation of the Principle of Maximum Entropy and the Principle of Minimum Cross-Entropy" by John E. Shore And Rodney w. Johnson (1980)
So I guess, if one wants to be consistent with his or hers credence as accurate probability assignments, then this is the way - the Principle of Maximum Entropy appears to be the correct, consistent and rational way here."

So yeah, why waste time on these conversations, when these have been already hold in the last century or so?
I also like to be efficient with my time and resources.