Bayesian Reflection Principles and Ignorance of the Future

post by crickets · 2024-01-25T19:00:16.463Z · LW · GW · No comments

This is a question post.

Contents

  Answers
    2 GuySrinivasan
None
No comments

Hi everyone,

New user here. I am learning Bayesian epistemology by working through Mike Titelbaum’s recent Fundamentals of Bayesian Epistemology. I’ve been learning about deference principles — principles concerning deferring to experts, where experts may include experts in the colloquial sense, the objective chances, and even one’s future self (provided one is sure that one’s opinions will update via conditionalization).

I have a question about the Reflection Principle. Roughly, Reflection tells us to adopt the credence now that we expect we would adopt at some future time (on the assumption that our opinions are updated by conditionalizing). More carefully, Reflection says:

Reflection: For any proposition A in L, real number x, and times ti and tj with j > i, rationality requires: cri(A | crj(A) = x) = x (Titelbaum, 143).

Obviously, I do not have perfect knowledge of what my future evidence will be, however, so I cannot be sure what my future credences will be, de re. (Of course, I could know them de dicto, as in: “my future credences will be whatever credences result from conditionalizing on the evidence I acquire.”)

Suppose that currently my evidence is E and cr2024(p) = 0.5. In 2050, I might have evidence E (i.e. my evidence won't have changed), E*, or E**. If In 2050, I have evidence E, then cr2050(p) = 0.5. But suppose that I instead 2050, I have evidence E*, and by conditionalizing on E* I have cr2050(p) = 0.3. Or suppose that in 2050, I have evidence E**, and by conditionalizing on E** I have cr2050(p) = 0.7. (Suppose these are the only three possibilities, just for the sake of ease.) 

I don't in 2024 know what evidence I'll have as to whether p in 2050. To what future self should my 2024 self defer, then? The one with E, E*, or E**? Is there a way to adjudicate this? The intuitive answer is that I should figure out which is likeliest: (i) that my future self has E, (ii) that my future self has E*, or (iii) that my future self has E**. I can then defer to the likeliest future self. But how can I determine this?

I apologize if this is a n00b question. Titelbaum does not discuss this issue in the book, and I'm not sure where to look. 

Thanks.

Answers

answer by SarahNibs (GuySrinivasan) · 2024-01-25T21:18:29.925Z · LW(p) · GW(p)

To what future self should my 2024 self defer, then? The one with E, E*, or E**?


To each with your current probability that that will be your future self. Take an expectation.

which is likeliest [...] defer to the likeliest

Any time you find yourself taking a point estimate and then doing further calculations with it, rather than multiplying out over all the possibilities, ask whether you should be doing the latter.

cr2024 = P2024(E) * 0.5 + P2024(E*) * 0.3 + P2024(E**) * 0.7

↑ comment by crickets · 2024-01-27T03:54:28.232Z · LW(p) · GW(p)

Hi! Thank you for the answer. Your answer is a little bit cryptic, though. Could you maybe provide some more detail or elaborate a bit on what you're saying?

Are you in essence saying something like "take the weighted average of the three possibilities and my probabilities that they'll occur" and use this to determine my expected future credence?

Replies from: GuySrinivasan
↑ comment by SarahNibs (GuySrinivasan) · 2024-01-27T03:59:21.665Z · LW(p) · GW(p)

Yes, exactly that.

No comments

Comments sorted by top scores.