[adamzerner]

I think a Bayesian perspective is helpful here. 1) How likely is it that Zuck makes that statement if he feels like he is responsible? 2) How likely is it if he doesn't feel responsible? I think the answers to those questions are quite similar, in which case hearing the statement doesn't actually tell us much.

[Enginerrrd]

Just to elaborate on this since I had to think about it and I might save someone else the effort:

Let M = Makes the statement Let R = Feels Responsible Notation: ~R = Not R

Answering OP's questions 1 & 2: I think it's safe to assume P(M|R) = P(M|~R) = 1 So P(M) = 1

So Bayes' theorem simplifies as follows: P(R|M) = P(M|R)P(R)/PM) = P(R)

Thus, OP's point is that the statement tells us essentially nothing about the probability he actually feels responsible or empathetic, and I agree.

[jsmith45]

To use Bayes to update here, you must determine the conditional probabilities as they were before you knew that M occurred, and could thus update to P(M)=1. If one did not already know that M happened then one certainly could not say `P(M|R) = P(M|~R) = 1`. One might be able to claim `P(M|R) = P(M|~R) = P(M)`, which is just saying the events are independent.

Certainly with a prior that the events are independent, then you won't be able to update your probability of R by knowing that M did happen, any more than knowing last nights lotto numbers would probability of R.

In reality, things are even worse, as assuming independence is not fully reasonable, so you will end up with uncertainly about how or if the variables relate. One could assume some form of meta probability distribution of various ways the variables could relate, but then direct application of Bayes formula not feasible. You would still in that scenario not be learning much if anything useful about P(R).

[abstractmath]

But both of your assumptions imply the conclusion, so the math doesn’t actually seem helpful at all.

[svnt]

You’ve identified the problem with many Bayesian approaches, not just this one. Without sufficient data to make the probabilities accurate, it just shifts the uncertainty to the process of choosing probabilities.

[adamzerner]

I disagree. The question at hand is how we should update our beliefs in response to the evidence of Zuck making the statement. Given the priors of P(M|R) and P(M|~R), it tells us that we shouldn't really update. Different priors would lead to a different update.

Sometimes this sort of thing happens where our priors don't allow for a belief update in response to evidence. For example, does me writing this comment change your best guess as to whether my favorite color is blue? That depends on what you think of P(favorite color blue | comment) and P(favorite color blue | ~comment). Both of those are probably the same right? If so, my comment doesn't allow you to update.

This excerpt from http://www.hpmor.com/chapter/20 is relevant:

  Professor Quirrell looked at Harry. "Mr. Potter," he said solemnly, with only a slight grin, "a word of advice. There is such a thing as a performance which is too perfect. Real people who have just been beaten and humiliated for fifteen minutes do not stand up and graciously forgive their enemies. It is the sort of thing you do when you're trying to convince everyone you're not Dark, not -"

  "I can't believe this! You can't have every possible observation confirm your theory! "

  "And that was a trifle too much indignation."

  "What on Earth do I have to do to convince you? "

  "To convince me that you harbor no ambitions of becoming a Dark Lord?" said Professor Quirrell, now looking outright amused. "I suppose you could just raise your right hand."

  "What?" Harry said blankly. "But I can raise my right hand whether or not I -" Harry stopped, feeling rather stupid.

  "Indeed," said Professor Quirrell. "You can just as easily do it either way. There is nothing you can do to convince me because I would know that was exactly what you were trying to do. And if we are to be even more precise, then while I suppose it is barely possible that perfectly good people exist even though I have never met one, it is nonetheless improbable that someone would be beaten for fifteen minutes and then stand up and feel a great surge of kindly forgiveness for his attackers. On the other hand it is less improbable that a young child would imagine this as the role to play in order to convince his teacher and classmates that he is not the next Dark Lord. The import of an act lies not in what that act resembles on the surface, Mr. Potter, but in the states of mind which make that act more or less probable."

  Harry blinked. He'd just had the dichotomy between the representativeness heuristic and the Bayesian definition of evidence explained to him by a wizard.

[abstractmath]

The interesting and important question is what are the priors.

Since all of the priors here are speculative, there’s no use in pulling out Bayes.

The outcome given Bayes is the same as the outcome without Bayes, just a bit noisier ;)

[adamzerner]

Yes, thank you! I appreciate that. I should have elaborated in my OP.

https://arbital.com/p/bayes_rule/ has some great follow up info.