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How to interpret subjective Bayesians who want to be hard-nosed Bayesians is often like swimming round and round in a funnel of currents where there’s nothing to hold on to. Well, I think I’ve recently stopped the flow and pegged it. Christian Hennig and I have often discussed this (on my regular blog) and something Gelman posted today, linked me to an earlier exchange between he and Christian.
Christian: I came across an exchange between you and Andrew because it was linked to by Andrew on a current blog post
It really brings out the confusion I have had, we both have had, and which I am writing about right now (in my book), as to what people like Gelman mean when they talk about posterior probabilities. First:
a posterior of .9 to
H: “θ is positive”
is identified with giving 9 to 1 odds on H.
Gelman had said: “it seems absurd to assign a 90% belief to the conclusion. I am not prepared to offer 9 to 1 odds on the basis of a pattern someone happened to see that could plausibly have occurred by chance”
Then Christian says, this would be to suggest “I don’t believe it” means “it doesn’t agree with my subjective probability” and Christian doubts Andrew could mean that. But I say he does mean that. His posterior probability is his subjective (however evidence-based) probability. ‘
Next the question is, what’s the probability assigned to? I think it is assigned to H:θ > 0
As for the meaning of “this event would occur 90% of the time in the long run under repeated trials” I’m guessing that “this event” is also H. The repeated “trials” allude to a repeated θ generating mechanism, or over different systems each with a θ. The outputs would be claims of form H (or not-H or different assertions about the θ for the case at hand ), and he’s saying 90% of the time the outputs would be H, or H would be the case. The outputs are not ordinary test results, but states of affairs, namely θ > 0.
Bottom line: It seems to me that all Bayesians all who assign posteriors to parameters (aside from empirical Bayesians) really mean the kind of odds statement that you and I and most people associate with partial -belief or subjective probability. “Epistemic probability” would do as well, but equivocal. It doesn’t matter how terrifically objectively warranted that subjective probability assignment is, we’re talking meaning. And when one finally realizes this is what they meant all along, everything they say is less baffling. What do you think?
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Background
Andrew Gelman:
First off, a claimed 90% probability that θ>0 seems too strong. Given that the p-value (adjusted for multiple comparisons) was only 0.2—that is, a result that strong would occur a full 20% of the time just by chance alone, even with no true difference—it seems absurd to assign a 90% belief to the conclusion. I am not prepared to offer 9 to 1 odds on the basis of a pattern someone happened to see that could plausibly have occurred by chance,
Christian Hennig says:
“Then the data under discussion (with a two-sided p-value of 0.2), combined with a uniform prior on θ, yields a 90% posterior probability that θ is positive. Do I believe this? No.”
What exactly would it mean to “believe” this? Are you referring to a “true unknown” posterior probability with which you compare the computed one? How would the “true” one be defined?
Later there’s this:
“I am not prepared to offer 9 to 1 odds on the basis of a pattern someone happened to see that could plausibly have occurred by chance, …”
…which kind of suggests that “I don’t believe it” means “it doesn’t agree with my subjective probability” – but knowing you a bit I’m pretty sure that’s not what you meant before. But what is it then?
Rejected Posts of D. Mayo 