## Friday, February 12, 2016

### The Principle of Sufficient Reason and Probability

I just posted the paper here. Forthcoming in Oxford Studies in Metaphysics.

Abstract: I shall argue that considerations about frequency-to-chance inferences make very plausible some a localized version of the Principle of Sufficient Reason (PSR). But a localized version isn’t enough, and so we should accept a full PSR.

IanS said...

I’m a bit worried by the application of Bayes to nonmeasurable events. Applying Bayes over an interval could make sense (with caveats) where the interval relates to a definite but unknown probability. (For example, under one hypothesis, a coin might have a bias that lies in some specified interval.) This is not the case for nonmeasurable events: there is indeed an interval, but there is no definite probability.

Alexander R Pruss said...

I'm thinking that Bayes works for interval-valued probabilities as follows: we look at the family of all the prior probability functions that fit with the intervals, apply Bayes with each one, and then take the smallest interval containing all the posteriors.

IanS said...

Yes, you can plug a range of numbers into Bayes. But what does the calculated interval mean?

Strictly, there is no posterior probability. It seems reasonable that you should not adopt a posterior credence outside the calculated interval. It’s not so clear what can be said about posterior credences inside the calculated interval.

An example. You have two N-sided dice, each with faces marked 1 to N. They look the same. You know that one is fair but you know nothing at all about the other. You model it by a complete range of probabilities for each side, subject only to the constraint that each set of probabilities adds up to 1. You choose one die on a fair coin toss. What is the probability that it is the fair one? 1/2, of course. You roll it and note the outcome. What should be your new credence that is it the fair one? Applying Bayes as you have suggested gives the range [1/(N+1), 1]. This is the “Bayesian divergence” of your paper. But surely the setup is symmetric and the outcome, whatever it was, gave you no useful information. So shouldn’t you stick with 1/2?.