## Thursday, February 23, 2017

### Flatness of priors

I. J. Good is said to have said that we can know someone’s priors by their posteriors. Suppose that Alice has the following disposition with respect to the measurement of an unknown quantity X: For some finite bound ϵ and finite interval [a, b], whenever Alice would learn that:

1. The value of X + F is x where x is in [a, b], where
2. F is a symmetric error independent of the actual value of X and certain to be no greater than ϵ according to her priors, and
3. the interval [x − ϵ, x + ϵ] is a subset of [a, b]

then Alice’s posterior epistemically expected value for X would be x.

Call this The Disposition. Many people seem to have The Disposition for some values of ϵ, a and b. For instance, suppose that you’re like Cavendish and you’re measuring the gravitational constant G. Then within some reasonable range of values, if your measurement gives you G plus some independent symmetric error F, your epistemically expected value for G will be probably be equal to the number you measure.

Fact. If Alice is a Bayesian agent who has The Disposition and X is measurable with respect to her priors, then Alice’s priors for X conditional on X being in [a, b] are uniform over [a, b].

So, by Good’s maxim about priors, someone like the Cavendish-like figure has a uniform distribution for the gravitational constant within some reasonable interval (there is a lower bound of zero for G, and an upper bound provided by the fact that even before the experiment we know that we don’t experience strong gravitational attraction to other people).