Bayesian Inference

Abstract

Following Carnap — part way — we shall construe the outcome of a statistical inference as the assignment of a probability to a statement. Sometimes this is done wholesale — we arrive at a distribution yielding probabilities for a whole family of statements, as when we infer that the length of a table is three feet one inch, with a standard deviation of half an inch. This does not mean, of course, that we suppose that the length of the table has a standard deviation of half an inch, nor, indeed, that the length of the table is somehow indeterminate. (On an atomic level, it may well be that the length of the table is indeterminate in some sense, or that this is one way of interpreting the formalism of atomic theory. This has nothing to do with our measurement of the table. If we restricted ourselves to length predicates corresponding to even multiples of one millimeter, our statistical statement about the length of the table would still — almost—hold.) Taken in conjunction with (a locution that should be taken quite literally) the conventional theory of measurement, the statement about the length of the table provides us with a shorthand for an infinite number of probability statements about the length of the table: the probability is 0.63 that it lies between $3'1{\frac{1}{2}^{\prime \prime }}$ and $3'{\frac{1}{2}^{\prime \prime }}$, and so on.