Evidential Reasoning

We consider the problem of inference on the basis of observations. Suppose Alice has two coins, F (with probability 1/2 of landing heads) and B (with probability 3/4 of landing heads). She chooses a coin, tosses it, and it lands heads. How likely was the coin to be F? How likely was it to be B? Suppose she tosses the coin 100 times, and it lands heads every time. How likely was it to be F? How likely was it to be B? Note that in the second case, the likelihood of the coin being B has increased—we have gained some evidence for the coin being B. How can we quantify this evidence? There are various ways of quantifying evidence, the subject of confirmation theory. Idea: we have an experiment (tossing a coin) whose outcomes (landing heads, tails) provide evidenc for an hypothesis (F, B). Most approaches rely on likelihood functions. The relevant information can be captured using an evidence space E = (H,O, μ) where H is a finite set of hypotheses, O is a finite set of observations (or outcomes), and μ is a iikelihood vector, assigning to every hypothesis h a likelihood function μh which is just a probability distribution on O. (Basically, this is the probability of the observations when the true hypothesis is h.) The plan of this talk is to (I) review the classical framework, (II) generalize the framework to uncertain likelihoods, and (III) deal with complex experiments.