Conditional Degree of Belief

The normative force of Bayesian inference is based on the existence of a uniquely rational degree of belief in evidence E given a hypothesis H, that is, p(E|H). To motivate this claim, think of what a Bayesian inference in science typically looks like. There is a competing set of hypotheses that correspond to different values of an unknown parameter, such as the chance θ ∈ [0, 1] of a coin to come up heads. Each of these hypotheses specifies the probability of a certain observation E (e.g., two heads in three tosses). This happens by means of a mathematical function, the probability density ρH(E). For example, in the case of tossing a coin N times, the Binomial distribution describes the probability of observing k heads and N − k tails if it is assumed that the tosses are independent and identically distributed (henceforth, i.i.d.). Assuming that the probability of heads on any particular throw is H : θ = 3/5, this would amount to ρH(E) = (k )(3/5) k(2/5)N−k. Based on these numbers, a Bayesian reasoner is interested in the posterior degree of belief in the hypothesis that θ equals a certain real number, or that θ falls within an interval [a, b]. Agreement on p(E|H) is required for a variety of reasons. First, the Bayesian’s primary measure of evidence is the Bayes factor (Kass and Raftery, 1995). It quantifies the evidence for a hypothesis H0 over a competitor H1 as the ratio of the probabilities p(E|H0)/p(E|H1)—or equivalently, as the ratio of posterior to prior odds. When these degrees of belief may rationally vary, it follows that one may not be able to agree on ∗Contact information: Tilburg Center for Logic, Ethics and Philosophy of Science (TiLPS), Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands. Email: [email protected]. Webpage: www.laeuferpaar.de