Bayesian inference

• Inductive inference according to Bayesian statistics is the process of updating the degree of certainty of a hypothesis on the basis of observed evidence.
• Bayes’ theorem allows us to calculate the posterior probabilities of hypotheses, P(H|E), from their likelihood, P(E|H), and prior probability, P(H).

Bayesian Statistics as Inductive Logic

• Howson and Urbach (2006): the Bayesian probability calculus provides a rule for inductive inferences, or a sort of inductive logic.

Bayesian Statistics as Internalist Epistemology

• The difficulty with seeing Bayesian statistics as inductive logic:
• The problem of justification
• Bayesian statistics as internalist justification

Problems with Internalist Epistemology

• Although today philosophers no longer require that justification must endow beliefs with absolute certainty, still, the concept of justification is expected to be truth-conducive. But is the internalist conception of justification truth-conducive?
• The regress problem: A coherent assignment of probabilities to beliefs is not enough to vindicate the correctness of the posterior probability, unless we have sufficient reason to believe in the legitimacy of the premises. So the problem is how the two major premises of Bayesian inference, prior probability and likelihood, are justified.

Justification of priors: washing out

• The importance of priors: base rate fallacy illustrates the need for an appropriate prior distribution in justifying the conclusion of Bayesian inference.
• But if Bayesian inferences depend on subjective priors, how can they reach objective conclusions?
• Washing out of priors with data: As the data accumulate, the posterior probabilities ($a_n$ vs. $b_n$) approach each other, even if they have very different prior distributions ($a_0$ vs. $b_0$).

Justification of priors: non-Informative Priors

• In practice, we never have an infinite amount of data and must choose an appropriate prior distribution as the starting point.
• The chosen prior is then expected to serve as a base premise, or foundation, for sustaining the subsequent updating process (epistemological foundationalism in philosophy).
  • How are these priors justified? A priori vs. a posteriori strategy.
• The principle of indifference is the most standard a priori strategy for justifying prior distributions without resorting to experience.