Bayesian Probability and Inference

Abstract

Bayes’ theorem is presented with its applications. Bayesian approach to probability is then introduced, with a discussion of possible applications. Prior and posterior probabilities are defined. The application to the continuous case is presented, and Bayesian inference is introduced, which can also be interpreted as learning process from multiple observations. The treatment of nuisance parameters with Bayesian inference is discussed. Credible intervals are defined, which determine uncertainty intervals with Bayesian inference. The assessment of different hypothesis with Bayes factors is introduced. The main critical issue with Bayesian inference is the dependence of the estimate result on assumed prior probability, which makes Bayesian probability intrinsically subjective. Jeffrey’s priors are introduced, which are invariant under variable transformation. The general approach to variable transformation and error propagation under the Bayesian approach is discussed.