ABSTRACT
The Bayesian approach underlies some of the earliest applications of statistical models and probability. The formalization of prior beliefs, typically in regard to a specific set of population characteristics, is a formal component of designing an experiment from the Bayesian perspective. While Markov Chain Monte Carlo based integration has widened the set of operational priors and posterior densities available for analyses, it is useful to note several standard densities that arise in the Normal sample case from a Bayesian perspective. The frequentist and Bayesian approaches differ in approach and interpretation for the testing of specific parameter value hypotheses. The normal approximation for the posterior distribution, sometimes referred to as the Bayesian Central Limit theorem, is related to a version of the frequentist central limit theorem. The Laplace approximation is another approach to estimation based on higher order Taylor expansions.
