Normal Approximations to Likelihoods and to Posteriors

Abstract

In this chapter, we review several elementary concepts and methods from mathematical statistics. In Section 2.1, the likelihood and loglikelihood functions are defined and illustrated. The definition and an illustration of the posterior density is also presented. In Section 2.2, the method of maximum likelihood is defined and the Newton-Raphson algorithm is presented as an algorithm for computing maximum likelihood estimates. Section 2.3 presents frequentist and Bayesian justification for using the normal approximation to the likelihood or to the posterior as the basis for inference. The delta method is defined and illustrated in Section 2.4 and Section 2.5 reviews the notion of the Highest Posterior Density region, a Bayesian approach to confidence regions and significance levels.