Normal Approximations to Likelihoods and to Posteriors
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 illustration of the posterior density are also presented. Noninformative and conjugate priors are discussed in Section 2.2. In Section 2.3, the method of maximum likelihood is defined and the Newton-Raphson algorithm is presented as an algorithm for locating the posterior/likelihood mode. Section 2.4 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 2.5 and Section 2.6 reviews the notion of the Highest Posterior Density region, a Bayesian approach to confidence regions and significance levels.
KeywordsPosterior Distribution Normal Approximation Posterior Density Item Response Model Likelihood Equation
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