Posterior Distribution

The notion of a posterior distribution comes from Bayesian statistics. Under the Bayesian approach, prior beliefs about parameters are combined with sample information to create updated, or posterior beliefs about the parameters. In the case of empirical Bayes estimators, the prior information comes from the sample data as well.

Posterior distributions have found a variety of applications. A couple simple examples include:

  • In student assessment student scores are often based on the posterior score distribution for the examinee. In this case, the prior distribution is often taken as the observed distribution of scores for the full sample of students, or some subset of the sample of which the individual student is a member. The sample information is given by the likelihood of the responses to test items;
  • In small area estimation estimates for a small geographic area from a larger survey are often based on posterior distributions. In these cases, estimates from the whole population, or a relevant subpopulation are taken as the prior, and the (often limited) sampled cases from the target small area provide the sampled information.

The posterior information is proportional to the product of the prior information and the sample information.

Figure 1 provides an heuristic example of how posterior distributions are formed in the case of student assessments. In that figure we present an hypothetical student with an easy form and a difficult form of the same test. The three Panels (A, B, and C) reflect the prior distribution (A) (which is the same for both tests), the measurement likelihood that constitutes the sample information (B), and posterior (C) distributions. The top row of graphs contains graphs based on the harder test, and the bottom row, the easier test. Panel A depicts the prior normal (population) distribution, which is, of course, the same regardless of the test administered. Panel B contains the measurement likelihood for our hypothetical student given the difficult (top row) and easier (bottom row) tests (both normalized to integrate to one). Notice that the likelihood given the difficult test only provides information that the student's ability is below the test's ability to measure. Taking the product of the prior and measurement distributions yields the posterior distribution.

Notice that the distributions change depending on the mix of items on the test--the prior distribution is less influential when there is more information from the sample. Figure 2 overlays the two curves to facilitate comparison.

This illustration is heuristic--it presents only one possible pair of response patterns from this examinee. A more accurate representation would "average over" all possible response patterns, given their likelihood from this examinee. However, the simple example makes the relevant point.

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