Bayesian Inference and Diagnostics for the Three Parameter Logistic Model.

Allan Birnbaum made two historical contributions to the theories of statistics and educational testing. The first was his famous 1962 proof that the sufficiency and conditionality principles together imply the likelihood principle, thus justifying conditional, e.g., Bayesian, inference when compared with frequentist inference. The second was his famous mental test model introduced in Lord and Novick (1968). This proposal attempts to follow in Birnbaum's tradition by using Bayesian ideas to show that his mental test model possesses even broader applicability than hitherto realized. A conditional maximization procedure is used to approximate numerical integrations thus broadening the scope of Bayesian inference and model checking. Additional keywords: Exchangeability, Computer programs, Normal distribution.