Some Formulas for Use with Bayesian Ability Estimates

Abstract

Relationships between Bayesian ability estimates and the parameters of a normal population distribution are derived in the context of classical test theory. Analogues are provided for use as approximations in work with item response theory. The following questions were addressed:

What is the relationship between the distribution of the latent ability variable in a population, and the distribution of ability estimates?

Because calculating Bayesian estimates typically requires knowing the population distribution, how should one proceed if it is not known?

What if Bayesian ability estimates have been calculated in accordance with a common population distribution, but it is later desired to estimate the distributions of specified subpopulations?

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