Hierarchical Models: Random and Fixed Effects

The article describes probability models that are in the form of a hierarchy; the basic quantities in a situation having their probabilities described in terms of parameters, which themselves are described in terms of other parameters, and so on. This is an arrangement which provides great flexibility in probability modeling. Distinction is made between random effects, to which are attached probabilities, and fixed effects, which have probabilities only in the Bayesian paradigm. Examples of hierarchical models are given, together with illustrations of calculations needed for their resolution. Alternative descriptions in terms of equations are described.