Prior Distribution

Bayesian Modeling Stages

To facilitate discussion of prior distributions, or priors, some Bayesian terms must first be introduced. There are three distinct modeling stages in the Bayesian approach, the first of which is specification of a model for the observed data, termed the likelihood, which represents the statistical relationship between the parameter and the observed data. Specification of prior information is the next stage. In the final stage, after the data are observed, prior information on the parameters is combined with the likelihood model to provide a distribution of parameter information. This combination of the likelihood and priors comes via Bayes’s theorem, often operationalized via Markov chain Monte Carlo methods. Because the combination occurs after the data are observed, the distribution of parameter values is known as the posterior parameter distribution. Posterior distributions specify the probability that each parameter equals a particular value or lies in a certain range of values.

[Page 1298]The posterior is determined by the amount of information contained in both the likelihood and the prior. The process of constructing a posterior distribution is a blending between the prior and the likelihood. That is, the prior acts as a weight for the likelihood in the formation of the posterior. In general, if the prior information is weak, then the posterior will be relatively unaffected by the form of the prior because the prior carries little weight in the blending process with the likelihood. Similarly, if the prior information is strong, then the posterior will be significantly affected by the form of the prior because the prior carries considerable weight in the posterior’s formation.

Prior Distributions

Priors define a probabilistic model for the parameters, and a researcher has several options for incorporating the prior information into the Bayesian modeling process. Each option has varying degrees of influence, or weight, on the formation of the posterior. What follows is an overview of prior distribution features.

Priors can be strong and narrowly focused. Conversely, they could be weak, reflecting a less focused range of inference, but still with some informative qualities. Typically, the strength of a prior distribution is controlled by the prior distribution’s variance, termed prior precision or informativeness. Smaller prior variances demonstrate more precision. For instance, a normal (0, var = 1) prior distribution would be considered more precise and informative than a normal (0,var = 100) prior distribution because the former has a smaller variance than the latter.

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