Definition of Term: Weak prior distribution

In Bayesian statistics, prior knowledge about a parameter is specified in the form of a probability distribution called its prior distribution (or simply its prior). The use of prior information is a feature of Bayesian statistics, and one which is contentious in the field of statistical inference. Most opponents of the Bayesian approach disagree with the use of prior information. In this context, there has been considerable study of the notion of a prior distribution that represents prior ignorance, or at least a state of very weak prior information, since by using such a prior it may be possible to evade this particular criticism of the Bayesian approach. In this toolkit, we will call such prior distributions “weak priors,” although they may be found answering to many other names in the literature (such as reference priors, noninformative priors, default priors or objective priors).

The use of weak priors has itself been criticised on various grounds. Strict adherents of the Bayesian view argue that genuine prior information invariably exists (i.e. a state of prior ignorance is unrealistic) and that to deny prior information is wasteful. On more pragmatic grounds, it is clear that despite all the research into weak priors there is nothing like a consensus on what is the weak prior to use in any situation, and there are numerous competing theories leading to alternative weak prior formulations. On theoretical grounds, others point to logical inconsistencies in any systematic use of weak priors.

In the MUCM toolkit, we adopt a Bayesian approach (or a variant known as the Bayes Linear approach) and so take the view that prior information should be used. Nevertheless we see a pragmatic value in weak priors. When prior information is genuinely weak relative to the information that may be gained from the data in a statistical analysis, it can be shown that the precise choice of prior distribution has little effect on the statistical results (inferences). Then the use of a weak prior can be justified as being an adequate replacement for spending unnecessary effort on specifying the prior. (And in this context, the existence of alternative weak prior formulations is not a problem because it should not matter which we use.)

Note that conventional weak priors are often improper.