From prior information to posterior inference
Brian J. Reich, Sujit Kumar Ghosh · 2019
This chapter discusses several general approaches for selecting prior distributions and deals with conjugate priors. Conjugate priors lead to simple expressions for the posterior distribution. The chapter illustrates how prior information affects the Bayesian analysis. Conjugate priors are the most convenient choice. A prior and likelihood pair are conjugate if the resulting posterior is a member of the same family of distributions as the prior. The beta prior is conjugate for both the binomial and negative binomial likelihood and both a gamma prior and a Bernoulli prior are conjugate for Poisson likelihood. A limitation of a conjugate prior is that restricting the prior to a parametric family limits how accurately prior uncertainty can be expressed. In many cases there is no prior information, leaving the analyst to select, say, conjugate uninformative priors in an ad hoc manner. In the absence of prior information, selecting the prior can be viewed as a nuisance to be avoided if possible.