Introduction to Imprecise Probabilities

7.4 Generalized Bayesian inference

This section discusses the application of the general framework of coherent lower previsions (as established in Chapter 2) for generalizing Bayesian inference to imprecise probabilities.³³ We confine presentation on approaches with imprecise prior distributions but still precise sampling models. First, we briefly recall some fundamental results from traditional Bayesian statistics needed later on. After introducing the general Bayesian setting, we then focus on the prototypical important case of generalized conjugate inference in exponential families. There we consider first applications to formulate near-ignorance priors, and secondly the situation of informative priors including their imprecise probability based opportunities in the context of prior-data-conflict. An outlook on some further aspects concludes this section.

7.4.1 Some selected results from traditional Bayesian statistics

7.4.1.1 Conjugate families of distributions

Traditional Bayesian inference is frequently based on so-called conjugate priors³⁴ (related to a specific likelihood). Such priors have the convenient property that the posterior resulting from (7.2) belongs to the same class of parametric distributions as the prior, and thus only the parameters have to be updated, which makes calculation of the posterior and thus the whole Bayesian inference easily tractable.

Fortunately, there are general results guiding the construction of conjugate priors for models used most ...