Generalized Bayesian inference with sets of conjugate priors for dealing with prior-data conflict: course at Lund University

In the Bayesian approach to statistical inference, possibly subjective knowledge on model parameters can be expressed by so-called prior distributions. A prior distribution is updated, via Bayes’ Rule, to the so-called posterior distribution, which combines prior information and information from data into a ‘complete picture’, thus expressing our state of knowledge about model parameters after having seen the data.

A problem that then can arise is called prior-data conflict: from the viewpoint of the prior, the observations seem very surprising, i.e., the information from data is in conflict with the prior assumptions. Unfortunately, models based on conjugate priors (which allow for straight-forward calculation of the posterior) are insensitive to prior-data conflict, in the sense that the spread of the posterior distribution does not increase in case of such a conflict. The posterior then conveys a false sense of certainty, by communicating that we can quantify uncertainty on model parameters quite precisely when in fact we cannot.

It is however possible to preserve tractability and have a meaningful reaction to prior-data conflict by using sets of conjugate priors for modelling prior information. This approach, which can be seen as imprecise probability method or robust Bayesian procedure, avoids the spurious over-precision of standard Bayesian methods and allows to adequately express vague or partial prior knowledge. With the precision of prior probability statements intuitively modelled through the magnitude of the set of priors, the posterior set appropriately reflects the prior precision, the amount of data, and prior-data conflict.