Robust Bayesian inference using Bayes spaces

In the world of geoanalytics and mining statistics we are often confronted with indirect and uncertain measurments, where there might be considerable bias in the observations itself. This contribution addresses a new view to robust Bayesian inference, which might be suitable to address such problems.

Bayes spaces are spaces of distributions and likelihoods, similar to compositions and the Aitchison simplex. In these spaces the updating a prior to a posterior is a vector addition. This simple structure allows it to reconsider questions from Bayesian statistical analysis. E.g. a posteriors from different priors have a constant difference independent of the actual observation.

In this contribution we will consider robustness against uncertainty in model assumptions and data errors. The structure allows it to introduce uncertainty about the prior knowledge by replacing a single prior by convex set of possible priors and a single model of likelihoods by convex sets of possible models. A Bayesian update is than convex Minkowski sum of the two sets, which can be explictly computed and analysed. We will show in examples how model uncertainty and possible data errors can be expressed.

We will also discuss the uncertainty introduced by this approach. While uncertainty measured in the geometry of the Bayes spaces measured as diameter convex result will diverge at rate n, we can typically observe a constant residual uncertainty in the model estimation.