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When it is acknowledged that all candidate parameterised statistical modelsare misspecified relative to the data generating process, the decision maker(DM) must currently concern themselves with inference for the parameter valueminimising the KL-divergence between the model and the process (Walker, 2013).However, it has long been known that minimising the KL-divergence places alarge weight on correctly capturing the tails of the sample distribution. As aresult the DM is required to worry about the robustness of their model to tailmisspecifications if they want to conduct principled inference. In this paperwe alleviate these concerns for the DM. We advance recent methodologicaldevelopments in general Bayesian updating (Bissiri, Holmes and Walker, 2016) topropose a statistically well principled Bayesian updating of beliefs targetingthe minimisation of more general divergence criteria. We improve both themotivation and the statistical foundations of existing Bayesian minimumdivergence estimation (Hooker and Vidyashankar, 2014; Ghosh and Basu, 2016),allowing the well principled Bayesian to target predictions from the model thatare close to the genuine model in terms of some alternative divergence measureto the KL-divergence. Our principled formulation allows us to consider abroader range of divergences than have previously been considered. In fact weargue defining the divergence measure forms an important, subjective part ofany statistical analysis, and aim to provide some decision theoretic rationalfor this selection. We illustrate how targeting alternative divergence measurescan impact the conclusions of simple inference tasks, and discuss then how ourmethods might apply to more complicated, high dimensional models.

Principles of Bayesian Inference using General Divergence Criteria