  When it is acknowledged that all candidate parameterised statistical models
are misspecified relative to the data generating process, the decision maker
(DM) must currently concern themselves with inference for the parameter value
minimising the KL-divergence between the model and the process (Walker, 2013).
However, it has long been known that minimising the KL-divergence places a
large weight on correctly capturing the tails of the sample distribution. As a
result the DM is required to worry about the robustness of their model to tail
misspecifications if they want to conduct principled inference. In this paper
we alleviate these concerns for the DM. We advance recent methodological
developments in general Bayesian updating (Bissiri, Holmes and Walker, 2016) to
propose a statistically well principled Bayesian updating of beliefs targeting
the minimisation of more general divergence criteria. We improve both the
motivation and the statistical foundations of existing Bayesian minimum
divergence estimation (Hooker and Vidyashankar, 2014; Ghosh and Basu, 2016),
allowing the well principled Bayesian to target predictions from the model that
are close to the genuine model in terms of some alternative divergence measure
to the KL-divergence. Our principled formulation allows us to consider a
broader range of divergences than have previously been considered. In fact we
argue defining the divergence measure forms an important, subjective part of
any statistical analysis, and aim to provide some decision theoretic rational
for this selection. We illustrate how targeting alternative divergence measures
can impact the conclusions of simple inference tasks, and discuss then how our
methods might apply to more complicated, high dimensional models.
