  In the usual Bayesian setting, a full probabilistic model is required to link
the data and parameters, and the form of this model and the inference and
prediction mechanisms are specified via de Finetti's representation. In
general, such a formulation is not robust to model mis-specification of its
component parts. An alternative approach is to draw inference based on loss
functions, where the quantity of interest is defined as a minimizer of some
expected loss, and to construct posterior distributions based on the loss-based
formulation; this strategy underpins the construction of the Gibbs posterior.
We develop a Bayesian non-parametric approach; specifically, we generalize the
Bayesian bootstrap, and specify a Dirichlet process model for the distribution
of the observables. We implement this using direct prior-to-posterior
calculations, but also using predictive sampling. We also study the assessment
of posterior validity for non-standard Bayesian calculations, and provide an
efficient way to calibrate the scaling parameter in the Gibbs posterior so that
it can achieve the desired coverage rate. We show that the developed
non-standard Bayesian updating procedures yield valid posterior distributions
in terms of consistency and asymptotic normality under model mis-specification.
Simulation studies show that the proposed methods can recover the true value of
the parameter efficiently and achieve frequentist coverage even when the sample
size is small. Finally, we apply our methods to evaluate the causal impact of
speed cameras on traffic collisions in England.
