Generalized Bayesian Updating and the Loss-Likelihood Bootstrap

In this paper, we revisit the weighted likelihood bootstrap and show that it is well-motivated for Bayesian inference under misspecified models. We extend the underlying idea to a wider family of inferential problems. This allows us to calibrate an analogue of the likelihood function in situations where little is known about the data-generating mechanism. We demonstrate our method on a number of examples. There are some problems that arise when Bayesian methods are applied in modern settings. The construction of a global probabilistic representation through a joint model of the environment is often an impossible task. If the data does not come from the ascribed probability model then the posterior uncertainty quantification is theoretically invalid; the coherence and rationality that is foundational to Bayesian theory is lost. Often there are a finite number of functionals (or parameters) of interest to the practitioner, or decisions to be made. In this case it would be desirable to target these parameters directly, making as few assumptions about the rest of the environment as possible.