A General Framework for Updating Belief Distributions

We propose a general framework for Bayesian inference that does not require the specification of a complete probability model, or likelihood, for the data. As data sets become larger and systems under investigation more complex it is increasingly challenging for Bayesian analysts to attempt to model the true data generating mechanism. Moreover, when the object of interest is a low dimensional statistic, such as a mean or median, it is cumbersome to have to achieve this via a complete model for the whole data distribution. If Bayesian analysis is to keep pace with modern applications it will need to forsake the notion that it is either possible or desirable to model the complete data distribution. Our proposed framework uses loss-functions to connect information in the data to statistics of interest. The updating of beliefs then follows from a decision theoretic approach involving cumulative loss functions. Importantly, the procedure coincides with Bayesian updating when a true likelihood is given, yet provides coherent subjective inference in much more general settings. We demonstrate our approach in important application areas for which Bayesian inference is problematic including variable selection in survival analysis models and inference on a set of quantiles of a sampling distribution. Connections to other inference frameworks are highlighted.