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. 1. Introduction. Data sets are increasing in size and modelling environments are becoming more complex. This presents opportunities for Bayesian statistics but also major challenges, perhaps the greatest of which is the requirement to define the true sampling distribution, or likelihood, for the data generator f 0 (x), regardless of the study objective. So even if the task is inference for a low-dimensional statistic of the population, the analyst is required to model the complete data distribution and, moreover, assume that the model is "true". We propose a coherent procedure for general Bayesian inference that does not require complete knowledge of f 0 (x) and which connects information in the data to the value of an unknown object or parameter of interest via the use of loss functions. By "coherent" we mean that all relevant information is contained in the posterior probability distribution. For ease of exposition we shall use the terminology "parameter of interest" and "statistic of interest" interchangeably. We show how the approach leads to conventional Bayesian updating when the true likelihood is known but allows for rational updating of beliefs in much more general settings. The central tenet of our paper is this: as applications get more complex Bayesian analysts will increasingly be forced to forsake the notion that they can precisely model all aspects of the data. Settling for a misspecified model undermines the traditional Bayesian approach leading to interpretability problems along with the reliability of the posterior distribution. If the analyst acknowledges this then they should seek an alternative coherent way to proceed. We aim to contribute to this task.
The definition of a parameter of interest through the loss l(y, θ) is also present in general Bayesian updating introduced by REF , where a coherent posterior over a parameter of interest is obtained without the need to specify a joint generative model.
A General Framework for Updating Belief Distributions
Mathematics Medicine