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General Bayesian updating and model misspecification

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Bayesian statistics provides a unified approach to the updating of beliefs but is challenged by modern applications through the formal requirement to define the true sampling distribution, or joint likelihood, for the whole data generating process regardless of the study objective. So even if the task is inference for a low-dimensional statistic Bayesian analysis is required to model the complete data distribution and, moreover, assume that the model is ``true’’. In this talk we present a coherent procedure for general Bayesian inference based on the use of loss functions to connect information in data to parameters of interest. The updating of a prior belief distribution to a posterior then follows from a decision theoretic foundation involving cumulative loss functions and a requirement for coherency. Sensitivity to model misspecification can be characterised via neighbourhoods in model space around the approximating model. Importantly, the procedure coincides with Bayesian updating when a true likelihood is known, yet provides coherent subjective inference in much more general settings. We demonstrate the approach on examples including model-free general Bayesian co-clustering of time series.

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