Abstract:
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The standard approach to Bayesian inference is based on the assumption that the distribution of the data belongs to the chosen model class. However, even a small violation of this assumption can have a large impact on the outcome of a Bayesian procedure, particularly when the data set is large. We introduce a simple, coherent approach to Bayesian inference that improves robustness to small departures from the model: rather than conditioning on the observed data exactly, one conditions on the event that the model generates data close to the observed data, with respect to a given statistical distance. When closeness is defined in terms of relative entropy, the resulting "coarsened posterior" can be approximated by simply raising the likelihood to a certain fractional power, making the method computationally efficient and easy to implement in practice. We illustrate with real and simulated data, and provide theoretical results.
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