Optimization-centric generalizations of Bayesian inference

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Abstract

The mathematical machinery underlying Bayesian inference is Bayes' Rule| an important and elegant result dating back to the 18th century (Bayes, 1763). But in statistical practice, mathematical elegance alone is not enough: for Bayes' Rule to be a useful practical device, we need to impose a number of stringent assumptions that often do not reflect the realities of modern statistical and Machine Learning applications. This thesis sets out to propose and apply formalisms that are useful in situations where the assumptions underlying Bayes' Rule are dramatically violated. These assumptions include the presumption of a correctly specified statistical model, prior information of sufficient quality to improve the posterior belief, and adequate computational power. The violations of these assumptions and the proposed remedies will be explored theoretically and methodologically, but also empirically on a number of Machine Learning applications.

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