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Abstract:
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In his remarkable 1971 paper, Larry Brown exhibited the interrelations between Bayes risks under squared error loss and the information in the marginal. The Fisher information being a variational operator in suitable Sobolev spaces is amenable to further analysis by using the tools of variational analysis and inequalities in analysis and mathematical physics. Examples of such tools are the intermediate derivative norm inequalities due to many eminent mathematicians, among them Hardy, Littlewood, Polya, Nash, Poincare, and Kolmogorov; examples from physics include the uncertainty principle of quantum physics. In this talk, we show methods to derive bounds on Bayes risks from both directions by bridging the techniques of Brown and the variational techniques of analysis and physics.
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