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Projective Limits of Bayes Equations

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Bayesian nonparametric models are essentially Bayesian models on infinite-dimensional spaces. Most work along these lines in statistics focusses on probability models over the simplex. In machine learning, the problem has recently received much attention as well, and attempts have been made to define models on a wider range of infinite-dimensional objects, including measures, functions and infinite permutations and graphs.

In my talk, I will discuss the construction of nonparametric Bayesian models from finite-dimensional Bayes equations, analogous to Daniell-Kolmogorov extension of measures to their projective limits. I will present an extension theorem applicable to regular conditional probabilities. This can be used to guarantee that “conditional” properties of the finite-dimensional marginal models, such as conjugacy and sufficiency, carry over to the infinite-dimensional projective limit model, and to determine the functional form of the nonparametric Bayesian posterior if the model is conjugate.

This talk is part of the Statistics series.

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