Abstract
A general approach to Bayesian learning revisits some classical results, which study which functionals on a prior distribution are expected to increase, in a preposterior sense. The results are applied to information functionals of the Shannon type and to a class of functionals based on expected distance. A close connection is made between the latter and a metric embedding theory due to Schoenberg and others. For the Shannon type, there is a connection to majorization theory for distributions. A computational method is described to solve generalized optimal experimental design problems arising from the learning framework based on a version of the well-known approximate Bayesian computation (ABC) method for carrying out the Bayesian analysis based on Monte Carlo simulation. Some simple examples are given.
Highlights
A Bayesian approach to the optimal design of experiments uses some measure of preposterior utility, or information, to assess the efficacy of an experimental design or, more generally, the choice of sampling distribution
We study two types of learning function, those of which we shall call the Shannon type and, in Section 3, those based on distances
We present a general method for performing optimum experimental design calculations, which, combined with the theory of learning outlined above, may provide a comprehensive approach
Summary
A Bayesian approach to the optimal design of experiments uses some measure of preposterior utility, or information, to assess the efficacy of an experimental design or, more generally, the choice of sampling distribution. Various versions of this approach have been developed by Blackwell [1], and Torgerson [2]. Renyi [3], Lindley [4] and Goel and DeGroot [5] use information-theoretic approaches to measure the value of an experiment; see the review paper by Ginebra [6]. There is wider interest in these issues in cognitive science and epistemology; see Chater and Oaksford [9].
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