Abstract

The concept of shrinking bet size in Kelly betting to minimize estimated frequentist risk has recently been mooted. This rescaling appears to conflict with Bayesian decision theory through the likelihood principle and the complete class theorem; the Bayesian solution should already be optimal. We show theoretically and through examples that when the model
determining the likelihood function is correct, the prior distribution (if not dominated by data) is `correct' in a frequentist sense, and the posterior distribution is proper, then no further rescaling is required. However, if the model or the prior distribution is incorrect, or the posterior distribution improper, frequentist risk minimization can be a useful technique. We discuss how it might best be exploited. Another example, from maintenance, is used to show the wider applicability of the methodology; these conclusions
apply generally to decision-making.

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