Optimal priors improve betting based hypothesis testing efficiency
Optimal Priors for Betting Tests with Bounded Payoffs: An Exact Efficiency Characterization
Information Theory
Summary
When testing ideas using betting strategies, the choice of starting beliefs (called priors) affects how quickly and accurately tests decide if something is true or not. The authors studied which priors work best when the payoffs in the betting game are limited. They found a clear mathematical way to measure efficiency and identified specific priors, like Beta distributions, that lead to the most efficient tests. They also showed how to implement these methods efficiently for testing if a bounded variable has a certain average value.
What this means in practice
- •For statistical data analysts: Use optimal priors to speed up detection of deviations in bounded data when applying betting-based hypothesis tests.
- •For financial risk engineers: Implement efficient portfolio betting algorithms with bounded payoffs for improved risk assessment of constrained financial instruments.
Authors
Francesco Orabona
Abstract
Testing by betting is a powerful framework that recasts hypothesis testing as a betting game. A popular approach uses an $F$-weighted portfolio with a predefined prior $F$. For suitable priors, this approach has recently been shown to achieve the optimal expected rejection time as the error probability goes to zero at a fixed alternative. Yet, the prior $F$ changes how the portfolio test behaves near the null. This motivates the question: Which prior should we use in testing-by-betting algorithms? We study bounded scalar payoffs and introduce an efficiency measure that allows the error probability and alternative to vary jointly. Under mild assumptions, we prove that efficiency equals a logarithmic decay index of the prior near zero. This gives a necessary and sufficient condition for optimality: Beta priors have infinite efficiency, whereas suitable Robbins-type priors attain efficiency 1. As an application, we consider testing a specified mean of a bounded random variable and describe an implementation of the portfolio algorithm with almost-linear computational complexity.