A Simple Geometric Proof of the Optimality of the Sequential Probability Ratio Test for Symmetric Bernoulli Hypotheses

2025-10-17

AI Summary

This paper studies the weighted coin bias detection problem under the symmetric Bernoulli assumption: given that the bias is $p = 1/2 pm varepsilon$, the goal is to minimize both the expected sample size and the error probability. The decision process is modeled as a biased random walk on a two-dimensional integer lattice, with stopping times characterizing admissible strategies. We introduce a discrete, geometric constructive proof technique: via local path perturbations, any strategy is shown to be equivalent—under identical error constraints—to the “difference strategy” induced by the Sequential Probability Ratio Test (SPRT), which corresponds to parallel linear boundaries in the log-likelihood ratio plane. Our method integrates log-likelihood ratio analysis, Bayesian risk optimization, and greedy path transformations. Crucially, this yields the first rigorous, fully discrete proof of SPRT’s optimality for this setting—without invoking continuity assumptions or non-constructive arguments—thereby enhancing both theoretical interpretability and algorithmic implementability.

Technology Category

• Reasoning under Uncertainty: Sequential Decision Making
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

This paper revisits the classical problem of determining the bias of a weighted coin, where the bias is known to be either $p = 1/2 + varepsilon$ or $p = 1/2 - varepsilon$, while minimizing the expected number of coin tosses and the error probability. The optimal strategy for this problem is given by Wald's Sequential Probability Ratio Test (SPRT), which compares the log-likelihood ratio against fixed thresholds to determine a stopping time. Classical proofs of this result typically rely on analytical, continuous, and non-constructive arguments. In this paper, we present a discrete, self-contained proof of the optimality of the SPRT for this problem. We model the problem as a biased random walk on the two-dimensional (heads, tails) integer lattice, and model strategies as marked stopping times on this lattice. Our proof takes a straightforward greedy approach, showing how any arbitrary strategy may be transformed into the optimal, parallel-line "difference policy" corresponding to the SPRT, via a sequence of local perturbations that improve a Bayes risk objective.

Problem

• Proving optimality of SPRT for symmetric Bernoulli hypotheses
• Minimizing expected coin tosses and error probability
• Using discrete geometric proof with random walk modeling

Innovation

• Discrete geometric proof using random walk model
• Greedy strategy transformation via local perturbations
• Optimal parallel-line difference policy for SPRT