Multi-Stage Active Sequential Hypothesis Testing with Clustered Hypotheses

TL;DR

This paper introduces an adaptive multi-stage hypothesis testing strategy that clusters similar hypotheses to reduce observations, achieving asymptotic optimality and significant efficiency improvements over existing methods.

Contribution

It proposes a novel deterministic, multi-stage hypothesis-elimination algorithm that clusters hypotheses based on distribution similarity, reducing observation requirements and improving efficiency.

Findings

Achieves asymptotically optimal Bayes risk as error probability approaches zero.

Reduces the mean number of observations by several orders of magnitude compared to existing algorithms.

Maintains bounded Bayes risk as the number of hypotheses increases.

Abstract

We consider the problem where an active Decision-Maker (DM) is tasked to identify the true hypothesis using as few as possible observations while maintaining accuracy. The DM collects observations according to its determined actions and knows the distributions under each hypothesis. We propose a deterministic and adaptive multi-stage hypothesis-elimination strategy where the DM selects an action, applies it repeatedly, and discards hypotheses in light of its obtained observations. The DM selects actions based on maximal separation expressed by the distance between the parameter vectors of each distribution under each hypothesis. Close distributions can be clustered, simplifying the search and significantly reducing the number of required observations. Our algorithms achieve vanishing Average Bayes Risk (ABR) as the error probability approaches zero, i.e., the algorithm is asymptotically optimal. Furthermore, we show that the ABR is bounded when the number of hypotheses grows. Simulations are carried out to evaluate the algorithm's performance compared to another multi-stage hypothesis-elimination algorithm, where an improvement of several orders of magnitude in the mean number of observations required is observed.