Asymptotic log-optimality of plug-in likelihood ratio e-variables for composite alternatives

Establish asymptotic log-optimality of the plug-in likelihood ratio e-variable E_n = ∏_{i=1}^n q_{i-1}(X_i)/p(X_i) for testing a simple null distribution P against a composite alternative set Q, where q_{i-1} is chosen (for example) as the posterior mean under a prior ν on Q based on X_1,…,X_{i-1}. Specifically, prove under weak regularity assumptions that lim_{n→∞} (1/n) E^{Q^n}[log E_n] equals the optimal rate achieved by the mixture likelihood ratio method, i.e., that the plug-in method attains the same asymptotic e-power when the mixture method is asymptotically log-optimal.

Background

In the composite alternative setting, two broad strategies for constructing e-variables were outlined: a mixture method that integrates likelihood ratios over a prior ν on Q, and a plug-in method that uses data-adaptive choices q_{i-1} for the next likelihood ratio factor. The mixture method is often asymptotically optimal under mild conditions, while the plug-in method is attractive computationally and conceptually. The authors note that one expects the plug-in method, with q_{i-1} chosen sensibly (e.g., posterior mean under ν), to share the same asymptotic log-optimality as the mixture method.

While this expectation is natural, the authors explicitly state that proving such a general result is currently beyond the available theory. They highlight that results of this generality remain conjectural, indicating an unresolved question regarding the precise asymptotic e-power (growth rate) of the plug-in construction.