The Uncertain Detector

Classical detection theory assumes you know the noise distribution. If the noise is Gaussian with known variance, the optimal detector is the matched filter. If the noise follows some other known distribution, the optimal detector follows from the Neyman-Pearson lemma. But what if you don’t know the distribution — not just its parameters, but its form?

Nonlinear expectation theory provides the framework. Instead of a single probability measure, you have a set of possible measures, and the optimal detector must work well across all of them. The expectation becomes nonlinear because it maximizes over the uncertainty set rather than integrating against a single distribution.

The paper derives optimal detectors for binary channels with two kinds of distributional uncertainty: variance-only uncertainty (you know the mean but not the variance) and mean-plus-variance uncertainty (you know neither).

The structural result: mean uncertainty changes the form of the optimal detector; variance uncertainty does not. When the mean is known, the detector’s structure is the same as in the classical case — only its threshold changes. When the mean is uncertain, the detector itself must change shape to hedge against the unknown bias.

This asymmetry is unexpected. Intuitively, both kinds of uncertainty should complicate the detector equally. But the optimal response to variance uncertainty is a scaling adjustment (change the threshold), while the optimal response to mean uncertainty is a geometric adjustment (change the decision boundary shape). The distinction parallels the difference between multiplicative and additive perturbations — they affect different aspects of the detection problem.

The detectors outperform classical ones in most scenarios with uncertain noise models, validating the theory experimentally.