Finite-trial discrimination

For two independent ensembles with $n_{\theta },n_0$ trials, consider the binary postprocessed event in a fixed setting. It can be realized by independent classical randomization when the scores $r_{\mathbf{j}}$ are not zero or one.

For $0<\beta <1$, put

With probability at least $1-\beta$,

where ${\Delta }_{\mathbf{a}}=|\mathrm{tr}{E}_{\mathbf{a}}({\rho }_{\theta }-{\rho }_0)|$. This follows from the Bernoulli form of Lemma lem:concentration and the two probability comparisons.

Conversely, an observed gap exceeding $e_{\theta }+e_0+u_{\theta }+u_0$ excludes equality of the two ideal test probabilities at that confidence level, under the stated preparation and apparatus assumptions.