Research proof index

There exists an optimal T for the hypothesis testing, that is, it's a minimum and not just an infimum. This tightens the T from exists_min' to a ⟪ρ,T⟫ = 1 - ε bound.

There exists an optimal T for the hypothesis testing, that is, it's a minimum and not just an infimum. This states we have 1 - ε ≤ ρ.exp_val T, but we can always "worsen" T to make that bound tight, which is exists_min.

The space of strategies T in OptimalHypothesisRate is compact.

The space of strategies T in OptimalHypothesisRate is convex.

On the 1D Hilbert space, the optimal hypothesis testing rate is simply 1 - ε, since there's nothing to learn. (More generally this would hold whenever ρ=σ.) -

When the allowed Type I error ε is less than 1 (so, we have some limit on our errors), and the kernel of the state ρ contains the kernel of some element in S, then the optimal hypothesis rate is positive - there is some lower bound on the type II errors we'll see. In other words, under these conditions, we cannot completely avoid type II errors.