MAP and Bayes tests in sparse vectors detection

The paper focuses on the statistical analysis of Bayes and Maximum Posterior Probability (MAP) tests in the problem of detecting sparse vectors. Our main goal is to test the simple hypothesis $${\mathcal {H}}_0: \Vert \theta \Vert =0$$ H 0 : ‖ θ ‖ = 0 versus the composite alternative $${\mathcal {H}}_1: \Vert \theta \Vert >0 $$ H 1 : ‖ θ ‖ > 0 based on the observations $$\begin{aligned} Y_k=\sum _{s=1}^p\ {\mathbf {1}}\bigl \{ I_s=k\bigr \} \theta _s+\sigma \xi _k, \quad k=1, 2,\ldots , \end{aligned}$$ Y k = ∑ s = 1 p 1 { I s = k } θ s + σ ξ k , k = 1 , 2 , … , where $$\theta =(\theta _1,\ldots , \theta _p)^\top \in {\mathbb {R}}^p$$ θ = ( θ 1 , … , θ p ) ⊤ ∈ R p is an unknown vector, $$\xi _k$$ ξ k are i.i.d. $${\mathcal {N}}(0,1)$$ N ( 0 , 1 ) , and $$\mathbf { I}=\{I_1,\ldots ,I_p\}$$ I = { I 1 , … , I p } is an unknown random multi-index with differing components and the probability distribution $$\begin{aligned} {\mathbf {P}}\bigl (\mathbf { I}\bigr )\propto \prod _{k\in \mathbf { I}}{\bar{\pi }}_{k}. \end{aligned}$$ P ( I ) ∝ ∏ k ∈ I π ¯ k . It is assumed that the known prior distribution $${\bar{\pi }}=({\bar{\pi }}_1,\ldots )$$ π ¯ = ( π ¯ 1 , … ) has a large entropy. In the case of a known p, we find the limiting distributions of MAP and Bayes tests statistics under $${\mathcal {H}}_0$$ H 0 . Under the alternative, we characterize the sensitivity of these tests by computing detectable sets of $$\theta $$ θ . Finally, the case of unknown p is considered. We construct a multiple MAP test and show that it adapts to p under mild assumptions. This test is based on the non-asymptotic law of the three times iterated logarithm for the cumulative mean of the Wiener process.