A Generalized Poor-Verdú Error Bound for Multihypothesis Testing

Abstract

A lower bound on the minimum error probability for multihypothesis testing is established. The bound, which is expressed in terms of the cumulative distribution function of the tilted posterior hypothesis distribution given the observation with tilting parameter , generalizes an earlier bound due the Poor and Verdú (1995). A sufficient condition is established under which the new bound (minus a multiplicative factor) provides the exact error probability asymptotically in . Examples illustrating the new bound are also provided.

Arimoto–Rényi Conditional Entropy and Bayesian $M$ -Ary Hypothesis Testing

“…13) One of the bounds by Han and Verdú [34] in Item 8) was generalized by Polyanskiy and Verdú [56] to give a lower bound on the α-mutual information ( [74], [85]). 14) In [73], Shayevitz gave a lower bound, in terms of the Rényi divergence, on the maximal worst-case missdetection exponent for a binary composite hypothesis testing problem when the false-alarm probability decays to zero with the number of i.i.d. observations.…”

“…where the outer expectations are with respect to Y ∼ P Y ; (284) follows from (14) and the equality βρ β−1 = α α−1 (see (282) and (283)); (285) follows from β β−1 > 0, 1 β > 0, Jensen's inequality and the concavity of t ρ on [0, ∞); (286) follows from (283) and (282); (287) holds due to the concavity of t θ on [0, ∞) and Jensen's inequality; (288) follows from (282); (289) follows from (14).…”

Bayesian Error-Based Sequences of Statistical Information Bounds

“…Note that for a given x, as n increases p n (θ m |x) approaches 1 from below for m = m * (x) = argmax m {p(θ m |x)} and 0 from above for all other m values. This increasing purification of the posterior p n (Θ|X), also known as the tilted posterior PMF of order n [14], with increasing n also implies a decreasing value of the GEE.…”

“…For the special choice, s m (X) = p(θ m |X), the RHS of inequality (14) reduces to −H(Θ|X), which is the FM bound. A more general choice involves the generalized posterior PMF, p n (θ m |X), defined earlier in (7), that becomes sharper the larger its order n,…”

“…The A related topic of bounding the MPE for multi-hypothesis testing in the Bayesian setting has received much attention for two main reasons, one being the need for computational efficiency and the other being its usefulness in developing and proving certain asymptotic results in coding theory. Bounds of this type include the Fano lower bound (FB) involving the equivocation entropy (EE) [2] and its generalization for nonuniform priors [10] and countably infinite alphabets [11], the Shannon bound [12], the Poor-Verdu bound [13], and the generalized Poor-Verdu bound [14]. Of these, only the FB and its generalizations yield bounds on the EE and thus on MI, which is the subject of main interest here.…”