Comments on “entropies of degree β and lower bounds for the average error rate”

Consider a two-dimensional random variable (X, θ) with θ ∊ {θ
1
…, θ
m
) and X ∊ R
^d
, and suppose that the distribution of (X, θ) is given by a) P(θ = θ
i
) = p
i
, 1 ≤ i ≤ m, and b) P{X < x/θ = θ
i
} has a probability density p(x/θ
i
)1 ≤ i ≤ m. We call θ the state of the observation X. Let R
^∗
denote the Bayes' rate, viz., R
^∗
= E{r
^∗
(x)}, and r
^∗
(x) = 1 − max
i
p(θ
i
/x). Then for any β > 1, a lower bound on R
^∗
due to Devijver
¹
is the inequality