A Note on Log-Concavity

This is a small observation concerning scale mixtures and their log-concavity. A function f(x) ≥ 0, x ∈ Rn is called log-concave if f (λx + (1− λ)y) ≥ f(x)f(y) (1) for all x,y ∈ Rn, λ ∈ [0, 1]. Log-concavity is important in applied Bayesian Statistics, since a distribution with a log-concave density is easy to treat with many different approximate inference techniques. For example, log-concavity implies unimodality. Log-concave distributions over few variables can be sampled from using a generic Markov chain Monte Carlo technique called adaptive rejection sampling [3]. For certain approximate inference techniques such as expectation propagation [5, 6], log-concavity of all sites means that the algorithm can be implemented in a numerically stable manner and tends to converge quickly, while in the absense of log-concavity it can fail badly. Many well-known densities are log-concave, for example the Gaussian or the Gamma ∝ xeI{x>0}, the latter for a ≥ 1. In the Bayesian context it is important to note that exponential family densities are in general log-concave in their natural parameters (but not necessarily in their data argument). A very important result concerning log-concave functions has been given by Prékopa (see [1], Sect. 1.8). Namely, if f : Rn1 × Rn2 → R is (jointly) log-concave, so is g(x) = ∫ f(x,y)dy , where the integral is over all of Rn2 . In this note, we are interested in scale mixture distributions [4] P (x) = E[N(x|h, s)] with some mixing distribution P (s) over the variance. We give a sufficient condition for P (x) to be log-concave, and show that the condition is not necessary. Since concavity is preserved under linear transformations, we can assume w.l.o.g. that h = 0. We have that