A Note on Log-Concavity
Matthias Seeger · Infoscience (Ecole Polytechnique Fédérale de Lausanne) · 2007
This is a small observation concerning scale mixtures and their log-concavity. A function f(x) ≥ 0, x ∈ R n is called log-concave if f (λx + (1 − λ)y) ≥ f(x) λ f(y) 1−λ (1) for all x,y ∈ R n, 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 ∝ x a−1 e −bx I {x>0}, the latter for a ≥ 1. In the Bayesian context it is important to note that