In convex analysis, a non-negative function f : Rn → R+ is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it satisfies the inequality for all x,y ∈ dom f and 0 < θ < 1 . If f is strictly positive, this is equivalent to saying that the logarithm of the function, log ∘ f , is concave; that is, for all x,y ∈ dom f and 0 < θ < 1 . Examples of log-concave functions are the 0-1 indicator functions of convex sets (which requires the more flexible definition), and the Gaussian function. Similarly, a function is log-convex if it satisfies the reverse inequality for all x,y ∈ dom f and 0 < θ < 1 .

Properties

f(x)

exp(−x2/2) which is log-concave since log f(x)

−x2/2 is a concave function of x . But f is not concave since the second derivative is positive for | x x satisfying f(x) > 0 ,

Operations preserving log-concavity

f and g are log-concave functions, then log f and log g are concave by definition. Therefore f g is log-concave. f(x,y) Rn+m → R is log-concave, then h(x,y)

f(x-y) g(y) is log-concave if f and g are log-concave, and therefore

Log-concave distributions

Log-concave distributions are necessary for a number of algorithms, e.g. adaptive rejection sampling. Every distribution with log-concave density is a maximum entropy probability distribution with specified mean μ and Deviation risk measure D. As it happens, many common probability distributions are log-concave. Some examples: Note that all of the parameter restrictions have the same basic source: The exponent of non-negative quantity must be non-negative in order for the function to be log-concave. The following distributions are non-log-concave for all parameters: Note that the cumulative distribution function (CDF) of all log-concave distributions is also log-concave. However, some non-log-concave distributions also have log-concave CDF's: The following are among the properties of log-concave distributions: