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
f ( θ x + ( 1 − θ ) y ) ≥ f ( x ) θ f ( y ) 1 − θ 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,
log f ( θ x + ( 1 − θ ) y ) ≥ θ log f ( x ) + ( 1 − θ ) log f ( y ) 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
f ( θ x + ( 1 − θ ) y ) ≤ f ( x ) θ f ( y ) 1 − θ for all x,y ∈ dom f and 0 < θ < 1.
A log-concave function is also quasi-concave. This follows from the fact that the logarithm is monotone implying that the superlevel sets of this function are convex.Every concave function that is nonnegative on its domain is log-concave. However, the reverse does not necessarily hold. An example is the Gaussian function 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| > 1:From above two points, concavity ⇒ log-concavity ⇒ quasiconcavity.A twice differentiable, nonnegative function with a convex domain is log-concave if and only if for all x satisfying f(x) > 0,i.e.negative semi-definite. For functions of one variable, this condition simplifies to
Products: The product of log-concave functions is also log-concave. Indeed, if f and g are log-concave functions, then log f and log g are concave by definition. Thereforeis concave, and hence also
f g is log-concave.
Marginals: if f(x,y) : Rn+m → R is log-concave, thenis log-concave (see
Prékopa–Leindler inequality).
This implies that convolution preserves log-concavity, since h(x,y) = f(x-y) g(y) is log-concave if f and g are log-concave, and thereforeis log-concave.
Log-concave distributions are necessary for a number of algorithms, e.g. adaptive rejection sampling.
As it happens, many common probability distributions are log-concave. Some examples:
The normal distribution and multivariate normal distributions.The exponential distribution.The uniform distribution over any convex set.The logistic distribution.The extreme value distribution.The Laplace distribution.The chi distribution.The Wishart distribution, where n >= p + 1.The Dirichlet distribution, where all parameters are >= 1.The gamma distribution if the shape parameter is >= 1.The chi-square distribution if the number of degrees of freedom is >= 2.The beta distribution if both shape parameters are >= 1.The Weibull distribution if the shape parameter is >= 1.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:
The Student's t-distribution.The Cauchy distribution.The Pareto distribution.The log-normal distribution.The F-distribution.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 log-normal distribution.The Pareto distribution.The Weibull distribution when the shape parameter < 1.The gamma distribution when the shape parameter < 1.The following are among the properties of log-concave distributions:
If a density is log-concave, so is its cumulative distribution function (CDF).If a multivariate density is log-concave, so is the marginal density over any subset of variables.The sum of two independent log-concave random variables is log-concave. This follows from the fact that the convolution of two log-concave functions is log-concave.The product of two log-concave functions is log-concave. This means that joint densities formed by multiplying two probability densities (e.g. the normal-gamma distribution, which always has a shape parameter >= 1) will be log-concave. This property is heavily used in general-purpose Gibbs sampling programs such as BUGS and JAGS, which are thereby able to use adaptive rejection sampling over a wide variety of conditional distributions derived from the product of other distributions. 