In convex analysis, a non-negative function is logarithmically concave if its domain is a convex set, and if it satisfies the inequality for all and. If is strictly positive, this is equivalent to saying that the logarithm of the function,, is concave; that is, for all and. Examples of log-concave functions are the 0-1 indicator functions of convex sets, and the Gaussian function. Similarly, a function is log-convex if it satisfies the reverse inequality for all and.
Properties
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 = which is log-concave since = is a concave function of. But is not concave since the second derivative is positive for || > 1:
A twice differentiable, nonnegative function with a convex domain is log-concave if and only if for all satisfying,
Operations preserving log-concavity
Products: The product of log-concave functions is also log-concave. Indeed, if and are log-concave functions, then and are concave by definition. Therefore
Marginals: if : is log-concave, then
This implies that convolution preserves log-concavity, since = is log-concave if and are log-concave, and therefore
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 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 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 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.
