Logarithmically concave function


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

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 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: