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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 ...
The Brunn–Minkowski inequality asserts that the Lebesgue measure is log-concave. The restriction of the Lebesgue measure to any convex set is also log-concave.. By a theorem of Borell, [2] a probability measure on R^d is log-concave if and only if it has a density with respect to the Lebesgue measure on some affine hyperplane, and this density is a logarithmically concave function.
The logarithm function is not defined for zero, so log probabilities can only represent non-zero probabilities. Since the logarithm of a number in (,) interval is negative, often the negative log probabilities are used. In that case the log probabilities in the following formulas would be inverted.
An extension of rejection sampling that can be used to overcome this difficulty and efficiently sample from a wide variety of distributions (provided that they have log-concave density functions, which is in fact the case for most of the common distributions—even those whose density functions are not concave themselves) is known as adaptive ...
But for practical purposes it is more convenient to work with the log-likelihood function in maximum likelihood estimation, in particular since most common probability distributions—notably the exponential family—are only logarithmically concave, [34] [35] and concavity of the objective function plays a key role in the maximization.
An estimation procedure that is often claimed to be part of Bayesian statistics is the maximum a posteriori (MAP) estimate of an unknown quantity, that equals the mode of the posterior density with respect to some reference measure, typically the Lebesgue measure.
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The rows of Pascal's triangle are examples for logarithmically concave sequences. In mathematics, a sequence a = (a 0, a 1, ..., a n) of nonnegative real numbers is called a logarithmically concave sequence, or a log-concave sequence for short, if a i 2 ≥ a i−1 a i+1 holds for 0 < i < n.