{
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  "slug": "logarithmically-concave-function",
  "title": "Logarithmically concave function",
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  "lead": "In convex analysis, a non-negative function is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it satisfies the inequality : f(\\theta x + (1 - \\theta) y) \\geq f(x)^{\\theta} f(y)^{1 - \\theta} for all and . If is strictly positive, this is equivalent to saying that the logarithm of the function, , is concave; that is, : \\log f(\\theta x + (1 - \\theta) y) \\geq \\theta \\log f(x) + (1-\\theta) \\log f(y) for all and . 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(\\theta x + (1 - \\theta) y) \\leq f(x)^{\\theta} f(y)^{1 - \\theta} for all and . For non-negative discrete functions , it is log-concave if : f(k)^2 \\geq f(k+1)f(k-1)",
  "body": "In convex analysis, a non-negative function is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it satisfies the inequality f(\\theta x + (1 - \\theta) y) \\geq f(x)^{\\theta} f(y)^{1 - \\theta} for all and . If is strictly positive, this is equivalent to saying that the logarithm of the function, , is concave; that is, \\log f(\\theta x + (1 - \\theta) y) \\geq \\theta \\log f(x) + (1-\\theta) \\log f(y) for all and . 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(\\theta x + (1 - \\theta) y) \\leq f(x)^{\\theta} f(y)^{1 - \\theta} for all and . For non-negative discrete functions , it is log-concave if f(k)^2 \\geq f(k+1)f(k-1) ==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: f(x)=e^{-\\frac{x^2}{2}} (x^2-1) \\nleq 0 From above two points, concavity \\Rightarrow log-concavity \\Rightarrow quasiconcavity. A twice differentiable, nonnegative function with a convex domain is log-concave if and only if for all satisfying , f(x)\\nabla^2f(x) \\preceq \\nabla f(x)\\nabla f(x)^T , i.e. f(x)\\nabla^2f(x) - \\nabla f(x)\\nabla f(x)^T is negative semi-definite. For functions of one variable, this condition simplifies to f(x)f(x) \\leq (f'(x))^2 ==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 \\log\\,f(x) + \\log\\,g(x) = \\log(f(x)g(x)) is concave, and hence also is log-concave. Marginals: if :  is log-concave, then g(x)=\\int f(x,y) dy is log-concave (see Prékopa–Leindler inequality). This implies that convolution preserves log-concavity, since  =  is log-concave if and are log-concave, and therefore (f*g)(x)=\\int f(x-y)g(y) dy = \\int h(x,y) dy is log-concave. ==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: the normal distribution and multivariate normal distributions, the exponential distribution, the uniform distribution over any convex set, the binomial distribution, the logistic distribution, the extreme value distribution, the Laplace distribution, the chi distribution, the hyperbolic secant distribution, the Wishart distribution, if n ≥ p + 1, the Dirichlet distribution, if 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, and 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, and 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 \\frac{d}{dx}\\log\\left(1-F(x)\\right) = -\\frac{f(x)}{1-F(x)} which is decreasing as it is the derivative of a concave function. ==See also== logarithmically concave sequence logarithmically concave measure logarithmically convex function convex function ==Notes== ==References==",
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    "title": "Logarithmically concave function",
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