Logarithmically concave function

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

$f(\theta x + (1 - \theta) y) \geq f(x)^{\theta} f(y)^{1 - \theta}$

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,

$\log f(\theta x + (1 - \theta) y) \geq \theta \log f(x) + (1-\theta) \log f(y)$

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

$f(\theta x + (1 - \theta) y) \leq f(x)^{\theta} f(y)^{1 - \theta}$

for all x,y ∈ dom f and 0 < θ < 1.

Properties

• A positive log-concave function is also quasi-concave.

• 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:

$f''(x)=e^{-\frac{x^2}{2}} (x^2-1) \nleq 0$

• A twice differentiable, nonnegative function with a convex domain is log-concave if and only if for all x satisfying f(x) > 0,

$f(x)\nabla^2f(x) \preceq \nabla f(x)\nabla f(x)^T$,

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 f and g are log-concave functions, then log f and log g are concave by definition. Therefore

$\log\,f(x) + \log\,g(x) = \log(f(x)g(x))$

is concave, and hence also f g is log-concave.

• Marginals: if f(x,y) : R^n+m → R 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 h(x,y) = f(x-y) g(y) is log-concave if f and g 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.

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.

See also

• logarithmically concave sequence
• logarithmically concave measure
• logarithmically convex function
• convex function

• Stephen Boyd and Lieven Vandenberghe, Convex Optimization (PDF) p.105
• See Mark Bagnoli and Ted Bergstrom (1989), "Log-Concave Probability and Its Applications", University of Michigan.[1]
• András Prékopa (1971), "Logarithmic concave measures with application to stochastic programming". Acta Scientiarum Mathematicarum, 32, pp. 301–316.