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

: <math>

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

</math>

for all and . If is strictly positive, this is equivalent to saying that the logarithm of the function, , is concave; that is,

: <math>

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

</math>

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

: <math>

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

</math>

for all and .

For non-negative discrete functions , it is log-concave if

: <math>

f(k)^2 \geq f(k+1)f(k-1)

</math>

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.

:i.e.

::<math>f(x)\nabla^2f(x) - \nabla f(x)\nabla f(x)^T</math> is

:negative semi-definite. For functions of one variable, this condition simplifies to

::<math>f(x)f(x) \leq (f'(x))^2</math>

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

::<math>\log\,f(x) + \log\,g(x) = \log(f(x)g(x))</math>

:is concave, and hence also is log-concave.

  • Marginals: if &nbsp;:&nbsp; is log-concave, then

::<math>g(x)=\int f(x,y) dy</math>

:is log-concave (see Prékopa–Leindler inequality).

  • This implies that convolution preserves log-concavity, since &nbsp;=&nbsp; is log-concave if and are log-concave, and therefore

::<math>(f*g)(x)=\int f(x-y)g(y) dy = \int h(x,y) dy</math>

: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,