# Logarithmically concave function

In convex analysis, a non-negative function f : RnR+ 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 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

• 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,
$f(x)\nabla^2f(x) \preceq \nabla f(x)\nabla f(x)^T$, [1]
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) : Rn+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-yg(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:[2]

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 (CDF) 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:

• 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 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.

## Notes

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

## References

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• Dharmadhikari, Sudhakar; Joag-Dev, Kumar (1988). Unimodality, convexity, and applications. Probability and Mathematical Statistics. Boston, MA: Academic Press, Inc. pp. xiv+278. ISBN 0-12-214690-5. MR 954608.
• Pfanzagl, Johann; with the assistance of R. Hamböker (1994). Parametric Statistical Theory. Walter de Gruyter. ISBN 3-11-013863-8. MR 1291393.
• Pečarić, Josip E.; Proschan, Frank; Tong, Y. L. (1992). Convex functions, partial orderings, and statistical applications. Mathematics in Science and Engineering 187. Boston, MA: Academic Press, Inc. pp. xiv+467 pp. ISBN 0-12-549250-2. MR 1162312.