Logarithmically Concave Function

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
: $f(\theta x + (1 - \theta) y) \geq f(x)^ f(y)^$
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
: $f(\theta x + (1 - \theta) y) \leq f(x)^ f(y)^$
for all and .

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:

::$f''(x)=e^ (x^2-1) \nleq 0$
* 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 logistic distribution.
*The extreme value distribution.
*The Laplace distribution.
*The chi distribution.
*The hyperbolic secant 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.
* If a density is log-concave, so is its survival function.See
* If a density is log-concave, it has a monotone hazard rate (MHR), and is a regular distribution since the derivative of the logarithm of the survival function is the negative hazard rate, and by concavity is monotone i.e.
::$\frac\log\left(1-F(x)\right) = -\frac$ 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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{{DEFAULTSORT:Logarithmically Concave Function
Mathematical analysis
Convex analysis