# Logarithmically concave function

In convex analysis, a non-negative function *f* : **R**^{n} → **R**_{+} is **logarithmically concave** (or **log-concave** for short) if its domain is a convex set, and if it satisfies the inequality

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,

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

for all *x*,*y* ∈ dom *f* and 0 < *θ* < 1.

## Properties[edit]

- 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.
^{[1]} - 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(−x^{2}/2) which is log-concave since log*f*(*x*) = −*x*^{2}/2 is a concave function of*x*. But*f*is not concave since the second derivative is positive for |*x*| > 1:

- From above two points, concavity log-concavity quasiconcavity.
- A twice differentiable, nonnegative function with a convex domain is log-concave if and only if for all
*x*satisfying*f*(*x*) > 0,

- ,
^{[1]}

- ,

- i.e.

- is

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

## Operations preserving log-concavity[edit]

- 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

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

- Marginals: if
*f*(*x*,*y*) :**R**^{n+m}→**R**is log-concave, then

- 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

- is log-concave.

## Log-concave distributions[edit]

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*.^{[2]}
As it happens, many common probability distributions are log-concave. Some examples:^{[3]}

- 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.^{[4]} - The Dirichlet distribution, where all parameters are >= 1.
^{[4]} - 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.
^{[5]} - 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.

- which is decreasing as it is the derivative of a concave function.

## See also[edit]

- logarithmically concave sequence
- logarithmically concave measure
- logarithmically convex function
- convex function

## Notes[edit]

- ^
^{a}^{b}Boyd, Stephen; Vandenberghe, Lieven (2004). "Log-concave and log-convex functions".*Convex Optimization*. Cambridge University Press. pp. 104–108. ISBN 0-521-83378-7. **^**Grechuk, B.; Molyboha, A.; Zabarankin, M. (2009). "Maximum Entropy Principle with General Deviation Measures".*Mathematics of Operations Research*.**34**(2): 445–467. doi:10.1287/moor.1090.0377.**^**See Bagnoli, Mark; Bergstrom, Ted (2005). "Log-Concave Probability and Its Applications" (PDF).*Economic Theory*.**26**(2): 445–469. doi:10.1007/s00199-004-0514-4.- ^
^{a}^{b}Prékopa, András (1971). "Logarithmic concave measures with application to stochastic programming".*Acta Scientiarum Mathematicarum*.**32**: 301–316. **^**See Bagnoli, Mark; Bergstrom, Ted (2005). "Log-Concave Probability and Its Applications" (PDF).*Economic Theory*.**26**(2): 445–469. doi:10.1007/s00199-004-0514-4.

## References[edit]

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*Parametric Statistical Theory*. Walter de Gruyter. ISBN 3-11-013863-8. MR 1291393.

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