Logarithmically concave function

Short description: Type of mathematical 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

[math]\displaystyle{ f(\theta x + (1 - \theta) y) \geq f(x)^{\theta} f(y)^{1 - \theta} }[/math]

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,

[math]\displaystyle{ \log f(\theta x + (1 - \theta) y) \geq \theta \log f(x) + (1-\theta) \log f(y) }[/math]

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

[math]\displaystyle{ f(\theta x + (1 - \theta) y) \leq f(x)^{\theta} f(y)^{1 - \theta} }[/math]

for all x,y ∈ dom f and 0 < θ < 1.

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 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:

[math]\displaystyle{ f''(x)=e^{-\frac{x^2}{2}} (x^2-1) \nleq 0 }[/math]

• From above two points, concavity [math]\displaystyle{ \Rightarrow }[/math] log-concavity [math]\displaystyle{ \Rightarrow }[/math] quasiconcavity.
• A twice differentiable, nonnegative function with a convex domain is log-concave if and only if for all x satisfying f(x) > 0,

[math]\displaystyle{ f(x)\nabla^2f(x) \preceq \nabla f(x)\nabla f(x)^T }[/math],

i.e.

[math]\displaystyle{ 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]\displaystyle{ 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 f and g are log-concave functions, then log f and log g are concave by definition. Therefore

[math]\displaystyle{ \log\,f(x) + \log\,g(x) = \log(f(x)g(x)) }[/math]

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

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

[math]\displaystyle{ g(x)=\int f(x,y) dy }[/math]

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

[math]\displaystyle{ (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 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.
• 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.

[math]\displaystyle{ \frac{d}{dx}\log\left(1-F(x)\right) = -\frac{f(x)}{1-F(x)} }[/math] 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

References

• Barndorff-Nielsen, Ole (1978). Information and exponential families in statistical theory. Wiley Series in Probability and Mathematical Statistics. Chichester: John Wiley \& Sons, Ltd.. pp. ix+238 pp. ISBN 0-471-99545-2. 
• 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. 

• Pfanzagl, Johann; with the assistance of R. Hamböker (1994). Parametric Statistical Theory. Walter de Gruyter. ISBN 3-11-013863-8. 

• 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.