Logarithmically concave 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

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

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

• A positive log-concave function is also quasi-concave.

• 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,

$f(x)\nabla ^{2}f(x)\preceq \nabla f(x)\nabla f(x)^{T}$,

i.e.

$f(x)\nabla ^{2}f(x)-\nabla f(x)\nabla f(x)^{T}$ 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) : R^n+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 {{{1}}} 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.

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. MR 0489333.

• 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 0954608. {{cite book}}: Cite has empty unknown parameter: |1= (help)

• Pfanzagl, Johann; with the assistance of R. Hamböker (1994). Parametric Statistical Theory. Walter de Gruyter. ISBN 3-11-01-3863-8. MR 1291393. {{cite book}}: Cite has empty unknown parameter: |1= (help)

• Pečarić, Josip E.; Proschan, Frank; Tong, Y. L. (1992). Convex functions, partial orderings, and statistical applications. Mathematics in Science and Engineering. Vol. 187. Boston, MA: Academic Press, Inc. pp. xiv+467 pp. ISBN 0-12-549250-2. MR 1162312. {{cite book}}: Cite has empty unknown parameters: |1= and |2= (help)

See also

• logarithmically convex function
• convex function
• logarithmically concave measure