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 ( θ x + ( 1 − θ ) y ) ≥ f ( x ) θ f ( y ) 1 − θ {\displaystyle 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 ( θ x + ( 1 − θ ) y ) ≥ θ log ⁡ f ( x ) + ( 1 − θ ) log ⁡ f ( y ) {\displaystyle \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 it satisfies the reverse inequality

f ( θ x + ( 1 − θ ) y ) ≤ f ( x ) θ f ( y ) 1 − θ {\displaystyle f(\theta x+(1-\theta )y)\leq f(x)^{\theta }f(y)^{1-\theta }}

for all x,y ∈ dom f and 0 < θ < 1.

For non-negative discrete functions f : Z → R+, it is log-concave if

f ( k ) 2 ≥ f ( k + 1 ) f ( k − 1 ) {\displaystyle f(k)^{2}\geq f(k+1)f(k-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:

f ″ ( x ) = e − x 2 2 ( x 2 − 1 ) ≰ 0 {\displaystyle f''(x)=e^{-{\frac {x^{2}}{2}}}(x^{2}-1)\nleq 0}

  • From above two points, concavity ⇒ {\displaystyle \Rightarrow } log-concavity ⇒ {\displaystyle \Rightarrow } quasiconcavity.
  • 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 ) ∇ 2 f ( x ) ⪯ ∇ f ( x ) ∇ f ( x ) T {\displaystyle f(x)\nabla ^{2}f(x)\preceq \nabla f(x)\nabla f(x)^{T}},

i.e.

f ( x ) ∇ 2 f ( x ) − ∇ f ( x ) ∇ f ( x ) T {\displaystyle 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 ) ≤ ( f ′ ( x ) ) 2 {\displaystyle 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 ) ) {\displaystyle \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 ) = ∫ f ( x , y ) d y {\displaystyle 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-y) g(y) is log-concave if f and g are log-concave, and therefore

( f ∗ g ) ( x ) = ∫ f ( x − y ) g ( y ) d y = ∫ h ( x , y ) d y {\displaystyle (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:

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 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.

d d x log ⁡ ( 1 − F ( x ) ) = − f ( x ) 1 − F ( x ) {\displaystyle {\frac {d}{dx}}\log \left(1-F(x)\right)=-{\frac {f(x)}{1-F(x)}}} which is decreasing as it is the derivative of a concave function.

See also

Notes

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