Inconvex analysis, anon-negative functionf :Rⁿ →R₊ islogarithmically concave (orlog-concave for short) if itsdomain is aconvex set, and if it satisfies the inequality

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

for allx,y ∈ domf and0 < θ < 1. Iff is strictly positive, this is equivalent to saying that thelogarithm of the function,log ∘f, isconcave; that is,

${\displaystyle \log f(\theta x+(1-\theta )y)\ge \theta \log f(x)+(1-\theta )\log f(y)}$

for allx,y ∈ domf and0 < θ < 1.

Examples of log-concave functions are the 0-1indicator functions of convex sets (which requires the more flexible definition), and theGaussian function.

Similarly, a function islog-convex if it satisfies the reverse inequality

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

for allx,y ∈ domf and0 < θ < 1.

• A log-concave function is alsoquasi-concave. This follows from the fact that the logarithm is monotone implying that thesuperlevel 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 theGaussian functionf(x) = exp(−x²/2) which is log-concave sincelogf(x) = −x²/2 is a concave function ofx. Butf is not concave since thesecond derivative is positive for |x| > 1:

${\displaystyle f^{″}(x)=e^{-\frac{x^2}{2}}(x^2-1)\nleqq 0}$

• From above two points,concavity${\displaystyle \Rightarrow }$ log-concavity${\displaystyle \Rightarrow }$quasiconcavity.
• A twice differentiable, nonnegative function with a convex domain is log-concaveif and only if for allx satisfyingf(x) > 0,

${\displaystyle f(x){\mathrm{\nabla }}^{2}f(x)⪯\mathrm{\nabla }f(x)\mathrm{\nabla }f(x{)}^T}$,^[1]

i.e.

${\displaystyle f(x){\mathrm{\nabla }}^{2}f(x)-\mathrm{\nabla }f(x)\mathrm{\nabla }f(x{)}^T}$ is

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

${\displaystyle f(x)f^{″}(x)\le (f^{\prime }(x){)}^2}$

• Products: The product of log-concave functions is also log-concave. Indeed, iff andg are log-concave functions, thenlog f andlog g are concave by definition. Therefore

${\displaystyle \log f(x)+\log g(x)=\log (f(x)g(x))}$

is concave, and hence alsof g is log-concave.

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

${\displaystyle g(x)=\int f(x,y)dy}$

is log-concave (seePrékopa–Leindler inequality).

• This implies thatconvolution preserves log-concavity, sinceh(x,y) = f(x-y) g(y) is log-concave iff andg are log-concave, and therefore

${\displaystyle (f\ast g)(x)=\int f(x-y)g(y)dy=\int h(x,y)dy}$

is log-concave.

Log-concave distributions are necessary for a number of algorithms, e.g.adaptive rejection sampling. Every distribution with log-concave density is amaximum entropy probability distribution with specified meanμ andDeviation risk measureD.^[2] As it happens, many commonprobability distributions are log-concave. Some examples:^[3]

• thenormal distribution andmultivariate normal distributions,
• theexponential distribution,
• theuniform distribution over anyconvex set,
• thelogistic distribution,
• theextreme value distribution,
• theLaplace distribution,
• thechi distribution,
• thehyperbolic secant distribution,
• theWishart distribution, ifn ≥p + 1,^[4]
• theDirichlet distribution, if all parameters are ≥ 1,^[4]
• thegamma distribution if theshape parameter is ≥ 1,
• thechi-square distribution if the number of degrees of freedom is ≥ 2,
• thebeta distribution if both shape parameters are ≥ 1, and
• theWeibull 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:

• theStudent's t-distribution,
• theCauchy distribution,
• thePareto distribution,
• thelog-normal distribution, and
• theF-distribution.

Note that thecumulative distribution function (CDF) of all log-concave distributions is also log-concave. However, some non-log-concave distributions also have log-concave CDF's:

• thelog-normal distribution,
• thePareto distribution,
• theWeibull distribution when the shape parameter < 1, and
• thegamma distribution when the shape parameter < 1.

The following are among the properties of log-concave distributions:

• If a density is log-concave, so is itscumulative distribution function (CDF).
• If a multivariate density is log-concave, so is themarginal density over any subset of variables.
• The sum of two independent log-concaverandom 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 thatjoint densities formed by multiplying two probability densities (e.g. thenormal-gamma distribution, which always has a shape parameter ≥ 1) will be log-concave. This property is heavily used in general-purposeGibbs sampling programs such asBUGS andJAGS, which are thereby able to useadaptive rejection sampling over a wide variety ofconditional distributions derived from the product of other distributions.
• If a density is log-concave, so is itssurvival function.^[3]
• If a density is log-concave, it has a monotonehazard rate (MHR), and is aregular distribution since the derivative of the logarithm of the survival function is the negative hazard rate, and by concavity is monotone i.e.

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

• logarithmically concave sequence
• logarithmically concave measure
• logarithmically convex function
• convex function

• 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.

• Pfanzagl, Johann; with the assistance of R. Hamböker (1994).Parametric Statistical Theory. Walter de Gruyter.ISBN 3-11-013863-8.MR 1291393.

• 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.

• Mathematical analysis
• Convex analysis

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