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| In [[convex analysis]], a [[non-negative]] function {{math|''f'' : '''R'''<sup>''n''</sup> → '''R'''<sub>+</sub>}} is '''logarithmically concave''' (or '''log-concave''' for short) if its [[domain of a function|domain]] is a [[convex set]], and if it satisfies the inequality
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| : <math>
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| f(\theta x + (1 - \theta) y) \geq f(x)^{\theta} f(y)^{1 - \theta}
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| </math>
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| for all {{math|''x'',''y'' ∈ dom ''f''}} and {{math|0 < ''θ'' < 1}}. If {{math|''f''}} is strictly positive, this is equivalent to saying that the [[logarithm]] of the function, {{math|log ∘ ''f''}}, is [[concave function|concave]]; that is,
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| : <math>
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| \log f(\theta x + (1 - \theta) y) \geq \theta \log f(x) + (1-\theta) \log f(y)
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| </math>
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| for all {{math|''x'',''y'' ∈ dom ''f''}} and {{math|0 < ''θ'' < 1}}.
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| Examples of log-concave functions are the 0-1 [[indicator function]]s of convex sets (which requires the more flexible definition), and the [[Gaussian function]].
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| Similarly, a function is '''[[log-convex]]''' if satisfies the reverse inequality
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| : <math>
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| f(\theta x + (1 - \theta) y) \leq f(x)^{\theta} f(y)^{1 - \theta}
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| </math>
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| for all {{math|''x'',''y'' ∈ dom ''f''}} and {{math|0 < ''θ'' < 1}}.
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| ==Properties==
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| * A positive log-concave function is also [[Quasi-concave_function | quasi-concave]].
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| * 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]] {{math|''f''(''x'')}} = {{math|exp(−x<sup>2</sup>/2)}} which is log-concave since {{math|log ''f''(''x'')}} = {{math|−''x''<sup>2</sup>/2}} is a concave function of {{math|''x''}}. But {{math|''f''}} is not concave since the second derivative is positive for |{{math|''x''}}| > 1:
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| ::<math>f''(x)=e^{-\frac{x^2}{2}} (x^2-1) \nleq 0</math>
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| * A twice differentiable, nonnegative function with a convex domain is log-concave if and only if for all {{math|''x''}} satisfying {{math|''f''(''x'') > 0}},
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| ::<math>f(x)\nabla^2f(x) \preceq \nabla f(x)\nabla f(x)^T</math>, <ref> Stephen Boyd and Lieven Vandenberghe, [http://www.stanford.edu/~boyd/cvxbook/ Convex Optimization] (PDF) p.105</ref>
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| :i.e.
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| ::<math>f(x)\nabla^2f(x) - \nabla f(x)\nabla f(x)^T</math> is
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| :[[positive-definite matrix|negative semi-definite]]. For functions of one variable, this condition simplifies to
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| ::<math>f(x)f''(x) \leq (f'(x))^2</math>
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| ==Operations preserving log-concavity==
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| * Products: The product of log-concave functions is also log-concave. Indeed, if {{math|''f''}} and {{math|''g''}} are log-concave functions, then {{math|log ''f''}} and {{math|log ''g''}} are concave by definition. Therefore
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| ::<math>\log\,f(x) + \log\,g(x) = \log(f(x)g(x))</math>
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| :is concave, and hence also {{math|''f'' ''g''}} is log-concave.
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| * [[marginal distribution|Marginals]]: if {{math|''f''(''x'',''y'')}} : {{math|'''R'''<sup>''n''+''m''</sup> → '''R'''}} is log-concave, then
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| ::<math>g(x)=\int f(x,y) dy</math>
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| :is log-concave (see [[Prékopa–Leindler inequality]]).
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| * This implies that [[convolution]] preserves log-concavity, since {{math|''h''(''x'',''y'')}} = {{math|''f''(''x''-''y'') ''g''(''y'')}} is log-concave if {{math|''f''}} and {{math|''g''}} are log-concave, and therefore
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| ::<math>(f*g)(x)=\int f(x-y)g(y) dy = \int h(x,y) dy</math>
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| :is log-concave.
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| ==Log-concave distributions==
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| Log-concave distributions are necessary for a number of algorithms, e.g. [[adaptive rejection sampling]].
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| As it happens, many common [[probability distribution]]s are log-concave. Some examples:<ref>See Mark Bagnoli and Ted Bergstrom (1989), "Log-Concave Probability and Its Applications", University of Michigan.[http://www.econ.ucsb.edu/~tedb/Theory/delta.pdf]</ref>
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| *The [[normal distribution]] and [[multivariate normal distribution]]s.
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| *The [[exponential distribution]].
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| *The [[uniform distribution (continuous)|uniform distribution]] over any [[convex set]].
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| *The [[logistic distribution]].
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| *The [[extreme value distribution]].
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| *The [[Laplace distribution]].
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| *The [[chi distribution]].
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| *The [[Wishart distribution]], where ''n'' >= ''p'' + 1.<ref name="prekopa">András Prékopa (1971), "Logarithmic concave measures with application to stochastic programming". ''Acta Scientiarum Mathematicarum'', 32, pp. 301–316.</ref>
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| *The [[Dirichlet distribution]], where all parameters are >= 1.<ref name="prekopa"/>
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| *The [[gamma distribution]] if the shape parameter is >= 1.
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| *The [[chi-square distribution]] if the number of degrees of freedom is >= 2.
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| *The [[beta distribution]] if both shape parameters are >= 1.
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| *The [[Weibull distribution]] if the shape parameter is >= 1.
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| 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.
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| The following distributions are non-log-concave for all parameters:
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| *The [[Student's t-distribution]].
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| *The [[Cauchy distribution]].
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| *The [[Pareto distribution]].
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| *The [[log-normal distribution]].
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| *The [[F-distribution]].
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| 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:
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| *The [[log-normal distribution]].
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| *The [[Pareto distribution]].
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| *The [[Weibull distribution]] when the shape parameter < 1.
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| *The [[gamma distribution]] when the shape parameter < 1.
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| The following are among the properties of log-concave distributions:
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| *If a density is log-concave, so is its [[cumulative distribution function]] (CDF).
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| *If a multivariate density is log-concave, so is the [[marginal density]] over any subset of variables.
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| *The sum of two log-concave [[random variable]]s is log-concave. This follows from the fact that the convolution of two log-concave functions is log-concave.
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| *The product of two log-concave functions is log-concave. This means that [[joint distribution|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 distribution]]s derived from the product of other distributions.
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * {{cite book|authorlink=Ole Barndorff-Nielsen|last=Barndorff-Nielsen|first=Ole|title=Information and exponential families in statistical theory|series=Wiley Series in Probability and Mathematical Statistics|publisher=John Wiley \& Sons, Ltd.|location=Chichester|year=1978|pages=ix+238 pp.|isbn=0-471-99545-2|mr=489333}}
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| * {{cite book|title=Unimodality, convexity, and applications
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| |last1=Dharmadhikari|first1=Sudhakar
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| |last2=Joag-Dev
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| |first2=Kumar|
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| |series=Probability and Mathematical Statistics
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| |publisher=Academic Press, Inc.
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| |location=Boston, MA
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| |year=1988
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| |pages=xiv+278
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| |isbn=0-12-214690-5|mr=954608}}
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| * {{cite book|title=Parametric Statistical Theory | last1=Pfanzagl | first1=Johann
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| |authorlink= <!-- Johann Pfanzagl -->
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| |last2=with the assistance of R. Hamböker
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| |year=1994|
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| |publisher=Walter de Gruyter
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| |isbn=3-11-013863-8
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| |mr=1291393}}
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| * {{cite book|title=Convex functions, partial orderings, and statistical applications|last1=Pečarić|first1=Josip E.|last2=Proschan|first2=Frank|last3=Tong|first3=Y. L.|<!-- authorlink2=Frank Proschan -->
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| |series=Mathematics in Science and Engineering|
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| |volume=187
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| |publisher=Academic Press, Inc.
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| |location=Boston, MA
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| |year=1992|pages=xiv+467 pp.
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| |isbn=0-12-549250-2
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| |mr=1162312}}
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| ==See also==
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| *[[logarithmically concave sequence]]
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| *[[logarithmically concave measure]]
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| *[[logarithmically convex function]]
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| *[[convex function]]
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| {{DEFAULTSORT:Logarithmically Concave Function}}
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| [[Category:Mathematical analysis]]
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| [[Category:Convex analysis]]
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