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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
: <math>
    f(\theta x + (1 - \theta) y) \geq f(x)^{\theta} f(y)^{1 - \theta}
  </math>
for all {{math|''x'',''y'' ∈ dom ''f''}} and {{math|0&nbsp;<&nbsp;''&theta;''&nbsp;<&nbsp;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,
: <math>
  \log  f(\theta x + (1 - \theta) y) \geq \theta \log f(x) + (1-\theta) \log f(y)
  </math>
for all {{math|''x'',''y'' ∈ dom ''f''}} and {{math|0&nbsp;<&nbsp;''&theta;''&nbsp;<&nbsp;1}}.


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]].


Similarly, a function is '''[[log-convex]]''' if satisfies the reverse inequality
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: <math>
    f(\theta x + (1 - \theta) y) \leq f(x)^{\theta} f(y)^{1 - \theta}
  </math>
for all {{math|''x'',''y'' ∈ dom ''f''}} and {{math|0&nbsp;<&nbsp;''&theta;''&nbsp;<&nbsp;1}}.
 
==Properties==
* A positive log-concave function is also [[Quasi-concave_function | 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]] {{math|''f''(''x'')}}&nbsp;=&nbsp;{{math|exp(&minus;x<sup>2</sup>/2)}} which is log-concave since {{math|log ''f''(''x'')}}&nbsp;=&nbsp;{{math|&minus;''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''}}|&nbsp;>&nbsp;1:
 
::<math>f''(x)=e^{-\frac{x^2}{2}} (x^2-1) \nleq 0</math>
 
* A twice differentiable, nonnegative function with a convex domain is log-concave if and only if for all {{math|''x''}} satisfying {{math|''f''(''x'')&nbsp;>&nbsp;0}},
 
::<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>
 
:i.e.
 
::<math>f(x)\nabla^2f(x) - \nabla f(x)\nabla f(x)^T</math> is
 
:[[positive-definite matrix|negative semi-definite]]. For functions of one variable, this condition simplifies to
 
::<math>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 {{math|''f''}} and {{math|''g''}} are log-concave functions, then {{math|log&nbsp;''f''}} and {{math|log&nbsp;''g''}} are concave by definition. Therefore
 
::<math>\log\,f(x) + \log\,g(x) = \log(f(x)g(x))</math>
 
:is concave, and hence also {{math|''f''&nbsp;''g''}} is log-concave.
 
* [[marginal distribution|Marginals]]: if {{math|''f''(''x'',''y'')}}&nbsp;:&nbsp;{{math|'''R'''<sup>''n''+''m''</sup>&nbsp;&rarr;&nbsp;'''R'''}} is log-concave, then
 
::<math>g(x)=\int f(x,y) dy</math>
 
:is log-concave (see [[Prékopa–Leindler inequality]]).
 
* This implies that [[convolution]] preserves log-concavity, since {{math|''h''(''x'',''y'')}}&nbsp;=&nbsp;{{math|''f''(''x''-''y'')&nbsp;''g''(''y'')}} is log-concave if {{math|''f''}} and {{math|''g''}} are log-concave, and therefore
 
::<math>(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]].
 
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>
*The [[normal distribution]] and [[multivariate normal distribution]]s.
*The [[exponential distribution]].
*The [[uniform distribution (continuous)|uniform distribution]] over any [[convex set]].
*The [[logistic distribution]].
*The [[extreme value distribution]].
*The [[Laplace distribution]].
*The [[chi distribution]].
*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>
*The [[Dirichlet distribution]], where all parameters are >= 1.<ref name="prekopa"/>
*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 log-concave [[random variable]]s 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 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.
 
==Notes==
{{Reflist}}
 
==References==
 
* {{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}}
 
* {{cite book|title=Unimodality, convexity, and applications
|last1=Dharmadhikari|first1=Sudhakar
|last2=Joag-Dev
|first2=Kumar|
|series=Probability and Mathematical Statistics
|publisher=Academic Press, Inc.
|location=Boston, MA
|year=1988
|pages=xiv+278
|isbn=0-12-214690-5|mr=954608}} 
 
* {{cite book|title=Parametric Statistical Theory | last1=Pfanzagl | first1=Johann
|authorlink= <!-- Johann Pfanzagl -->
|last2=with the assistance of R. Hamböker
|year=1994|
|publisher=Walter de Gruyter
|isbn=3-11-013863-8
|mr=1291393}}
 
* {{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 -->
|series=Mathematics in Science and Engineering|
|volume=187
|publisher=Academic Press, Inc.
|location=Boston, MA
|year=1992|pages=xiv+467 pp.
|isbn=0-12-549250-2
|mr=1162312}}
 
==See also==
*[[logarithmically concave sequence]]
*[[logarithmically concave measure]]
*[[logarithmically convex function]]
*[[convex function]]
 
{{DEFAULTSORT:Logarithmically Concave Function}}
[[Category:Mathematical analysis]]
[[Category:Convex analysis]]

Latest revision as of 09:45, 17 December 2014


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