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{{Short description|Type of mathematical function}}
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>
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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,
: <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 < ''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]].
Similarly, a function is
: <math>
f(\theta x + (1 - \theta) y) \leq f(x)^{\theta} f(y)^{1 - \theta}
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==Properties==
* A log-concave function is also [[Quasi-concave function|quasi-concave]]. This follows from the fact that the logarithm is monotone implying that the [[Level set#Sublevel and superlevel sets|superlevel sets]] of this function are convex.<ref name=":0" />
* 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:
::<math>f''(x)=e^{-\frac{x^2}{2}} (x^2-1) \nleq 0</math>
* From above two points, [[
* 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}},
::<math>f(x)\nabla^2f(x) \preceq \nabla f(x)\nabla f(x)^T</math>,<ref name=":0">{{cite book |first1=Stephen |last1=Boyd
:i.e.
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==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''
| last1=Grechuk | last2=Molyboha | last3=Zabarankin | | title=Maximum Entropy Principle with General Deviation Measures | journal=[[Mathematics of Operations Research | volume=34
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>▼
| issue=2
| pages=445–467
*The [[exponential distribution]].▼
| doi=10.1287/moor.1090.0377
*The [[uniform distribution (continuous)|uniform distribution]] over any [[convex set]].▼
| url=https://www.researchgate.net/profile/Bogdan-Grechuk/publication/220442393_Maximum_Entropy_Principle_with_General_Deviation_Measures/links/59132b61a6fdcc963e7ed4fd/Maximum-Entropy-Principle-with-General-Deviation-Measures.pdf}}</ref>
▲As it happens, many common [[probability distribution]]s are log-concave. Some examples:<ref name=":1">See {{cite journal |first1=Mark |last1=Bagnoli
*
*
*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]], if 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, and
*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:
*
*
*
*
*
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:
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*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 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
* If a density is log-concave, so is its [[survival function]].<ref name=":1" />
* If a density is log-concave, it has a monotone [[hazard rate]] (MHR), and is a [[Regular distribution (economics)|regular distribution]] since the derivative of the logarithm of the survival function is the negative hazard rate, and by concavity is monotone i.e.
::<math>\frac{d}{dx}\log\left(1-F(x)\right) = -\frac{f(x)}{1-F(x)}</math> which is decreasing as it is the derivative of a concave function.
==See also==▼
*[[logarithmically concave sequence]]▼
*[[logarithmically concave measure]]▼
*[[logarithmically convex function]]▼
*[[convex function]]▼
==Notes==
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==References==
* {{cite book|
▲* {{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
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* {{cite book|title=Parametric Statistical Theory | last1=Pfanzagl | first1=Johann
|
|last2=with the assistance of R. Hamböker
|year=1994|publisher=Walter de Gruyter
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|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.
|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}}
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