Logarithmically concave function: Difference between revisions

Revision as of 00:12, 18 April 2019

Revision as of 00:13, 18 April 2019

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*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 >= 1) will be log-concave. This property is heavily used in general-purpose [[Gibbs sampling]] programs such as [[Bayesian inference using Gibbs sampling|BUGS]] and [[Just another Gibbs sampler|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.

==See also==

*[[logarithmically concave sequence]]

*[[logarithmically concave measure]]

*[[logarithmically convex function]]

*[[convex function]]

==Notes==

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==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

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|isbn=0-12-549250-2

|mr=1162312}}

==See also==

*[[logarithmically concave sequence]]

*[[logarithmically concave measure]]

*[[logarithmically convex function]]

*[[convex function]]

{{DEFAULTSORT:Logarithmically Concave Function}}