Logarithmically concave function: Difference between revisions

Revision as of 13:06, 17 May 2017

Revision as of 09:40, 20 June 2017

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::<math>f''(x)=e^{-\frac{x^2}{2}} (x^2-1) \nleq 0</math>

* From above two points, [[Concave_functionConcave function|concavity]] <math>\Rightarrow</math> log-concavity <math>\Rightarrow</math> [[Quasiconcave_functionQuasiconcave function|quasiconcavity]].

* 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}},

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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'' .<ref name="Grechuk1">Grechuk, B., Molyboha, A., Zabarankin, M. (2009) [https://www.researchgate.net/publication/220442393_Maximum_Entropy_Principle_with_General_Deviation_Measures Maximum Entropy Principle with General Deviation Measures], Mathematics of Operations Research 34(2), 445--467, 2009.</ref>.

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.