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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">{{cite journal |last1=Grechuk |first1=B. |last2=Molyboha |first2=A. |last3=Zabarankin |first3=M. |year=2009 |title=Maximum Entropy Principle with General Deviation Measures |journal=Mathematics of Operations Research |volume=34 |issue=2 |pages=445–467 |doi=10.1287/moor.1090.0377 }}</ref>
As it happens, many common [[probability distribution]]s are log-concave. Some examples:<ref name=":1">See {{cite journal |first1=Mark |last1=Bagnoli |first2=Ted |last2=Bergstrom |year=2005 |title=Log-Concave Probability and Its Applications |journal=Economic Theory |volume=26 |issue=2 |pages=445–469 |doi=10.1007/s00199-004-0514-4 |s2cid=1046688 |url=http://www.econ.ucsb.edu/~tedb/Theory/delta.pdf }}</ref>
*The [[normal distribution]] and [[multivariate normal distribution]]s.
*The [[exponential distribution]].
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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.
* 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|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.
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