Logarithmically concave function: Difference between revisions

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| last2=Molyboha | first2=Anton

| last3=Zabarankin | first3=Michael

| date=May 2009

| title=Maximum Entropy Principle with General Deviation Measures

| journal=[[Mathematics of Operations Research]]

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| issue=2

| pages=445–467

| doi=10.1287/moor.1090.0377}}</ref>

| 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 |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>

*Thethe [[normal distribution]] and [[multivariate normal distribution]]s.,

*Thethe [[exponential distribution]].,

*Thethe [[uniform distribution (continuous)|uniform distribution]] over any [[convex set]].,

*Thethe [[logistic distribution]].,

*Thethe [[extreme value distribution]].,

*Thethe [[Laplace distribution]].,

*Thethe [[chi distribution]].,

*Thethe [[hyperbolic secant distribution]].,

*Thethe [[Wishart distribution]], whereif ''n'' >=≥ ''p'' + 1.,<ref name="prekopa">{{cite journal | last1 = Prékopa | first1 = András | author-link = András Prékopa | year = 1971 | title = Logarithmic concave measures with application to stochastic programming | journal = [[Acta Scientiarum Mathematicarum]] | volume = 32 | issue = 3-4 | pages = 301–316 | url = http://rutcor.rutgers.edu/~prekopa/SCIENT1.pdf}}</ref>

*Thethe [[Dirichlet distribution]], whereif all parameters are >=≥ 1.,<ref name="prekopa"/>

*Thethe [[gamma distribution]] if the shape parameter is >=≥ 1.,

*Thethe [[chi-square distribution]] if the number of degrees of freedom is >=≥ 2.,

*Thethe [[beta distribution]] if both shape parameters are >=≥ 1., and

*Thethe [[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:

*Thethe [[Student's t-distribution]].,

*Thethe [[Cauchy distribution]].,

*Thethe [[Pareto distribution]].,

*Thethe [[log-normal distribution]]., and

*Thethe [[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:

*Thethe [[log-normal distribution]].,

*Thethe [[Pareto distribution]].,

*Thethe [[Weibull distribution]] when the shape parameter < 1., and

*Thethe [[gamma distribution]] when the shape parameter < 1.

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 >=≥ 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]] (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.