Editing Logarithmically concave function

{{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>
    f(\theta x + (1 - \theta) y) \geq f(x)^{\theta} f(y)^{1 - \theta}
  </math>
for all {{math|''x'',''y'' ∈ dom ''f''}} and {{math|0&nbsp;<&nbsp;''&theta;''&nbsp;<&nbsp;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&nbsp;<&nbsp;''&theta;''&nbsp;<&nbsp;1}}.

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 ''[[log-convex]]'' if it satisfies the reverse inequality
: <math>
    f(\theta x + (1 - \theta) y) \leq f(x)^{\theta} f(y)^{1 - \theta}
  </math>
for all {{math|''x'',''y'' ∈ dom ''f''}} and {{math|0&nbsp;<&nbsp;''&theta;''&nbsp;<&nbsp;1}}.

==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'')}}&nbsp;=&nbsp;{{math|exp(&minus;''x''<sup>2</sup>/2)}} which is log-concave since {{math|log ''f''(''x'')}}&nbsp;=&nbsp;{{math|&minus;''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''}}|&nbsp;>&nbsp;1:

::<math>f''(x)=e^{-\frac{x^2}{2}} (x^2-1) \nleq 0</math>
* From above two points, [[Concave function|concavity]] <math>\Rightarrow</math> log-concavity <math>\Rightarrow</math> [[Quasiconcave 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}},

::<math>f(x)\nabla^2f(x) \preceq \nabla f(x)\nabla f(x)^T</math>,<ref name=":0">{{cite book |first1=Stephen |last1=Boyd |author-link=Stephen P. Boyd |first2=Lieven |last2=Vandenberghe |chapter=Log-concave and log-convex functions |title=Convex Optimization |publisher=Cambridge University Press |year=2004 |isbn=0-521-83378-7 |chapter-url=https://web.stanford.edu/~boyd/cvxbook/ |pages=104–108 }}</ref>

:i.e.

::<math>f(x)\nabla^2f(x) - \nabla f(x)\nabla f(x)^T</math> is

:[[positive-definite matrix|negative semi-definite]]. For functions of one variable, this condition simplifies to

::<math>f(x)f''(x) \leq (f'(x))^2</math>

==Operations preserving log-concavity==

* Products: The product of log-concave functions is also log-concave. Indeed, if {{math|''f''}} and {{math|''g''}} are log-concave functions, then {{math|log&nbsp;''f''}} and {{math|log&nbsp;''g''}} are concave by definition. Therefore

::<math>\log\,f(x) + \log\,g(x) = \log(f(x)g(x))</math>

:is concave, and hence also {{math|''f''&nbsp;''g''}} is log-concave.

* [[marginal distribution|Marginals]]: if {{math|''f''(''x'',''y'')}}&nbsp;:&nbsp;{{math|'''R'''<sup>''n''+''m''</sup>&nbsp;&rarr;&nbsp;'''R'''}} is log-concave, then

::<math>g(x)=\int f(x,y) dy</math>

:is log-concave (see [[Prékopa–Leindler inequality]]).

* This implies that [[convolution]] preserves log-concavity, since {{math|''h''(''x'',''y'')}}&nbsp;=&nbsp;{{math|''f''(''x''-''y'')&nbsp;''g''(''y'')}} is log-concave if {{math|''f''}} and {{math|''g''}} are log-concave, and therefore

::<math>(f*g)(x)=\int f(x-y)g(y) dy = \int h(x,y) dy</math>

:is log-concave.

==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=Bogdan
| last2=Molyboha | first2=Anton
| last3=Zabarankin | first3=Michael
| date=May 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
| 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>
*the [[normal distribution]] and [[multivariate normal distribution]]s,
*the [[exponential distribution]],
*the [[uniform distribution (continuous)|uniform distribution]] over any [[convex set]],
*the [[logistic distribution]],
*the [[extreme value distribution]],
*the [[Laplace distribution]],
*the [[chi distribution]],
*the [[hyperbolic secant distribution]],
*the [[Wishart distribution]], if ''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>
*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:
*the [[Student's t-distribution]],
*the [[Cauchy distribution]],
*the [[Pareto distribution]],
*the [[log-normal distribution]], and
*the [[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:
*the [[log-normal distribution]],
*the [[Pareto distribution]],
*the [[Weibull distribution]] when the shape parameter < 1, and
*the [[gamma distribution]] when the shape parameter < 1.

The following are among the properties of log-concave distributions:
*If a density is log-concave, so is its [[cumulative distribution function]] (CDF).
*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.
::<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==
{{Reflist}}

==References==
* {{cite book|author-link=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
|last2=Joag-Dev
|first2=Kumar|series=Probability and Mathematical Statistics
|publisher=Academic Press, Inc.
|location=Boston, MA
|year=1988
|pages=xiv+278
|isbn=0-12-214690-5|mr=954608}}

* {{cite book|title=Parametric Statistical Theory | last1=Pfanzagl | first1=Johann 
|author-link= <!-- Johann Pfanzagl --> 
|last2=with the assistance of R. Hamböker 
|year=1994|publisher=Walter de Gruyter
|isbn=3-11-013863-8
|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.<!-- |author-link2=Frank Proschan -->
|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}}

{{DEFAULTSORT:Logarithmically Concave Function}}
[[Category:Mathematical analysis]]
[[Category:Convex analysis]]