Content deleted Content added
→Log-concave distributions: Added the Hyperbolic secant distribution |
→Log-concave distributions: I add a supplement that binomial distribution is also a log concave distribution |
||
| (18 intermediate revisions by 15 users not shown) | |||
Line 1:
{{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>
Line 11 ⟶ 12:
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
: <math>
f(\theta x + (1 - \theta) y) \leq f(x)^{\theta} f(y)^{1 - \theta}
Line 19 ⟶ 20:
==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'')}} = {{math|exp(−''x''<sup>2</sup>/2)}} which is log-concave since {{math|log ''f''(''x'')}} = {{math|−''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''}}| > 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'') > 0}},
::<math>f(x)\nabla^2f(x) \preceq \nabla f(x)\nabla f(x)^T</math>,<ref name=":0">{{cite book |
:i.e.
Line 56 ⟶ 57:
==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
| | | | | title=Maximum Entropy Principle with General Deviation Measures | journal=[[Mathematics of Operations Research | volume=34 | | | doi=10.1287/moor.1090.0377
As it happens, many common [[probability distribution]]s are log-concave. Some examples:<ref>See {{cite journal |first=Mark |last=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 |url=http://www.econ.ucsb.edu/~tedb/Theory/delta.pdf }}</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 |
*The [[exponential distribution]].▼
*
*
*
*
*
*
*The [[Wishart distribution]], where ''n'' >= ''p'' + 1.<ref name="prekopa">{{cite journal | last1 = Prékopa | first1 = András | year = 1971 | title = Logarithmic concave measures with application to stochastic programming | url = | journal = Acta Scientiarum Mathematicarum | volume = 32 | issue = | pages = 301–316 }}</ref>▼
*
*
▲*
*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:
*
*
*
*
*
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 following are among the properties of log-concave distributions:
Line 92 ⟶ 105:
*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
* 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==
Line 104 ⟶ 120:
==References==
* {{cite book|
* {{cite book|title=Unimodality, convexity, and applications
|last1=Dharmadhikari|first1=Sudhakar
Line 116 ⟶ 132:
* {{cite book|title=Parametric Statistical Theory | last1=Pfanzagl | first1=Johann
|
|last2=with the assistance of R. Hamböker
|year=1994|publisher=Walter de Gruyter
Line 122 ⟶ 138:
|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.
|series=Mathematics in Science and Engineering|volume=187
|publisher=Academic Press, Inc.
| |||