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Maximum entropy probability distribution

In statistics and information theory, a maximum entropy probability distribution has entropy that is at least as great as that of all other members of a specified class of probability distributions. According to the principle of maximum entropy, if nothing is known about a distribution except that it belongs to a certain class (usually defined in terms of specified properties or measures), then the distribution with the largest entropy should be chosen as the least-informative default. The motivation is twofold: first, maximizing entropy minimizes the amount of prior information built into the distribution; second, many physical systems tend to move towards maximal entropy configurations over time.

Contents

Definition of entropy and differential entropyEdit

If X is a discrete random variable with distribution given by

 

then the entropy of X is defined as

 

If X is a continuous random variable with probability density p(x), then the differential entropy of X is defined as[1][2][3]

 

p(x) log p(x) is understood to be zero whenever p(x) = 0.

This is a special case of more general forms described in the articles Entropy (information theory), Principle of maximum entropy, and differential entropy. In connection with maximum entropy distributions, this is the only one needed, because maximizing   will also maximize the more general forms.

The base of the logarithm is not important as long as the same one is used consistently: change of base merely results in a rescaling of the entropy. Information theorists may prefer to use base 2 in order to express the entropy in bits; mathematicians and physicists will often prefer the natural logarithm, resulting in a unit of nats for the entropy.

Distributions with measured constantsEdit

Many statistical distributions of applicable interest are those for which the moments or other measurable quantities are constrained to be constants. The following theorem by Ludwig Boltzmann gives the form of the probability density under these constraints.

Continuous versionEdit

Suppose S is a closed subset of the real numbers R and we choose to specify n measurable functions f1,...,fn and n numbers a1,...,an. We consider the class C of all real-valued random variables which are supported on S (i.e. whose density function is zero outside of S) and which satisfy the n expected value conditions

 

If there is a member in C whose density function is positive everywhere in S, and if there exists a maximal entropy distribution for C, then its probability density p(x) has the following shape:

 

where the constants c and λj have to be determined so that the integral of p(x) over S is 1 and the above conditions for the expected values are satisfied. Conversely, if constants c and λj like this can be found, then p(x) is indeed the density of the (unique) maximum entropy distribution for our class C.

Discrete versionEdit

Suppose S = {x1,x2,...} is a (finite or infinite) discrete subset of the reals and we choose to specify n functions f1,...,fn and n numbers a1,...,an. We consider the class C of all discrete random variables X which are supported on S and which satisfy the n conditions

 

If there exists a member of C which assigns positive probability to all members of S and if there exists a maximum entropy distribution for C, then this distribution has the following shape:

 

where the constants c and λj have to be determined so that the sum of the probabilities is 1 and the above conditions for the expected values are satisfied. Conversely, if constants c and λj like this can be found, then the above distribution is indeed the maximum entropy distribution for our class C.

ProofEdit

This theorem is proved with the calculus of variations and Lagrange multipliers. The constraints can be written as

 

We consider the functional

 

where the   are the Lagrange multipliers. The zeroth constraint ensures the second axiom of probability. The other constraints are that the measurements of the function are given constants up to order  . The entropy attains an extremum when the functional derivative is equal to zero:

 

It is an exercise for the reader that this extremum is indeed a maximum. Therefore, the maximum entropy probability distribution in this case must be of the form

 

The proof of the discrete version is essentially the same.

Uniqueness of MaximumEdit

Suppose  ,   are distributions satisfying the expectation-constraints. Letting   and considering the distribution   it is clear that this distribution satisfies the expectation-constraints remain and furthermore has as support  . From basic facts about entropy, it holds that  . Taking limits   and   respectively yields  .

It follows that a distribution satisfying the expectation-constraints and maximising entropy must necessarily have full support—i. e. the distribution is almost everywhere positive. It follows that the maximising distribution must be an internal point in the space of distributions satisfying the expectation-constraints, that is, it must be a local extreme. Thus it suffices to show that the local extreme is unique, in order to show both that the entropy-maximising distribution is unique (and this also shows that the local extreme is the global maximum).

Suppose   are local extremes. Reformulating the above computations these are characterised by parameters   via   and similarly for  , where  . We now note a series of identities: Via the satisfaction of the expectation-constraints and utilising gradients/directional derivatives, one has   and similarly for  . Letting   one obtains:

 

where   for some  . Computing further one has

 

where   is similar to the distribution above, only parameterised by  . Assuming that no non-trivial linear combination of the observables is a.e. constant, (which e. g. holds if the observables are independent and not a.e. constant), it holds that   has non-zero variance, unless  . By the above equation it is thus clear, that the latter must be the case. Hence  , so the parameters characterising the local extrema   are identical, which means that the distributions themselves are identical. Thus, the local extreme is unique and by the above discussion, the maximum is unique—provided a local extreme actually exists.

CaveatsEdit

Note that not all classes of distributions contain a maximum entropy distribution. It is possible that a class contain distributions of arbitrarily large entropy (e.g. the class of all continuous distributions on R with mean 0 but arbitrary standard deviation), or that the entropies are bounded above but there is no distribution which attains the maximal entropy (e.g. the class of all continuous distributions X on R with E(X) = 0 and E(X2) = E(X3) = 1 (See Cover, Ch 12)). It is also possible that the expected value restrictions for the class C force the probability distribution to be zero in certain subsets of S. In that case our theorem doesn't apply, but one can work around this by shrinking the set S.

ExamplesEdit

Every probability distribution is trivially a maximum entropy probability distribution under the constraint that the distribution have its own entropy. To see this, rewrite the density as   and compare to the expression of the theorem above. By choosing   to be the measurable function and

 

to be the constant,   is the maximum entropy probability distribution under the constraint

 .

Nontrivial examples are distributions that are subject to multiple constraints that are different from the assignment of the entropy. These are often found by starting with the same procedure   and finding that   can be separated into parts.

A table of examples of maximum entropy distributions is given in Lisman (1972) [4] and Park & Bera (2009)[5]

Uniform and piecewise uniform distributionsEdit

The uniform distribution on the interval [a,b] is the maximum entropy distribution among all continuous distributions which are supported in the interval [a, b], and thus the probability density is 0 outside of the interval. This uniform density can be related to Laplace's principle of indifference, sometimes called the principle of insufficient reason. More generally, if we're given a subdivision a=a0 < a1 < ... < ak = b of the interval [a,b] and probabilities p1,...,pk which add up to one, then we can consider the class of all continuous distributions such that

 

The density of the maximum entropy distribution for this class is constant on each of the intervals [aj-1,aj). The uniform distribution on the finite set {x1,...,xn} (which assigns a probability of 1/n to each of these values) is the maximum entropy distribution among all discrete distributions supported on this set.

Positive and specified mean: the exponential distributionEdit

The exponential distribution, for which the density function is

 

is the maximum entropy distribution among all continuous distributions supported in [0,∞] that have a specified mean of 1/λ.

Specified variance: the normal distributionEdit

The normal distribution N(μ,σ2), for which the density function is

 

has maximum entropy among all real-valued distributions supported on (-∞,∞) with a specified variance σ2 (a particular moment). Therefore, the assumption of normality imposes the minimal prior structural constraint beyond this moment. (See the differential entropy article for a derivation.)

In the case of distributions supported on [0,∞], the maximum entropy distribution depends on relationships between the first and second moments. In specific cases, it may be the exponential distribution, or may be another distribution, or may be undefinable.[6]

Discrete distributions with specified meanEdit

Among all the discrete distributions supported on the set {x1,...,xn} with a specified mean μ, the maximum entropy distribution has the following shape:

 

where the positive constants C and r can be determined by the requirements that the sum of all the probabilities must be 1 and the expected value must be μ.

For example, if a large number N of dice are thrown, and you are told that the sum of all the shown numbers is S. Based on this information alone, what would be a reasonable assumption for the number of dice showing 1, 2, ..., 6? This is an instance of the situation considered above, with {x1,...,x6} = {1,...,6} and μ = S/N.

Finally, among all the discrete distributions supported on the infinite set {x1,x2,...} with mean μ, the maximum entropy distribution has the shape:

 

where again the constants C and r were determined by the requirements that the sum of all the probabilities must be 1 and the expected value must be μ. For example, in the case that xk = k, this gives

 

such that respective maximum entropy distribution is the geometric distribution.

Circular random variablesEdit

For a continuous random variable   distributed about the unit circle, the Von Mises distribution maximizes the entropy when the real and imaginary parts of the first circular moment are specified[7] or, equivalently, the circular mean and circular variance are specified.

When the mean and variance of the angles   modulo   are specified, the wrapped normal distribution maximizes the entropy.[7]

Maximizer for specified mean, variance and skewEdit

There exists an upper bound on the entropy of continuous random variables on   with a specified mean, variance, and skew. However, there is no distribution which achieves this upper bound, because   is unbounded except when   (see Cover & Thomas (2006: chapter 12)).[clarification needed (explanation)]

However, the maximum entropy is ε-achievable: a distribution's entropy can be arbitrarily close to the upper bound. Start with a normal distribution of the specified mean and variance. To introduce a positive skew, perturb the normal distribution upward by a small amount at a value many σ larger than the mean. The skewness, being proportional to the third moment, will be affected more than the lower order moments.

Maximizer for specified mean and deviation risk measureEdit

Every distribution with log-concave density is a maximal entropy distribution with specified mean μ and Deviation risk measure D.[8]

In particular, the maximal entropy distribution with specified mean   and deviation   is:

  • The normal distribution  , if   is the standard deviation;
  • The Laplace distribution, if   is the average absolute deviation;[4]
  • The distribution with density of the form   if   is the standard lower semi-deviation, where  , and a,b,c are constants.[8]

Other examplesEdit

In the table below, each listed distribution maximizes the entropy for a particular set of functional constraints listed in the third column, and the constraint that x be included in the support of the probability density, which is listed in the fourth column.[4][5] Several examples (Bernoulli, geometric, exponential, Laplace, Pareto) listed are trivially true because their associated constraints are equivalent to the assignment of their entropy. They are included anyway because their constraint is related to a common or easily measured quantity. For reference,   is the gamma function,   is the digamma function,   is the beta function, and γE is the Euler-Mascheroni constant.

Table of probability distributions and corresponding maximum entropy constraints
Distribution Name Probability density/mass function Maximum Entropy Constraint Support
Uniform (discrete)   None  
Uniform (continuous)   None  
Bernoulli      
Geometric      
Exponential      
Laplace      
Asymmetric Laplace      
Pareto      
Normal      
von Mises      
Rayleigh      
Beta   for    
 
 
Cauchy      
Chi      
Chi-squared      
Erlang      
Gamma      
Lognormal      
Maxwell–Boltzmann      
Weibull      
Multivariate normal  
 
   
Binomial    [9]  
Poisson    [9]  

See alsoEdit

NotesEdit

  1. ^ Williams, D. (2001), Weighing the Odds, Cambridge University Press, ISBN 0-521-00618-X (pages 197-199).
  2. ^ Bernardo, J. M., Smith, A. F. M. (2000), Bayesian Theory, Wiley. ISBN 0-471-49464-X (pages 209, 366)
  3. ^ O'Hagan, A. (1994), Kendall's Advanced Theory of Statistics, Vol 2B, Bayesian Inference, Edward Arnold. ISBN 0-340-52922-9 (Section 5.40)
  4. ^ a b c Lisman, J. H. C.; van Zuylen, M. C. A. (1972). "Note on the generation of most probable frequency distributions". Statistica Neerlandica. 26 (1): 19–23. doi:10.1111/j.1467-9574.1972.tb00152.x. 
  5. ^ a b Park, Sung Y.; Bera, Anil K. (2009). "Maximum entropy autoregressive conditional heteroskedasticity model" (PDF). Journal of Econometrics. Elsevier: 219–230. Retrieved 2011-06-02. 
  6. ^ Dowson, D.; Wragg, A. (September 1973). "Maximum-entropy distributions having prescribed first and second moments (Corresp.)". IEEE Transactions on Information Theory. 19 (5): 689–693. doi:10.1109/tit.1973.1055060. ISSN 0018-9448. 
  7. ^ a b Jammalamadaka, S. Rao; SenGupta, A. (2001). Topics in circular statistics. New Jersey: World Scientific. ISBN 981-02-3778-2. Retrieved 2011-05-15. 
  8. ^ a b Grechuk, B., Molyboha, A., Zabarankin, M. (2009) Maximum Entropy Principle with General Deviation Measures, Mathematics of Operations Research 34(2), 445--467, 2009.
  9. ^ a b Harremös, Peter (2001), "Binomial and Poisson distributions as maximum entropy distributions", IEEE Transactions on Information Theory, 47 (5): 2039-2041, doi:10.1109/18.930936 .

ReferencesEdit