↑ arvau

Wikipedia / 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.

Definition of entropy and differential entropy

is a continuous random variable with probability density, then the differential entropy is defined as

is a discrete random variable

This is a special case of more general forms described in the articles Entropy (information theory), Principle of maximum entropy 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 base2 in order to express the entropy in bits; mathematicians and physicists often prefer the natural logarithm, resulting in a unit of "nat"ss for the entropy.

However, the chosen measure is crucial, even though the typical use of the Lebesgue measure is often defended as a "natural" choice: Which measure is chosen determines the entropy and the consequent maximum entropy distribution.

Distributions with measured constants

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 case

is a continuous, closed subset of the real numbers measurable functions moment conditions:

whose density function has the following form:

Lagrange multipliers integrates to unity):

Using the Karush–Kuhn–Tucker conditions

Note that when the moment constraints are equalities (instead of inequalities), that is,

can be dropped, which makes optimization over the Lagrange multipliers unconstrained.

Discrete case

moment conditions

then this distribution has the following shape:

:

is not present in the optimization.

Proof in the case of equality constraints

In the case of equality constraints, this theorem is proved with the calculus of variations and Lagrange multipliers. The constraints can be written as

We consider the functional

are the Lagrange multipliers. The zeroth constraint ensures the second axiom of probability. The entropy attains an extremum when the functional derivative is equal to zero:

),

. It can be verified that this is the maximal solution by checking that the variation around this solution is always negative.

Uniqueness of the maximum

It follows that a distribution satisfying the expectation-constraints and maximising entropy must necessarily have full support — i. e. the distribution is almost everywhere strictly 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).

We now note a series of identities: Via the satisfaction of the expectation-constraints and utilising gradients / directional derivatives, one has

one obtains:

Computing further, one has

Assuming that no non-trivial linear combination of the observables is almost everywhere (a.e.) constant, (which e.g. 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.

Caveats

Note that not all classes of distributions contain a maximum entropy distribution. It is possible that a class contains 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. 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.

Examples

to be the measurable function and

is the maximum entropy probability distribution under the constraint

can be separated into parts.

A table of examples of maximum entropy distributions is given in Lisman (1972) and Park & Bera (2009).

Uniform and piecewise uniform distributions

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 are given a subdivision a=a0 < a1 < ... < ak = b of the interval [a,b] and probabilities p1,...,pk 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 distribution

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.

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.

Specified mean and variance: the normal distribution

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). The same is true when the mean μ and the variance σ2 is specified (the first two moments), since entropy is translation invariant on (∞,∞). Therefore, the assumption of normality imposes the minimal prior structural constraint beyond these moments. (See the differential entropy article for a derivation.)

Discrete distributions with specified mean

Among all the discrete distributions supported on the set {x1,...,xn 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.

with mean μ 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

such that respective maximum entropy distribution is the geometric distribution.

Circular random variables

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 or, equivalently, the circular mean and circular variance are specified.

are specified, the wrapped normal distribution maximizes the entropy.

Maximizer for specified mean, variance and skew

with a specified mean, variance, and skew. However, there is no distribution which achieves this upper bound (see Cover& Thomas (2006: chapter 12)).

However, the maximum entropy is -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 larger than the mean. The skewness, being proportional to the third moment, will be affected more than the lower order moments.

This is a special case of the general case in which the exponential of any odd-order polynomial in x, but when the support is limited to a bounded or semi-bounded interval the upper entropy bound may be achieved (e.g. if x lies in the interval [0,] and <0, the exponential distribution will result).

Maximizer for specified mean and deviation risk measure

Every distribution with log-concave density is a maximal entropy distribution with specified mean and deviation risk measure .

is:

Other examples

be included in the support of the probability density, which is listed in the fourth column.

Several listed examples (Bernoulli, geometric, exponential, Laplace, Pareto) 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.

is the gamma function is the digamma function is the beta function is the Euler-Mascheroni constant.The maximum entropy principle can be used to upper bound the entropy of statistical mixtures.

See also

Notes

Citations

References