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.
If formula_1 is a discrete random variable with distribution given by
then the entropy of formula_1 is defined as
If formula_1 is a continuous random variable with probability density formula_6, then the differential entropy of formula_1 is defined as
The quantity formula_9 is understood to be zero whenever formula_10.
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 formula_11 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.
The choice of the measure formula_12 is however crucial in determining the entropy and the resulting maximum entropy distribution, even though the usual recourse to the Lebesgue measure is often defended as "natural"
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.
Suppose "S" is a closed subset of the real numbers R and we choose to specify "n" measurable functions "f"1...,"f""n" and "n" numbers "a"1...,"a""n". 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" moment 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 we assume that formula_15. The constant formula_16 and the "n" Lagrange multipliers formula_17 solve the constrained optimization problem with formula_18 (this condition ensures that formula_19 integrates to unity):
Using the Karush–Kuhn–Tucker conditions, it can be shown that the optimization problem has a unique solution, because the objective function in the optimization is concave in formula_21.
Note that if the moment conditions are equalities (instead of inequalities), that is, 
then the constraint condition formula_23 is dropped, making the optimization over the Lagrange multipliers unconstrained.
Discrete case.
Suppose formula_24 is a (finite or infinite) discrete subset of the reals and we choose to specify formula_25 functions "f"1...,"f""n" and "n" numbers "a"1...,"a""n". We consider the class "C" of all discrete random variables "X" which are supported on "S" and which satisfy the "n" moment 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 we assume that formula_28 and the constants formula_29 solve the constrained optimization problem with formula_18:
Again, if the moment conditions are equalities (instead of inequalities), then the constraint condition formula_23 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
where formula_35 and formula_36 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 formula_25. 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 (formula_39)
The proof of the discrete version is essentially the same.
Uniqueness of the maximum.
Suppose formula_41, formula_42 are distributions satisfying the expectation-constraints. Letting formula_43 and considering the distribution formula_44 it is clear that this distribution satisfies the expectation-constraints and furthermore has as support formula_45. From basic facts about entropy, it holds that formula_46. Taking limits formula_47 and formula_48 respectively yields formula_49.
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 formula_50 are local extremes. Reformulating the above computations these are characterised by parameters formula_51 via formula_52 and similarly for formula_42, where formula_54. We now note a series of identities: Via the satisfaction of the expectation-constraints and utilising gradients/directional derivatives, one has formula_55 and similarly for formula_56. Letting formula_57 one obtains:
where formula_59 for some formula_60. Computing further one has
where formula_62 is similar to the distribution above, only parameterised by formula_63. "Assuming" that no non-trivial linear combination of the observables is almost everywhere (a.e.) constant, (which "e.g." holds if the observables are independent and not a.e. constant), it holds that formula_64 has non-zero variance, unless formula_65. By the above equation it is thus clear, that the latter must be the case. Hence formula_66, so the parameters characterising the local extrema formula_50 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 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. 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.
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 formula_68 and compare to the expression of the theorem above. By choosing formula_69 to be the measurable function and
to be the constant, formula_6 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 formula_69 and finding that formula_74 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're given a subdivision "a"="a"0 < "a"1 < ... < "a""k" = "b" of the interval ["a","b"] and probabilities "p"1...,"p""k" 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 ["a""j"-1,"a""j"). The uniform distribution on the finite set {"x"1...,"x""n"} (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.
Specified 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 "σ"2 (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.
Discrete distributions with specified mean.
Among all the discrete distributions supported on the set {"x"1...,"x""n"} 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 {"x"1...,"x"6} = {1...,6} and μ = "S"/"N".
Finally, among all the discrete distributions supported on the infinite set formula_79 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 variables.
For a continuous random variable formula_82 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.
When the mean and variance of the angles formula_82 modulo formula_84 are specified, the wrapped normal distribution maximizes the entropy.
Maximizer for specified mean, variance and skew.
There exists an upper bound on the entropy of continuous random variables on formula_85 with a specified mean, variance, and skew. However, there is "no distribution which achieves this upper bound", because formula_86 is unbounded except when formula_87 (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.
Maximizer for specified mean and deviation risk measure.
Every distribution with log-concave density is a maximal entropy distribution with specified mean "μ" and Deviation risk measure "D".
In particular, the maximal entropy distribution with specified mean formula_88 and deviation formula_89 is:
Other examples.
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. 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, formula_96 is the gamma function, formula_97 is the digamma function, formula_98 is the beta function, and is the Euler-Mascheroni constant.