The principle of maximum entropy states that the probability distribution which best represents the current state of knowledge about a system is the one with largest entropy , in the context of precisely stated prior data (such as a proposition that expresses testable information ).
147-417: Another way of stating this: Take precisely stated prior data or testable information about a probability distribution function. Consider the set of all trial probability distributions that would encode the prior data. According to this principle, the distribution with maximal information entropy is the best choice. The principle was first expounded by E. T. Jaynes in two papers in 1957, where he emphasized
294-1169: A Bernoulli process . The entropy of the unknown result of the next toss of the coin is maximized if the coin is fair (that is, if heads and tails both have equal probability 1/2). This is the situation of maximum uncertainty as it is most difficult to predict the outcome of the next toss; the result of each toss of the coin delivers one full bit of information. This is because H ( X ) = − ∑ i = 1 n p ( x i ) log b p ( x i ) = − ∑ i = 1 2 1 2 log 2 1 2 = − ∑ i = 1 2 1 2 ⋅ ( − 1 ) = 1. {\displaystyle {\begin{aligned}\mathrm {H} (X)&=-\sum _{i=1}^{n}{p(x_{i})\log _{b}p(x_{i})}\\&=-\sum _{i=1}^{2}{{\frac {1}{2}}\log _{2}{\frac {1}{2}}}\\&=-\sum _{i=1}^{2}{{\frac {1}{2}}\cdot (-1)}=1.\end{aligned}}} However, if we know
441-420: A sigma-algebra on X {\displaystyle X} . The entropy of M {\displaystyle M} is H μ ( M ) = sup P ⊆ M H μ ( P ) . {\displaystyle \mathrm {H} _{\mu }(M)=\sup _{P\subseteq M}\mathrm {H} _{\mu }(P).} Finally, the entropy of the probability space
588-709: A smooth manifold of dimension m . {\displaystyle \ m~.} Suppose that we wish to find the stationary points x {\displaystyle \ x\ } of a smooth function f : M → R {\displaystyle \ f:M\to \mathbb {R} \ } when restricted to the submanifold N {\displaystyle \ N\ } defined by g ( x ) = 0 , {\displaystyle \ g(x)=0\ ,} where g : M → R {\displaystyle \ g:M\to \mathbb {R} \ }
735-478: A statistical ensemble . In ordinary language, the principle of maximum entropy can be said to express a claim of epistemic modesty, or of maximum ignorance. The selected distribution is the one that makes the least claim to being informed beyond the stated prior data, that is to say the one that admits the most ignorance beyond the stated prior data. The principle of maximum entropy is useful explicitly only when applied to testable information . Testable information
882-445: A Euclidean space, or even a Riemannian manifold. All appearances of the gradient ∇ {\displaystyle \ \nabla \ } (which depends on a choice of Riemannian metric) can be replaced with the exterior derivative d . {\displaystyle \ \operatorname {d} ~.} Let M {\displaystyle \ M\ } be
1029-490: A coin with probability p of landing on heads and probability 1 − p of landing on tails. The maximum surprise is when p = 1/2 , for which one outcome is not expected over the other. In this case a coin flip has an entropy of one bit . (Similarly, one trit with equiprobable values contains log 2 3 {\displaystyle \log _{2}3} (about 1.58496) bits of information because it can have one of three values.) The minimum surprise
1176-498: A competing measure in structures dual to that of subsets of a universal set. Information is quantified as "dits" (distinctions), a measure on partitions. "Dits" can be converted into Shannon's bits , to get the formulas for conditional entropy, and so on. Another succinct axiomatic characterization of Shannon entropy was given by Aczél , Forte and Ng, via the following properties: It was shown that any function H {\displaystyle \mathrm {H} } satisfying
1323-485: A given constant c {\displaystyle c} . The following is known as the Lagrange multiplier theorem. Let f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } be the objective function, g : R n → R c {\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} ^{c}} be
1470-409: A large number of quanta of probability. Rather than actually carry out, and possibly have to repeat, the rather long random experiment, the protagonist decides to simply calculate and use the most probable result. The probability of any particular result is the multinomial distribution , where is sometimes known as the multiplicity of the outcome. The most probable result is the one which maximizes
1617-428: A logarithm mediates between these two operations. The conditional entropy and related quantities inherit simple relation, in turn. The measure theoretic definition in the previous section defined the entropy as a sum over expected surprisals μ ( A ) ⋅ ln μ ( A ) {\displaystyle \mu (A)\cdot \ln \mu (A)} for an extremal partition. Here
SECTION 10
#17328919770351764-523: A measure of 'uncertainty', 'uninformativeness', or any other imprecisely defined concept. The information entropy function is not assumed a priori , but rather is found in the course of the argument; and the argument leads naturally to the procedure of maximizing the information entropy, rather than treating it in some other way. Suppose an individual wishes to make a probability assignment among m {\displaystyle m} mutually exclusive propositions. He has some testable information, but
1911-634: A message, as in data compression . For example, consider the transmission of sequences comprising the 4 characters 'A', 'B', 'C', and 'D' over a binary channel. If all 4 letters are equally likely (25%), one cannot do better than using two bits to encode each letter. 'A' might code as '00', 'B' as '01', 'C' as '10', and 'D' as '11'. However, if the probabilities of each letter are unequal, say 'A' occurs with 70% probability, 'B' with 26%, and 'C' and 'D' with 2% each, one could assign variable length codes. In this case, 'A' would be coded as '0', 'B' as '10', 'C' as '110', and 'D' as '111'. With this representation, 70% of
2058-533: A natural correspondence between statistical mechanics and information theory . In particular, Jaynes argued that the Gibbsian method of statistical mechanics is sound by also arguing that the entropy of statistical mechanics and the information entropy of information theory are the same concept. Consequently, statistical mechanics should be considered a particular application of a general tool of logical inference and information theory. In most practical cases,
2205-547: A new variable ( λ {\displaystyle \lambda } ) called a Lagrange multiplier (or Lagrange undetermined multiplier ) and study the Lagrange function (or Lagrangian or Lagrangian expression ) defined by L ( x , y , λ ) = f ( x , y ) + λ ⋅ g ( x , y ) , {\displaystyle {\mathcal {L}}(x,y,\lambda )=f(x,y)+\lambda \cdot g(x,y),} where
2352-478: A particular number will win a lottery has high informational value because it communicates the occurrence of a very low probability event. The information content , also called the surprisal or self-information, of an event E {\displaystyle E} is a function which increases as the probability p ( E ) {\displaystyle p(E)} of an event decreases. When p ( E ) {\displaystyle p(E)}
2499-546: A perfectly noiseless channel. Shannon strengthened this result considerably for noisy channels in his noisy-channel coding theorem . Entropy in information theory is directly analogous to the entropy in statistical thermodynamics . The analogy results when the values of the random variable designate energies of microstates, so Gibbs's formula for the entropy is formally identical to Shannon's formula. Entropy has relevance to other areas of mathematics such as combinatorics and machine learning . The definition can be derived from
2646-451: A quantity x which takes values in some interval of the real numbers (all integrals below are over this interval). We assume this information has the form of m constraints on the expectations of the functions f k , i.e. we require our probability density function to satisfy the inequality (or purely equality) moment constraints: where the F k {\displaystyle F_{k}} are observables. We also require
2793-414: A set of axioms establishing that entropy should be a measure of how informative the average outcome of a variable is. For a continuous random variable, differential entropy is analogous to entropy. The definition E [ − log p ( X ) ] {\displaystyle \mathbb {E} [-\log p(X)]} generalizes the above. The core idea of information theory
2940-423: A similar argument. Consider a paraboloid subject to two line constraints that intersect at a single point. As the only feasible solution, this point is obviously a constrained extremum. However, the level set of f {\displaystyle f} is clearly not parallel to either constraint at the intersection point (see Figure 3); instead, it is a linear combination of the two constraints' gradients. In
3087-426: A similar equivalence for these two ways of specifying the testable information in the maximum entropy method. The maximum entropy principle is also needed to guarantee the uniqueness and consistency of probability assignments obtained by different methods, statistical mechanics and logical inference in particular. The maximum entropy principle makes explicit our freedom in using different forms of prior data . As
SECTION 20
#17328919770353234-498: A special case, a uniform prior probability density (Laplace's principle of indifference , sometimes called the principle of insufficient reason), may be adopted. Thus, the maximum entropy principle is not merely an alternative way to view the usual methods of inference of classical statistics, but represents a significant conceptual generalization of those methods. However these statements do not imply that thermodynamical systems need not be shown to be ergodic to justify treatment as
3381-521: A unique Lagrange multiplier λ ⋆ ∈ R c {\displaystyle \lambda _{\star }\in \mathbb {R} ^{c}} such that D f ( x ⋆ ) = λ ⋆ T D g ( x ⋆ ) . {\displaystyle \operatorname {D} f(x_{\star })=\lambda _{\star }^{\mathsf {T}}\operatorname {D} g(x_{\star })~.} (Note that this
3528-409: A variable. The concept of information entropy was introduced by Claude Shannon in his 1948 paper " A Mathematical Theory of Communication ", and is also referred to as Shannon entropy . Shannon's theory defines a data communication system composed of three elements: a source of data, a communication channel , and a receiver. The "fundamental problem of communication" – as expressed by Shannon –
3675-444: Is H μ ( Σ ) {\displaystyle \mathrm {H} _{\mu }(\Sigma )} , that is, the entropy with respect to μ {\displaystyle \mu } of the sigma-algebra of all measurable subsets of X {\displaystyle X} . Consider tossing a coin with known, not necessarily fair, probabilities of coming up heads or tails; this can be modelled as
3822-553: Is σ μ ( A ) = − ln μ ( A ) . {\displaystyle \sigma _{\mu }(A)=-\ln \mu (A).} The expected surprisal of A {\displaystyle A} is h μ ( A ) = μ ( A ) σ μ ( A ) . {\displaystyle h_{\mu }(A)=\mu (A)\sigma _{\mu }(A).} A μ {\displaystyle \mu } -almost partition
3969-1245: Is g ( x , y ) . {\displaystyle \ g(x,y)~.} To summarize ∇ x , y , λ L ( x , y , λ ) = 0 ⟺ { ∇ x , y f ( x , y ) = − λ ∇ x , y g ( x , y ) g ( x , y ) = 0 {\displaystyle \nabla _{x,y,\lambda }{\mathcal {L}}(x,y,\lambda )=0\iff {\begin{cases}\nabla _{x,y}f(x,y)=-\lambda \,\nabla _{x,y}g(x,y)\\g(x,y)=0\end{cases}}} The method generalizes readily to functions on n {\displaystyle n} variables ∇ x 1 , … , x n , λ L ( x 1 , … , x n , λ ) = 0 {\displaystyle \nabla _{x_{1},\dots ,x_{n},\lambda }{\mathcal {L}}(x_{1},\dots ,x_{n},\lambda )=0} which amounts to solving n + 1 equations in n + 1 unknowns. The constrained extrema of f are critical points of
4116-400: Is a stationary point for the Lagrange function (stationary points are those points where the first partial derivatives of L {\displaystyle {\mathcal {L}}} are zero). The assumption ∇ g ≠ 0 {\displaystyle \nabla g\neq 0} is called constraint qualification. However, not all stationary points yield a solution of
4263-1238: Is a regular value . Let N {\displaystyle N} be the submanifold of M {\displaystyle \ M\ } defined by G ( x ) = 0 . {\displaystyle \ G(x)=0~.} x {\displaystyle \ x\ } is a stationary point of f | N {\displaystyle f|_{N}} if and only if ker ( d f x ) {\displaystyle \ \ker(\operatorname {d} f_{x})\ } contains ker ( d G x ) . {\displaystyle \ \ker(\operatorname {d} G_{x})~.} For convenience let L x = d f x {\displaystyle \ L_{x}=\operatorname {d} f_{x}\ } and K x = d G x , {\displaystyle \ K_{x}=\operatorname {d} G_{x}\ ,} where d G {\displaystyle \ \operatorname {d} G} denotes
4410-538: Is a set family P ⊆ P ( X ) {\displaystyle P\subseteq {\mathcal {P}}(X)} such that μ ( ∪ P ) = 1 {\displaystyle \mu (\mathop {\cup } P)=1} and μ ( A ∩ B ) = 0 {\displaystyle \mu (A\cap B)=0} for all distinct A , B ∈ P {\displaystyle A,B\in P} . (This
4557-455: Is a relaxation of the usual conditions for a partition.) The entropy of P {\displaystyle P} is H μ ( P ) = ∑ A ∈ P h μ ( A ) . {\displaystyle \mathrm {H} _{\mu }(P)=\sum _{A\in P}h_{\mu }(A).} Let M {\displaystyle M} be
Principle of maximum entropy - Misplaced Pages Continue
4704-473: Is a smooth function for which 0 is a regular value . Let d f {\displaystyle \ \operatorname {d} f\ } and d g {\displaystyle \ \operatorname {d} g\ } be the exterior derivatives of f {\displaystyle \ f\ } and g {\displaystyle \ g\ } . Stationarity for
4851-481: Is a somewhat conventional thing where λ ⋆ {\displaystyle \lambda _{\star }} is clearly treated as a column vector to ensure that the dimensions match. But, we might as well make it just a row vector without taking the transpose.) The Lagrange multiplier theorem states that at any local maximum (or minimum) of the function evaluated under the equality constraints, if constraint qualification applies (explained below), then
4998-409: Is a statement about a probability distribution whose truth or falsity is well-defined. For example, the statements and (where p 2 {\displaystyle p_{2}} and p 3 {\displaystyle p_{3}} are probabilities of events) are statements of testable information. Given testable information, the maximum entropy procedure consists of seeking
5145-438: Is approximately 0.693 n nats or 0.301 n decimal digits. The meaning of the events observed (the meaning of messages ) does not matter in the definition of entropy. Entropy only takes into account the probability of observing a specific event, so the information it encapsulates is information about the underlying probability distribution , not the meaning of the events themselves. Another characterization of entropy uses
5292-401: Is central to the definition of information entropy. The connection between thermodynamics and what is now known as information theory was first made by Ludwig Boltzmann and expressed by his equation : where S {\displaystyle S} is the thermodynamic entropy of a particular macrostate (defined by thermodynamic parameters such as temperature, volume, energy, etc.), W
5439-419: Is close to 1, the surprisal of the event is low, but if p ( E ) {\displaystyle p(E)} is close to 0, the surprisal of the event is high. This relationship is described by the function log ( 1 p ( E ) ) , {\displaystyle \log \left({\frac {1}{p(E)}}\right),} where log {\displaystyle \log }
5586-408: Is conventionally called the partition function . (The Pitman–Koopman theorem states that the necessary and sufficient condition for a sampling distribution to admit sufficient statistics of bounded dimension is that it have the general form of a maximum entropy distribution.) The λ k parameters are Lagrange multipliers. In the case of equality constraints their values are determined from
5733-415: Is fairly predictable. We can be fairly certain that, for example, 'e' will be far more common than 'z', that the combination 'qu' will be much more common than any other combination with a 'q' in it, and that the combination 'th' will be more common than 'z', 'q', or 'qu'. After the first few letters one can often guess the rest of the word. English text has between 0.6 and 1.3 bits of entropy per character of
5880-401: Is for the receiver to be able to identify what data was generated by the source, based on the signal it receives through the channel. Shannon considered various ways to encode, compress, and transmit messages from a data source, and proved in his source coding theorem that the entropy represents an absolute mathematical limit on how well data from the source can be losslessly compressed onto
6027-548: Is however a source of criticisms of the approach since this dominating measure is in fact arbitrary. The following argument is the result of a suggestion made by Graham Wallis to E. T. Jaynes in 1962. It is essentially the same mathematical argument used for the Maxwell–Boltzmann statistics in statistical mechanics , although the conceptual emphasis is quite different. It has the advantage of being strictly combinatorial in nature, making no reference to information entropy as
Principle of maximum entropy - Misplaced Pages Continue
6174-469: Is interpreted as being proportional to the amount of further Shannon information needed to define the detailed microscopic state of the system, that remains uncommunicated by a description solely in terms of the macroscopic variables of classical thermodynamics, with the constant of proportionality being just the Boltzmann constant . Adding heat to a system increases its thermodynamic entropy because it increases
6321-626: Is necessary and sufficient that the following system of 1 2 m ( m − 1 ) {\displaystyle \ {\tfrac {1}{2}}m(m-1)\ } equations holds: d f x ∧ d g x = 0 ∈ Λ 2 ( T x ∗ M ) {\displaystyle \operatorname {d} f_{x}\wedge \operatorname {d} g_{x}=0\in \Lambda ^{2}(T_{x}^{\ast }M)} where ∧ {\displaystyle \ \wedge \ } denotes
6468-603: Is not necessary to explicitly find the Lagrange multipliers, the numbers λ 1 , … , λ p {\displaystyle \ \lambda _{1},\ldots ,\lambda _{p}\ } such that d f x = ∑ i = 1 p λ i d ( g i ) x . {\displaystyle \ \operatorname {d} f_{x}=\sum _{i=1}^{p}\lambda _{i}\operatorname {d} (g_{i})_{x}~.} In this section, we modify
6615-602: Is not sure how to go about including this information in his probability assessment. He therefore conceives of the following random experiment. He will distribute N {\displaystyle N} quanta of probability (each worth 1 / N {\displaystyle 1/N} ) at random among the m {\displaystyle m} possibilities. (One might imagine that he will throw N {\displaystyle N} balls into m {\displaystyle m} buckets while blindfolded. In order to be as fair as possible, each throw
6762-438: Is now dedicated to the elicitation of maximum entropy priors and links with channel coding . Maximum entropy is a sufficient updating rule for radical probabilism . Richard Jeffrey 's probability kinematics is a special case of maximum entropy inference . However, maximum entropy is not a generalisation of all such sufficient updating rules. Alternatively, the principle is often invoked for model specification: in this case
6909-456: Is only defined for discrete probability spaces. Instead Edwin Jaynes (1963, 1968, 2003) gave the following formula, which is closely related to the relative entropy (see also differential entropy ). where q ( x ), which Jaynes called the "invariant measure", is proportional to the limiting density of discrete points . For now, we shall assume that q is known; we will discuss it further after
7056-1335: Is perpendicular to ∇ f ( x ) {\displaystyle \nabla f(\mathbf {x} )} (otherwise we could increase f {\displaystyle f} by moving along that allowable direction). In other words, ∇ f ( x ) ∈ A ⊥ = S . {\displaystyle \nabla f(\mathbf {x} )\in A^{\perp }=S\,.} Thus there are scalars λ 1 , λ 2 , … , λ M {\displaystyle \lambda _{1},\lambda _{2},\ \dots ,\lambda _{M}} such that ∇ f ( x ) = ∑ k = 1 M λ k ∇ g k ( x ) ⟺ ∇ f ( x ) − ∑ k = 1 M λ k ∇ g k ( x ) = 0 . {\displaystyle \nabla f(\mathbf {x} )=\sum _{k=1}^{M}\lambda _{k}\,\nabla g_{k}(\mathbf {x} )\quad \iff \quad \nabla f(\mathbf {x} )-\sum _{k=1}^{M}{\lambda _{k}\nabla g_{k}(\mathbf {x} )}=0~.} These scalars are
7203-471: Is shown separately rather than being included in g {\displaystyle g} , in which case the constraint is written g ( x , y ) = c , {\displaystyle g(x,y)=c,} as in Figure 1.) We assume that both f {\displaystyle f} and g {\displaystyle g} have continuous first partial derivatives . We introduce
7350-565: Is that it allows the optimization to be solved without explicit parameterization in terms of the constraints. As a result, the method of Lagrange multipliers is widely used to solve challenging constrained optimization problems. Further, the method of Lagrange multipliers is generalized by the Karush–Kuhn–Tucker conditions , which can also take into account inequality constraints of the form h ( x ) ≤ c {\displaystyle h(\mathbf {x} )\leq c} for
7497-530: Is that the "informational value" of a communicated message depends on the degree to which the content of the message is surprising. If a highly likely event occurs, the message carries very little information. On the other hand, if a highly unlikely event occurs, the message is much more informative. For instance, the knowledge that some particular number will not be the winning number of a lottery provides very little information, because any particular chosen number will almost certainly not win. However, knowledge that
SECTION 50
#17328919770357644-408: Is that the constraint gradients at the relevant point are linearly independent. The problem of finding the local maxima and minima subject to constraints can be generalized to finding local maxima and minima on a differentiable manifold M . {\displaystyle \ M~.} In what follows, it is not necessary that M {\displaystyle M} be
7791-414: Is the base of the logarithm used. Common values of b are 2, Euler's number e , and 10, and the corresponding units of entropy are the bits for b = 2 , nats for b = e , and bans for b = 10 . In the case of p ( x ) = 0 {\displaystyle p(x)=0} for some x ∈ X {\displaystyle x\in {\mathcal {X}}} ,
7938-533: Is the expected value operator , and I is the information content of X . I ( X ) {\displaystyle \operatorname {I} (X)} is itself a random variable. The entropy can explicitly be written as: H ( X ) = − ∑ x ∈ X p ( x ) log b p ( x ) , {\displaystyle \mathrm {H} (X)=-\sum _{x\in {\mathcal {X}}}p(x)\log _{b}p(x),} where b
8085-727: Is the logarithm , which gives 0 surprise when the probability of the event is 1. In fact, log is the only function that satisfies а specific set of conditions defined in section § Characterization . Hence, we can define the information, or surprisal, of an event E {\displaystyle E} by I ( E ) = − log 2 ( p ( E ) ) , {\displaystyle I(E)=-\log _{2}(p(E)),} or equivalently, I ( E ) = log 2 ( 1 p ( E ) ) . {\displaystyle I(E)=\log _{2}\left({\frac {1}{p(E)}}\right).} Entropy measures
8232-565: Is the marginal cost of the constraint, and is referred to as the shadow price . Sufficient conditions for a constrained local maximum or minimum can be stated in terms of a sequence of principal minors (determinants of upper-left-justified sub-matrices) of the bordered Hessian matrix of second derivatives of the Lagrangian expression. Suppose we wish to maximize f ( x , y ) = x + y {\displaystyle \ f(x,y)=x+y\ } subject to
8379-405: Is the trace . At an everyday practical level, the links between information entropy and thermodynamic entropy are not evident. Physicists and chemists are apt to be more interested in changes in entropy as a system spontaneously evolves away from its initial conditions, in accordance with the second law of thermodynamics , rather than an unchanging probability distribution. As the minuteness of
8526-407: Is the uniform distribution , The principle of maximum entropy is commonly applied in two ways to inferential problems: The principle of maximum entropy is often used to obtain prior probability distributions for Bayesian inference . Jaynes was a strong advocate of this approach, claiming the maximum entropy distribution represented the least informative distribution. A large amount of literature
8673-553: Is the method of Lagrange multipliers. Note that ∇ λ L ( x , y , λ ) = 0 {\displaystyle \ \nabla _{\lambda }{\mathcal {L}}(x,y,\lambda )=0\ } implies g ( x , y ) = 0 , {\displaystyle \ g(x,y)=0\ ,} as the partial derivative of L {\displaystyle {\mathcal {L}}} with respect to λ {\displaystyle \lambda }
8820-467: Is the number of microstates (various combinations of particles in various energy states) that can yield the given macrostate, and k B is the Boltzmann constant . It is assumed that each microstate is equally likely, so that the probability of a given microstate is p i = 1/ W . When these probabilities are substituted into the above expression for the Gibbs entropy (or equivalently k B times
8967-413: Is the probability of the i {\displaystyle i} proposition, while n i is the number of quanta that were assigned to the i {\displaystyle i} proposition (i.e. the number of balls that ended up in bucket i {\displaystyle i} ). Now, in order to reduce the 'graininess' of the probability assignment, it will be necessary to use quite
SECTION 60
#17328919770359114-472: Is the rate of change of the quantity being optimized as a function of the constraint parameter. As examples, in Lagrangian mechanics the equations of motion are derived by finding stationary points of the action , the time integral of the difference between kinetic and potential energy. Thus, the force on a particle due to a scalar potential, F = −∇ V , can be interpreted as a Lagrange multiplier determining
9261-492: Is to be independent of any other, and every bucket is to be the same size.) Once the experiment is done, he will check if the probability assignment thus obtained is consistent with his information. (For this step to be successful, the information must be a constraint given by an open set in the space of probability measures). If it is inconsistent, he will reject it and try again. If it is consistent, his assessment will be where p i {\displaystyle p_{i}}
9408-419: Is when p = 0 or p = 1 , when the event outcome is known ahead of time, and the entropy is zero bits. When the entropy is zero bits, this is sometimes referred to as unity, where there is no uncertainty at all – no freedom of choice – no information . Other values of p give entropies between zero and one bits. Information theory is useful to calculate the smallest amount of information required to convey
9555-532: Is worth noting that if we drop the "small for small probabilities" property, then H {\displaystyle \mathrm {H} } must be a non-negative linear combination of the Shannon entropy and the Hartley entropy . The Shannon entropy satisfies the following properties, for some of which it is useful to interpret entropy as the expected amount of information learned (or uncertainty eliminated) by revealing
9702-409: The λ k {\displaystyle \lambda _{k}} parameters are determined by the system of nonlinear equations: In the case with inequality moment constraints the Lagrange multipliers are determined from the solution of a convex optimization program. The invariant measure function q ( x ) can be best understood by supposing that x is known to take values only in
9849-754: The λ {\displaystyle \lambda } term may be either added or subtracted. If f ( x 0 , y 0 ) {\displaystyle f(x_{0},y_{0})} is a maximum of f ( x , y ) {\displaystyle f(x,y)} for the original constrained problem and ∇ g ( x 0 , y 0 ) ≠ 0 , {\displaystyle \nabla g(x_{0},y_{0})\neq 0,} then there exists λ 0 {\displaystyle \lambda _{0}} such that ( x 0 , y 0 , λ 0 {\displaystyle x_{0},y_{0},\lambda _{0}} )
9996-417: The Boltzmann constant k B indicates, the changes in S / k B for even tiny amounts of substances in chemical and physical processes represent amounts of entropy that are extremely large compared to anything in data compression or signal processing . In classical thermodynamics, entropy is defined in terms of macroscopic measurements and makes no reference to any probability distribution, which
10143-424: The Boltzmann constant , and p i is the probability of a microstate . The Gibbs entropy was defined by J. Willard Gibbs in 1878 after earlier work by Boltzmann (1872). The Gibbs entropy translates over almost unchanged into the world of quantum physics to give the von Neumann entropy introduced by John von Neumann in 1927: where ρ is the density matrix of the quantum mechanical system and Tr
10290-847: The base for the logarithm . Thus, entropy is characterized by the above four properties. This differential equation leads to the solution I ( u ) = k log u + c {\displaystyle \operatorname {I} (u)=k\log u+c} for some k , c ∈ R {\displaystyle k,c\in \mathbb {R} } . Property 2 gives c = 0 {\displaystyle c=0} . Property 1 and 2 give that I ( p ) ≥ 0 {\displaystyle \operatorname {I} (p)\geq 0} for all p ∈ [ 0 , 1 ] {\displaystyle p\in [0,1]} , so that k < 0 {\displaystyle k<0} . The different units of information ( bits for
10437-415: The binary logarithm log 2 , nats for the natural logarithm ln , bans for the decimal logarithm log 10 and so on) are constant multiples of each other. For instance, in case of a fair coin toss, heads provides log 2 (2) = 1 bit of information, which is approximately 0.693 nats or 0.301 decimal digits. Because of additivity, n tosses provide n bits of information, which
10584-406: The bounded interval ( a , b ), and that no other information is given. Then the maximum entropy probability density function is where A is a normalization constant. The invariant measure function is actually the prior density function encoding 'lack of relevant information'. It cannot be determined by the principle of maximum entropy, and must be determined by some other logical method, such as
10731-401: The derivative test of an unconstrained problem can still be applied. The relationship between the gradient of the function and gradients of the constraints rather naturally leads to a reformulation of the original problem, known as the Lagrangian function or Lagrangian. In the general case, the Lagrangian is defined as for functions f , g {\displaystyle f,g} ;
10878-437: The entropy of a random variable quantifies the average level of uncertainty or information associated with the variable's potential states or possible outcomes. This measures the expected amount of information needed to describe the state of the variable, considering the distribution of probabilities across all potential states. Given a discrete random variable X {\displaystyle X} , which takes values in
11025-487: The exterior product . The stationary points x {\displaystyle \ x\ } are the solutions of the above system of equations plus the constraint g ( x ) = 0 . {\displaystyle \ g(x)=0~.} Note that the 1 2 m ( m − 1 ) {\displaystyle \ {\tfrac {1}{2}}m(m-1)\ } equations are not independent, since
11172-406: The gradient of the function (at that point) can be expressed as a linear combination of the gradients of the constraints (at that point), with the Lagrange multipliers acting as coefficients . This is equivalent to saying that any direction perpendicular to all gradients of the constraints is also perpendicular to the gradient of the function. Or still, saying that the directional derivative of
11319-411: The method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables ). It is named after the mathematician Joseph-Louis Lagrange . The basic idea is to convert a constrained problem into a form such that
11466-427: The principle of transformation groups or marginalization theory . For several examples of maximum entropy distributions, see the article on maximum entropy probability distributions . Proponents of the principle of maximum entropy justify its use in assigning probabilities in several ways, including the following two arguments. These arguments take the use of Bayesian probability as given, and are thus subject to
11613-425: The probability distribution which maximizes information entropy , subject to the constraints of the information. This constrained optimization problem is typically solved using the method of Lagrange multipliers . Entropy maximization with no testable information respects the universal "constraint" that the sum of the probabilities is one. Under this constraint, the maximum entropy discrete probability distribution
11760-2139: The Lagrange multipliers. We now have M {\displaystyle M} of them, one for every constraint. As before, we introduce an auxiliary function L ( x 1 , … , x n , λ 1 , … , λ M ) = f ( x 1 , … , x n ) − ∑ k = 1 M λ k g k ( x 1 , … , x n ) {\displaystyle {\mathcal {L}}\left(x_{1},\ldots ,x_{n},\lambda _{1},\ldots ,\lambda _{M}\right)=f\left(x_{1},\ldots ,x_{n}\right)-\sum \limits _{k=1}^{M}{\lambda _{k}g_{k}\left(x_{1},\ldots ,x_{n}\right)}\ } and solve ∇ x 1 , … , x n , λ 1 , … , λ M L ( x 1 , … , x n , λ 1 , … , λ M ) = 0 ⟺ { ∇ f ( x ) − ∑ k = 1 M λ k ∇ g k ( x ) = 0 g 1 ( x ) = ⋯ = g M ( x ) = 0 {\displaystyle \nabla _{x_{1},\ldots ,x_{n},\lambda _{1},\ldots ,\lambda _{M}}{\mathcal {L}}(x_{1},\ldots ,x_{n},\lambda _{1},\ldots ,\lambda _{M})=0\iff {\begin{cases}\nabla f(\mathbf {x} )-\sum _{k=1}^{M}{\lambda _{k}\,\nabla g_{k}(\mathbf {x} )}=0\\g_{1}(\mathbf {x} )=\cdots =g_{M}(\mathbf {x} )=0\end{cases}}} which amounts to solving n + M {\displaystyle n+M} equations in n + M {\displaystyle \ n+M\ } unknowns. The constraint qualification assumption when there are multiple constraints
11907-453: The Lagrangian L {\displaystyle {\mathcal {L}}} , but they are not necessarily local extrema of L {\displaystyle {\mathcal {L}}} (see § Example 2 below). One may reformulate the Lagrangian as a Hamiltonian , in which case the solutions are local minima for the Hamiltonian. This is done in optimal control theory, in
12054-1510: The Lagrangian expression L {\displaystyle {\mathcal {L}}} . Often the Lagrange multipliers have an interpretation as some quantity of interest. For example, by parametrising the constraint's contour line, that is, if the Lagrangian expression is L ( x 1 , x 2 , … ; λ 1 , λ 2 , … ; c 1 , c 2 , … ) = f ( x 1 , x 2 , … ) + λ 1 ( c 1 − g 1 ( x 1 , x 2 , … ) ) + λ 2 ( c 2 − g 2 ( x 1 , x 2 , … ) ) + ⋯ {\displaystyle {\begin{aligned}&{\mathcal {L}}(x_{1},x_{2},\ldots ;\lambda _{1},\lambda _{2},\ldots ;c_{1},c_{2},\ldots )\\[4pt]={}&f(x_{1},x_{2},\ldots )+\lambda _{1}(c_{1}-g_{1}(x_{1},x_{2},\ldots ))+\lambda _{2}(c_{2}-g_{2}(x_{1},x_{2},\dots ))+\cdots \end{aligned}}} then ∂ L ∂ c k = λ k . {\displaystyle \ {\frac {\partial {\mathcal {L}}}{\partial c_{k}}}=\lambda _{k}~.} So, λ k
12201-501: The Lagrangian function, L ( x , y , λ ) = f ( x , y ) + λ ⋅ g ( x , y ) = x + y + λ ( x 2 + y 2 − 1 ) , {\displaystyle {\begin{aligned}{\mathcal {L}}(x,y,\lambda )&=f(x,y)+\lambda \cdot g(x,y)\\[4pt]&=x+y+\lambda (x^{2}+y^{2}-1)\ ,\end{aligned}}}
12348-407: The Shannon entropy), Boltzmann's equation results. In information theoretic terms, the information entropy of a system is the amount of "missing" information needed to determine a microstate, given the macrostate. In the view of Jaynes (1957), thermodynamic entropy, as explained by statistical mechanics , should be seen as an application of Shannon's information theory: the thermodynamic entropy
12495-493: The above properties must be a constant multiple of Shannon entropy, with a non-negative constant. Compared to the previously mentioned characterizations of entropy, this characterization focuses on the properties of entropy as a function of random variables (subadditivity and additivity), rather than the properties of entropy as a function of the probability vector p 1 , … , p n {\displaystyle p_{1},\ldots ,p_{n}} . It
12642-629: The above section regarding the case of a single constraint. Rather than the function g {\displaystyle g} described there, now consider a smooth function G : M → R p ( p > 1 ) , {\displaystyle \ G:M\to \mathbb {R} ^{p}(p>1)\ ,} with component functions g i : M → R , {\displaystyle \ g_{i}:M\to \mathbb {R} \ ,} for which 0 ∈ R p {\displaystyle 0\in \mathbb {R} ^{p}}
12789-430: The case of multiple constraints, that will be what we seek in general: The method of Lagrange seeks points not at which the gradient of f {\displaystyle f} is a multiple of any single constraint's gradient necessarily, but in which it is a linear combination of all the constraints' gradients. Concretely, suppose we have M {\displaystyle M} constraints and are walking along
12936-401: The change in action (transfer of potential to kinetic energy) following a variation in the particle's constrained trajectory. In control theory this is formulated instead as costate equations . Moreover, by the envelope theorem the optimal value of a Lagrange multiplier has an interpretation as the marginal effect of the corresponding constraint constant upon the optimal attainable value of
13083-1280: The coin is not fair, but comes up heads or tails with probabilities p and q , where p ≠ q , then there is less uncertainty. Every time it is tossed, one side is more likely to come up than the other. The reduced uncertainty is quantified in a lower entropy: on average each toss of the coin delivers less than one full bit of information. For example, if p = 0.7, then H ( X ) = − p log 2 ( p ) − q log 2 ( q ) = − 0.7 log 2 ( 0.7 ) − 0.3 log 2 ( 0.3 ) ≈ − 0.7 ⋅ ( − 0.515 ) − 0.3 ⋅ ( − 1.737 ) = 0.8816 < 1. {\displaystyle {\begin{aligned}\mathrm {H} (X)&=-p\log _{2}(p)-q\log _{2}(q)\\&=-0.7\log _{2}(0.7)-0.3\log _{2}(0.3)\\&\approx -0.7\cdot (-0.515)-0.3\cdot (-1.737)\\&=0.8816<1.\end{aligned}}} Uniform probability yields maximum uncertainty and therefore maximum entropy. Entropy, then, can only decrease from
13230-517: The constraint x 2 + y 2 = 1 . {\displaystyle \ x^{2}+y^{2}=1~.} The feasible set is the unit circle, and the level sets of f are diagonal lines (with slope −1), so we can see graphically that the maximum occurs at ( 1 2 , 1 2 ) , {\displaystyle \ \left({\tfrac {1}{\sqrt {2}}},{\tfrac {1}{\sqrt {2}}}\right)\ ,} and that
13377-450: The constraint The probability distribution with maximum information entropy subject to these inequality/equality constraints is of the form: for some λ 1 , … , λ m {\displaystyle \lambda _{1},\ldots ,\lambda _{m}} . It is sometimes called the Gibbs distribution . The normalization constant is determined by: and
13524-485: The constraint equations from the form g i ( x ) = 0 {\displaystyle g_{i}({\bf {x}})=0} to the form g i ( x ) = c i , {\displaystyle \ g_{i}({\bf {x}})=c_{i}\ ,} where the c i {\displaystyle \ c_{i}\ } are m real constants that are considered to be additional arguments of
13671-1195: The constraints function, both belonging to C 1 {\displaystyle C^{1}} (that is, having continuous first derivatives). Let x ⋆ {\displaystyle x_{\star }} be an optimal solution to the following optimization problem such that, for the matrix of partial derivatives [ D g ( x ⋆ ) ] j , k = ∂ g j ∂ x k {\displaystyle {\Bigl [}\operatorname {D} g(x_{\star }){\Bigr ]}_{j,k}={\frac {\ \partial g_{j}\ }{\partial x_{k}}}} , rank ( D g ( x ⋆ ) ) = c ≤ n {\displaystyle \operatorname {rank} (\operatorname {D} g(x_{\star }))=c\leq n} : maximize f ( x ) subject to: g ( x ) = 0 {\displaystyle {\begin{aligned}&{\text{maximize }}f(x)\\&{\text{subject to: }}g(x)=0\end{aligned}}} Then there exists
13818-490: The demon himself must increase thermodynamic entropy in the process, by at least the amount of Shannon information he proposes to first acquire and store; and so the total thermodynamic entropy does not decrease (which resolves the paradox). Landauer's principle imposes a lower bound on the amount of heat a computer must generate to process a given amount of information, though modern computers are far less efficient. Lagrange multiplier In mathematical optimization ,
13965-449: The distribution with the maximum entropy allowed by our information, the argument goes, we are choosing the most uninformative distribution possible. To choose a distribution with lower entropy would be to assume information we do not possess. Thus the maximum entropy distribution is the only reasonable distribution. The dependence of the solution on the dominating measure represented by m ( x ) {\displaystyle m(x)}
14112-543: The efficiency of a source set with n symbols can be defined simply as being equal to its n -ary entropy. See also Redundancy (information theory) . The characterization here imposes an additive property with respect to a partition of a set . Meanwhile, the conditional probability is defined in terms of a multiplicative property, P ( A ∣ B ) ⋅ P ( B ) = P ( A ∩ B ) {\displaystyle P(A\mid B)\cdot P(B)=P(A\cap B)} . Observe that
14259-435: The expected (i.e., average) amount of information conveyed by identifying the outcome of a random trial. This implies that rolling a die has higher entropy than tossing a coin because each outcome of a die toss has smaller probability ( p = 1 / 6 {\displaystyle p=1/6} ) than each outcome of a coin toss ( p = 1 / 2 {\displaystyle p=1/2} ). Consider
14406-695: The exterior product of the columns of the matrix of K x ∗ , {\displaystyle \ K_{x}^{\ast }\ ,} the stationary condition for f | N {\displaystyle \ f|_{N}\ } at x {\displaystyle \ x\ } becomes L x ∧ ω x = 0 ∈ Λ p + 1 ( T x ∗ M ) {\displaystyle L_{x}\wedge \omega _{x}=0\in \Lambda ^{p+1}\left(T_{x}^{\ast }M\right)} Once again, in this formulation it
14553-1182: The first possibility (we touch a contour line of f ), notice that since the gradient of a function is perpendicular to the contour lines, the tangents to the contour lines of f and g are parallel if and only if the gradients of f and g are parallel. Thus we want points ( x , y ) where g ( x , y ) = c and ∇ x , y f = λ ∇ x , y g , {\displaystyle \nabla _{x,y}f=\lambda \,\nabla _{x,y}g,} for some λ {\displaystyle \lambda } where ∇ x , y f = ( ∂ f ∂ x , ∂ f ∂ y ) , ∇ x , y g = ( ∂ g ∂ x , ∂ g ∂ y ) {\displaystyle \nabla _{x,y}f=\left({\frac {\partial f}{\partial x}},{\frac {\partial f}{\partial y}}\right),\qquad \nabla _{x,y}g=\left({\frac {\partial g}{\partial x}},{\frac {\partial g}{\partial y}}\right)} are
14700-413: The following properties. We denote p i = Pr( X = x i ) and Η n ( p 1 , ..., p n ) = Η( X ) . The rule of additivity has the following consequences: for positive integers b i where b 1 + ... + b k = n , Choosing k = n , b 1 = ... = b n = 1 this implies that the entropy of a certain outcome is zero: Η 1 (1) = 0 . This implies that
14847-562: The form of Pontryagin's minimum principle . The fact that solutions of the method of Lagrange multipliers are not necessarily extrema of the Lagrangian, also poses difficulties for numerical optimization. This can be addressed by minimizing the magnitude of the gradient of the Lagrangian, as these minima are the same as the zeros of the magnitude, as illustrated in Example 5: Numerical optimization . The method of Lagrange multipliers can be extended to solve problems with multiple constraints using
14994-433: The form of m constraints on the expectations of the functions f k ; that is, we require our probability distribution to satisfy the moment inequality/equality constraints: where the F k {\displaystyle F_{k}} are observables. We also require the probability density to sum to one, which may be viewed as a primitive constraint on the identity function and an observable equal to 1 giving
15141-539: The function is 0 in every feasible direction. For the case of only one constraint and only two choice variables (as exemplified in Figure 1), consider the optimization problem maximize x , y f ( x , y ) subject to g ( x , y ) = 0. {\displaystyle {\begin{aligned}{\underset {x,y}{\text{maximize}}}\quad &f(x,y)\\{\text{subject to}}\quad &g(x,y)=0.\end{aligned}}} (Sometimes an additive constant
15288-401: The image of K x ∗ : R p ∗ → T x ∗ M . {\displaystyle \ K_{x}^{\ast }:\mathbb {R} ^{p\ast }\to T_{x}^{\ast }M~.} Computationally speaking, the condition is that L x {\displaystyle L_{x}} belongs to the row space of
15435-681: The kernel ker ( d f x ) {\displaystyle \ \ker(\operatorname {d} f_{x})\ } contains T x N = ker ( d g x ) . {\displaystyle \ T_{x}N=\ker(\operatorname {d} g_{x})~.} In other words, d f x {\displaystyle \ \operatorname {d} f_{x}\ } and d g x {\displaystyle \ \operatorname {d} g_{x}\ } are proportional 1-forms. For this it
15582-853: The left-hand side of the equation belongs to the subvariety of Λ 2 ( T x ∗ M ) {\displaystyle \ \Lambda ^{2}(T_{x}^{\ast }M)\ } consisting of decomposable elements . In this formulation, it is not necessary to explicitly find the Lagrange multiplier, a number λ {\displaystyle \ \lambda \ } such that d f x = λ ⋅ d g x . {\displaystyle \ \operatorname {d} f_{x}=\lambda \cdot \operatorname {d} g_{x}~.} Let M {\displaystyle \ M\ } and f {\displaystyle \ f\ } be as in
15729-407: The logarithm is ad hoc and the entropy is not a measure in itself. At least in the information theory of a binary string, log 2 {\displaystyle \log _{2}} lends itself to practical interpretations. Motivated by such relations, a plethora of related and competing quantities have been defined. For example, David Ellerman 's analysis of a "logic of partitions" defines
15876-494: The matrix of K x , {\displaystyle \ K_{x}\ ,} or equivalently the column space of the matrix of K x ∗ {\displaystyle K_{x}^{\ast }} (the transpose). If ω x ∈ Λ p ( T x ∗ M ) {\displaystyle \ \omega _{x}\in \Lambda ^{p}(T_{x}^{\ast }M)\ } denotes
16023-425: The maximum entropy principle may require the solution to a quadratic programming problem, and thus provide a sparse mixture model as the optimal density estimator. One important advantage of the method is its ability to incorporate prior information in the density estimation. We have some testable information I about a quantity x taking values in { x 1 , x 2 ,..., x n }. We assume this information has
16170-536: The maximum or minimum of a function f {\displaystyle f} subject to the equality constraint g ( x ) = 0 {\displaystyle g(x)=0} , find the stationary points of L {\displaystyle {\mathcal {L}}} considered as a function of x {\displaystyle x} and the Lagrange multiplier λ {\displaystyle \lambda ~} . This means that all partial derivatives should be zero, including
16317-600: The maximum. Viewed in this way, it is an exact analogue to testing if the derivative of an unconstrained function is 0 , that is, we are verifying that the directional derivative is 0 in any relevant (viable) direction. We can visualize contours of f given by f ( x , y ) = d for various values of d , and the contour of g given by g ( x , y ) = c . Suppose we walk along the contour line with g = c . We are interested in finding points where f almost does not change as we walk, since these points might be maxima. There are two ways this could happen: To check
16464-926: The message. Named after Boltzmann's Η-theorem , Shannon defined the entropy Η (Greek capital letter eta ) of a discrete random variable X {\textstyle X} , which takes values in the set X {\displaystyle {\mathcal {X}}} and is distributed according to p : X → [ 0 , 1 ] {\displaystyle p:{\mathcal {X}}\to [0,1]} such that p ( x ) := P [ X = x ] {\displaystyle p(x):=\mathbb {P} [X=x]} : H ( X ) = E [ I ( X ) ] = E [ − log p ( X ) ] . {\displaystyle \mathrm {H} (X)=\mathbb {E} [\operatorname {I} (X)]=\mathbb {E} [-\log p(X)].} Here E {\displaystyle \mathbb {E} }
16611-502: The minimum occurs at ( − 1 2 , − 1 2 ) . {\displaystyle \ \left(-{\tfrac {1}{\sqrt {2}}},-{\tfrac {1}{\sqrt {2}}}\right)~.} For the method of Lagrange multipliers, the constraint is g ( x , y ) = x 2 + y 2 − 1 = 0 , {\displaystyle g(x,y)=x^{2}+y^{2}-1=0\ ,} hence
16758-474: The multiplicity W {\displaystyle W} . Rather than maximizing W {\displaystyle W} directly, the protagonist could equivalently maximize any monotonic increasing function of W {\displaystyle W} . He decides to maximize At this point, in order to simplify the expression, the protagonist takes the limit as N → ∞ {\displaystyle N\to \infty } , i.e. as
16905-426: The notation ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } denotes an inner product . The value λ {\displaystyle \lambda } is called the Lagrange multiplier. In simple cases, where the inner product is defined as the dot product , the Lagrangian is The method can be summarized as follows: in order to find
17052-447: The number of possible microscopic states of the system that are consistent with the measurable values of its macroscopic variables, making any complete state description longer. (See article: maximum entropy thermodynamics ). Maxwell's demon can (hypothetically) reduce the thermodynamic entropy of a system by using information about the states of individual molecules; but, as Landauer (from 1961) and co-workers have shown, to function
17199-431: The observation of event i follows from Shannon's solution of the fundamental properties of information : Given two independent events, if the first event can yield one of n equiprobable outcomes and another has one of m equiprobable outcomes then there are mn equiprobable outcomes of the joint event. This means that if log 2 ( n ) bits are needed to encode the first value and log 2 ( m ) to encode
17346-430: The observed data itself is assumed to be the testable information. Such models are widely used in natural language processing . An example of such a model is logistic regression , which corresponds to the maximum entropy classifier for independent observations. One of the main applications of the maximum entropy principle is in discrete and continuous density estimation . Similar to support vector machine estimators,
17493-586: The only possible values of I {\displaystyle \operatorname {I} } are I ( u ) = k log u {\displaystyle \operatorname {I} (u)=k\log u} for k < 0 {\displaystyle k<0} . Additionally, choosing a value for k is equivalent to choosing a value x > 1 {\displaystyle x>1} for k = − 1 / log x {\displaystyle k=-1/\log x} , so that x corresponds to
17640-395: The optimal profit to a player is calculated subject to a constrained space of actions, where a Lagrange multiplier is the change in the optimal value of the objective function (profit) due to the relaxation of a given constraint (e.g. through a change in income); in such a context λ ⋆ k {\displaystyle \ \lambda _{\star k}\ }
17787-839: The original objective function: If we denote values at the optimum with a star ( ⋆ {\displaystyle \star } ), then it can be shown that d f ( x 1 ⋆ ( c 1 , c 2 , … ) , x 2 ⋆ ( c 1 , c 2 , … ) , … ) d c k = λ ⋆ k . {\displaystyle {\frac {\ \operatorname {d} f\left(\ x_{1\star }(c_{1},c_{2},\dots ),\ x_{2\star }(c_{1},c_{2},\dots ),\ \dots \ \right)\ }{\operatorname {d} c_{k}}}=\lambda _{\star k}~.} For example, in economics
17934-453: The original problem, as the method of Lagrange multipliers yields only a necessary condition for optimality in constrained problems. Sufficient conditions for a minimum or maximum also exist , but if a particular candidate solution satisfies the sufficient conditions, it is only guaranteed that that solution is the best one locally – that is, it is better than any permissible nearby points. The global optimum can be found by comparing
18081-405: The partial derivative with respect to λ {\displaystyle \lambda ~} . or equivalently The solution corresponding to the original constrained optimization is always a saddle point of the Lagrangian function, which can be identified among the stationary points from the definiteness of the bordered Hessian matrix . The great advantage of this method
18228-792: The principle of maximum entropy are completely compatible and can be seen as special cases of the "method of maximum relative entropy". They state that this method reproduces every aspect of orthodox Bayesian inference methods. In addition this new method opens the door to tackling problems that could not be addressed by either the maximal entropy principle or orthodox Bayesian methods individually. Moreover, recent contributions (Lazar 2003, and Schennach 2005) show that frequentist relative-entropy-based inference approaches (such as empirical likelihood and exponentially tilted empirical likelihood – see e.g. Owen 2001 and Kitamura 2006) can be combined with prior information to perform Bayesian posterior analysis. Information entropy In information theory ,
18375-400: The probability density to integrate to one, which may be viewed as a primitive constraint on the identity function and an observable equal to 1 giving the constraint The probability density function with maximum H c subject to these constraints is: with the partition function determined by As in the discrete case, in the case where all moment constraints are equalities, the values of
18522-484: The probability levels go from grainy discrete values to smooth continuous values. Using Stirling's approximation , he finds All that remains for the protagonist to do is to maximize entropy under the constraints of his testable information. He has found that the maximum entropy distribution is the most probable of all "fair" random distributions, in the limit as the probability levels go from discrete to continuous. Giffin and Caticha (2007) state that Bayes' theorem and
18669-572: The propositions over the others. In that case, the only reasonable probability distribution would be uniform, and then the information entropy would be equal to its maximum possible value, log m {\displaystyle \log m} . The information entropy can therefore be seen as a numerical measure which describes how uninformative a particular probability distribution is, ranging from zero (completely informative) to log m {\displaystyle \log m} (completely uninformative). By choosing to use
18816-545: The remaining randomness in the random variable X {\displaystyle X} given the random variable Y {\displaystyle Y} . Entropy can be formally defined in the language of measure theory as follows: Let ( X , Σ , μ ) {\displaystyle (X,\Sigma ,\mu )} be a probability space . Let A ∈ Σ {\displaystyle A\in \Sigma } be an event . The surprisal of A {\displaystyle A}
18963-430: The respective gradients. The constant λ {\displaystyle \lambda } is required because although the two gradient vectors are parallel, the magnitudes of the gradient vectors are generally not equal. This constant is called the Lagrange multiplier. (In some conventions λ {\displaystyle \lambda } is preceded by a minus sign). Notice that this method also solves
19110-451: The restriction f | N {\displaystyle \ f|_{N}\ } at x ∈ N {\displaystyle \ x\in N\ } means d ( f | N ) x = 0 . {\displaystyle \ \operatorname {d} (f|_{N})_{x}=0~.} Equivalently,
19257-406: The same postulates. Consider a discrete probability distribution among m {\displaystyle m} mutually exclusive propositions . The most informative distribution would occur when one of the propositions was known to be true. In that case, the information entropy would be equal to zero. The least informative distribution would occur when there is no reason to favor any one of
19404-968: The second possibility, that f is level: if f is level, then its gradient is zero, and setting λ = 0 {\displaystyle \lambda =0} is a solution regardless of ∇ x , y g {\displaystyle \nabla _{x,y}g} . To incorporate these conditions into one equation, we introduce an auxiliary function L ( x , y , λ ) ≡ f ( x , y ) + λ ⋅ g ( x , y ) , {\displaystyle {\mathcal {L}}(x,y,\lambda )\equiv f(x,y)+\lambda \cdot g(x,y)\,,} and solve ∇ x , y , λ L ( x , y , λ ) = 0 . {\displaystyle \nabla _{x,y,\lambda }{\mathcal {L}}(x,y,\lambda )=0~.} Note that this amounts to solving three equations in three unknowns. This
19551-477: The second, one needs log 2 ( mn ) = log 2 ( m ) + log 2 ( n ) to encode both. Shannon discovered that a suitable choice of I {\displaystyle \operatorname {I} } is given by: I ( p ) = log ( 1 p ) = − log ( p ) . {\displaystyle \operatorname {I} (p)=\log \left({\tfrac {1}{p}}\right)=-\log(p).} In fact,
19698-594: The set X {\displaystyle {\mathcal {X}}} and is distributed according to p : X → [ 0 , 1 ] {\displaystyle p\colon {\mathcal {X}}\to [0,1]} , the entropy is H ( X ) := − ∑ x ∈ X p ( x ) log p ( x ) , {\displaystyle \mathrm {H} (X):=-\sum _{x\in {\mathcal {X}}}p(x)\log p(x),} where Σ {\displaystyle \Sigma } denotes
19845-426: The set of points satisfying g i ( x ) = 0 , i = 1 , … , M . {\displaystyle g_{i}(\mathbf {x} )=0,i=1,\dots ,M\,.} Every point x {\displaystyle \mathbf {x} } on the contour of a given constraint function g i {\displaystyle g_{i}} has a space of allowable directions:
19992-515: The solution equations are given. A closely related quantity, the relative entropy, is usually defined as the Kullback–Leibler divergence of p from q (although it is sometimes, confusingly, defined as the negative of this). The inference principle of minimizing this, due to Kullback, is known as the Principle of Minimum Discrimination Information . We have some testable information I about
20139-414: The solution of the nonlinear equations In the case of inequality constraints, the Lagrange multipliers are determined from the solution of a convex optimization program with linear constraints. In both cases, there is no closed form solution , and the computation of the Lagrange multipliers usually requires numerical methods . For continuous distributions , the Shannon entropy cannot be used, as it
20286-426: The space of vectors perpendicular to ∇ g i ( x ) . {\displaystyle \nabla g_{i}(\mathbf {x} )\,.} The set of directions that are allowed by all constraints is thus the space of directions perpendicular to all of the constraints' gradients. Denote this space of allowable moves by A {\displaystyle \ A\ } and denote
20433-674: The span of the constraints' gradients by S . {\displaystyle S\,.} Then A = S ⊥ , {\displaystyle A=S^{\perp }\,,} the space of vectors perpendicular to every element of S . {\displaystyle S\,.} We are still interested in finding points where f {\displaystyle f} does not change as we walk, since these points might be (constrained) extrema. We therefore seek x {\displaystyle \mathbf {x} } such that any allowable direction of movement away from x {\displaystyle \mathbf {x} }
20580-468: The stated prior data or testable information is given by a set of conserved quantities (average values of some moment functions), associated with the probability distribution in question. This is the way the maximum entropy principle is most often used in statistical thermodynamics . Another possibility is to prescribe some symmetries of the probability distribution. The equivalence between conserved quantities and corresponding symmetry groups implies
20727-419: The sum over the variable's possible values. The choice of base for log {\displaystyle \log } , the logarithm , varies for different applications. Base 2 gives the unit of bits (or " shannons "), while base e gives "natural units" nat , and base 10 gives units of "dits", "bans", or " hartleys ". An equivalent definition of entropy is the expected value of the self-information of
20874-1496: The tangent map or Jacobian T M → T R p {\displaystyle \ TM\to T\mathbb {R} ^{p}~} ( T x R p {\displaystyle \ T_{x}\mathbb {R} ^{p}} can be canonically identified with R p {\displaystyle \ \mathbb {R} ^{p}} ). The subspace ker ( K x ) {\displaystyle \ker(K_{x})} has dimension smaller than that of ker ( L x ) {\displaystyle \ker(L_{x})} , namely dim ( ker ( L x ) ) = n − 1 {\displaystyle \ \dim(\ker(L_{x}))=n-1\ } and dim ( ker ( K x ) ) = n − p . {\displaystyle \ \dim(\ker(K_{x}))=n-p~.} ker ( K x ) {\displaystyle \ker(K_{x})} belongs to ker ( L x ) {\displaystyle \ \ker(L_{x})\ } if and only if L x ∈ T x ∗ M {\displaystyle L_{x}\in T_{x}^{\ast }M} belongs to
21021-439: The time only one bit needs to be sent, 26% of the time two bits, and only 4% of the time 3 bits. On average, fewer than 2 bits are required since the entropy is lower (owing to the high prevalence of 'A' followed by 'B' – together 96% of characters). The calculation of the sum of probability-weighted log probabilities measures and captures this effect. English text, treated as a string of characters, has fairly low entropy; i.e. it
21168-536: The value associated with uniform probability. The extreme case is that of a double-headed coin that never comes up tails, or a double-tailed coin that never results in a head. Then there is no uncertainty. The entropy is zero: each toss of the coin delivers no new information as the outcome of each coin toss is always certain. To understand the meaning of −Σ p i log( p i ) , first define an information function I in terms of an event i with probability p i . The amount of information acquired due to
21315-399: The value of a random variable X : The inspiration for adopting the word entropy in information theory came from the close resemblance between Shannon's formula and very similar known formulae from statistical mechanics . In statistical thermodynamics the most general formula for the thermodynamic entropy S of a thermodynamic system is the Gibbs entropy where k B is
21462-1486: The value of the corresponding summand 0 log b (0) is taken to be 0 , which is consistent with the limit : lim p → 0 + p log ( p ) = 0. {\displaystyle \lim _{p\to 0^{+}}p\log(p)=0.} One may also define the conditional entropy of two variables X {\displaystyle X} and Y {\displaystyle Y} taking values from sets X {\displaystyle {\mathcal {X}}} and Y {\displaystyle {\mathcal {Y}}} respectively, as: H ( X | Y ) = − ∑ x , y ∈ X × Y p X , Y ( x , y ) log p X , Y ( x , y ) p Y ( y ) , {\displaystyle \mathrm {H} (X|Y)=-\sum _{x,y\in {\mathcal {X}}\times {\mathcal {Y}}}p_{X,Y}(x,y)\log {\frac {p_{X,Y}(x,y)}{p_{Y}(y)}},} where p X , Y ( x , y ) := P [ X = x , Y = y ] {\displaystyle p_{X,Y}(x,y):=\mathbb {P} [X=x,Y=y]} and p Y ( y ) = P [ Y = y ] {\displaystyle p_{Y}(y)=\mathbb {P} [Y=y]} . This quantity should be understood as
21609-413: The values of the original objective function at the points satisfying the necessary and locally sufficient conditions. The method of Lagrange multipliers relies on the intuition that at a maximum, f ( x , y ) cannot be increasing in the direction of any such neighboring point that also has g = 0 . If it were, we could walk along g = 0 to get higher, meaning that the starting point wasn't actually
#34965