Kullback–Leibler divergence

In mathematical statistics, the Kullback–Leibler divergence, (also called relative entropy), is a statistical distance: a measure of how one probability distribution Q is different from a second, reference probability distribution P.[1][2] A simple interpretation of the divergence of P from Q is the expected excess surprise from using Q as a model when the actual distribution is P. While it is a distance, it is not a metric, the most familiar type of distance: it is asymmetric in the two distributions (in contrast to variation of information), and does not satisfy the triangle inequality. Instead, in terms of information geometry, it is a divergence,[3] a generalization of squared distance, and for certain classes of distributions (notably an exponential family), it satisfies a generalized Pythagorean theorem (which applies to squared distances).[4]

In the simple case, a relative entropy of 0 indicates that the two distributions in question have identical quantities of information. It has diverse applications, both theoretical, such as characterizing the relative (Shannon) entropy in information systems, randomness in continuous time-series, and information gain when comparing statistical models of inference; and practical, such as applied statistics, fluid mechanics, neuroscience and bioinformatics.

Introduction and contextEdit

Consider two probability distributions   and  . Usually,   represents the data, the observations, or a measured probability distribution. Distribution   represents instead a theory, a model, a description or an approximation of  . The Kullback–Leibler divergence is then interpreted as the average difference of the number of bits required for encoding samples of   using a code optimized for   rather than one optimized for  . Note that the roles of   and   can be reversed in some situations where that is easier to compute, such as with the Expectation–maximization (EM) algorithm and Evidence lower bound (ELBO) computations.


The relative entropy was introduced by Solomon Kullback and Richard Leibler in Kullback & Leibler (1951) as "the mean information for discrimination between   and   per observation from  ",[5] where one is comparing two probability measures  , and   are the hypotheses that one is selecting from measure   (respectively). They denoted this by  , and defined the "'divergence' between   and  " as the symmetrized quantity  , which had already been defined and used by Harold Jeffreys in 1948.[6] In Kullback (1959), the symmetrized form is again referred to as the "divergence", and the relative entropies in each direction are referred to as a "directed divergences" between two distributions;[7] Kullback preferred the term discrimination information.[8] The term "divergence" is in contrast to a distance (metric), since the symmetrized divergence does not satisfy the triangle inequality.[9] Numerous references to earlier uses of the symmetrized divergence and to other statistical distances are given in Kullback (1959, pp. 6–7, 1.3 Divergence). The asymmetric "directed divergence" has come to be known as the Kullback–Leibler divergence, while the symmetrized "divergence" is now referred to as the Jeffreys divergence.


For discrete probability distributions   and   defined on the same probability space,  , the relative entropy from   to   is defined[10] to be


which is equivalent to


In other words, it is the expectation of the logarithmic difference between the probabilities   and  , where the expectation is taken using the probabilities  . Relative entropy is defined only if for all  ,   implies   (absolute continuity). Whenever   is zero the contribution of the corresponding term is interpreted as zero because


For distributions   and   of a continuous random variable, relative entropy is defined to be the integral:[11]: p. 55 


where   and   denote the probability densities of   and  .

More generally, if   and   are probability measures over a set  , and   is absolutely continuous with respect to  , then the relative entropy from   to   is defined as


where   is the Radon–Nikodym derivative of   with respect to  , and provided the expression on the right-hand side exists. Equivalently (by the chain rule), this can be written as


which is the entropy of   relative to  . Continuing in this case, if   is any measure on   for which   and   exist (meaning that   and   are absolutely continuous with respect to  ), then the relative entropy from   to   is given as


The logarithms in these formulae are taken to base 2 if information is measured in units of bits, or to base   if information is measured in nats. Most formulas involving relative entropy hold regardless of the base of the logarithm.

Various conventions exist for referring to   in words. Often it is referred to as the divergence between   and  , but this fails to convey the fundamental asymmetry in the relation. Sometimes, as in this article, it may be described as the divergence of   from   or as the divergence from   to  . This reflects the asymmetry in Bayesian inference, which starts from a prior   and updates to the posterior  . Another common way to refer to   is as the relative entropy of   with respect to  .

Basic exampleEdit

Kullback[2] gives the following example (Table 2.1, Example 2.1). Let P and Q be the distributions shown in the table and figure. P is the distribution on the left side of the figure, a binomial distribution with   and  . Q is the distribution on the right side of the figure, a discrete uniform distribution with the three possible outcomes  , 1, or 2 (i.e.  ), each with probability  .

Two distributions to illustrate relative entropy

Relative entropies   and   are calculated as follows. This example uses the natural log with base e, designated ln to get results in nats (see units of information).



The relative entropy from   to   is often denoted  .

In the context of machine learning,   is often called the information gain achieved if   would be used instead of   which is currently used. By analogy with information theory, it is called the relative entropy of   with respect to  . In the context of coding theory,   can be constructed by measuring the expected number of extra bits required to code samples from   using a code optimized for   rather than the code optimized for  .

Expressed in the language of Bayesian inference,   is a measure of the information gained by revising one's beliefs from the prior probability distribution   to the posterior probability distribution  . In other words, it is the amount of information lost when   is used to approximate  .[12] In applications,   typically represents the "true" distribution of data, observations, or a precisely calculated theoretical distribution, while   typically represents a theory, model, description, or approximation of  . In order to find a distribution   that is closest to  , we can minimize KL divergence and compute an information projection.

While it is a statistical distance, it is not a metric, the most familiar type of distance, but instead it is a divergence.[3] While metrics are symmetric and generalize linear distance, satisfying the triangle inequality, divergences are asymmetric and generalize squared distance, in some cases satisfying a generalized Pythagorean theorem. In general   does not equal  , and the asymmetry is an important part of the geometry.[3] The infinitesimal form of relative entropy, specifically its Hessian, gives a metric tensor that equals the Fisher information metric; see § Fisher information metric. Relative entropy satisfies a generalized Pythagorean theorem for exponential families (geometrically interpreted as dually flat manifolds), and this allows one to minimize relative entropy by geometric means, for example by information projection and in maximum likelihood estimation.[4]

Relative entropy is a special case of a broader class of statistical divergences called f-divergences as well as the class of Bregman divergences, and it is the only such divergence over probabilities that is a member of both classes.

Arthur Hobson proved that relative entropy is the only measure of difference between probability distributions that satisfies some desired properties, which are the canonical extension to those appearing in a commonly used characterization of entropy.[13] Consequently, mutual information is the only measure of mutual dependence that obeys certain related conditions, since it can be defined in terms of Kullback–Leibler divergence.


Illustration of the relative entropy for two normal distributions. The typical asymmetry is clearly visible.

In information theory, the Kraft–McMillan theorem establishes that any directly decodable coding scheme for coding a message to identify one value   out of a set of possibilities   can be seen as representing an implicit probability distribution   over  , where   is the length of the code for   in bits. Therefore, relative entropy can be interpreted as the expected extra message-length per datum that must be communicated if a code that is optimal for a given (wrong) distribution   is used, compared to using a code based on the true distribution  : it is the excess entropy.


where   is the cross entropy of   and  , and   is the entropy of   (which is the same as the cross-entropy of P with itself).

The relative entropy   can be thought of geometrically as a statistical distance, a measure of how far the distribution Q is from the distribution P. Geometrically it is a divergence: an asymmetric, generalized form of squared distance. The cross-entropy   is itself such a measurement (formally a loss function), but it cannot be thought of as a distance, since   isn't zero. This can be fixed by subtracting   to make   agree more closely with our notion of distance, as the excess loss. The resulting function is asymmetric, and while this can be symmetrized (see § Symmetrised divergence), the asymmetric form is more useful. See § Interpretations for more on the geometric interpretation.

Relative entropy relates to "rate function" in the theory of large deviations.[14][15]


a result known as Gibbs' inequality, with   equals zero if and only if   almost everywhere. The entropy   thus sets a minimum value for the cross-entropy  , the expected number of bits required when using a code based on   rather than  ; and the Kullback–Leibler divergence therefore represents the expected number of extra bits that must be transmitted to identify a value   drawn from  , if a code is used corresponding to the probability distribution  , rather than the "true" distribution  .
  • no upper-bound exists for the general case. However, it is shown that if   and   are two discrete probability distributions built by distributing the same discrete quantity, then the maximum value of   can be calculated.[16]
  • Relative entropy remains well-defined for continuous distributions, and furthermore is invariant under parameter transformations. For example, if a transformation is made from variable   to variable  , then, since   and   the relative entropy may be rewritten:
where   and  . Although it was assumed that the transformation was continuous, this need not be the case. This also shows that the relative entropy produces a dimensionally consistent quantity, since if   is a dimensioned variable,   and   are also dimensioned, since e.g.   is dimensionless. The argument of the logarithmic term is and remains dimensionless, as it must. It can therefore be seen as in some ways a more fundamental quantity than some other properties in information theory[17] (such as self-information or Shannon entropy), which can become undefined or negative for non-discrete probabilities.
  • Relative entropy is additive for independent distributions in much the same way as Shannon entropy. If   are independent distributions, with the joint distribution  , and   likewise, then
  • Relative entropy   is convex in the pair of probability mass functions  , i.e. if   and   are two pairs of probability mass functions, then

Duality Formula for Variational InferenceEdit

The following result, due to Donsker and Varadhan,[18] is known as Donsker and Varadhan's variational formula.

Theorem [Duality Formula for Variational Inference] Let   be a set endowed with an appropriate  -field  , and two probability measures   and  , which formulate two probability spaces   and   with  . (  indicates that   is absolutely continuous with respect to  .) Assume that there is a common dominating probability measure   such that   and  . Let   denote any real-valued random variable on   that satisfies  . Then the following equality holds


Further, the supremum on the right-hand side is attained if and only if it holds


almost surely with respect to probability measure  , where   and   denote the Radon-Nikodym derivatives of the probability measures   and   with respect to  , respectively.

Proof. Let   and  . By simple calculations we have




where the last inequality follows from  , for which equality occurs if and only if  . The conclusion follows.


Multivariate normal distributionsEdit

Suppose that we have two multivariate normal distributions, with means   and with (non-singular) covariance matrices   If the two distributions have the same dimension,  , then the relative entropy between the distributions is as follows:[19]: p. 13 


The logarithm in the last term must be taken to base e since all terms apart from the last are base-e logarithms of expressions that are either factors of the density function or otherwise arise naturally. The equation therefore gives a result measured in nats. Dividing the entire expression above by   yields the divergence in bits.

A special case, and a common quantity in variational inference, is the relative entropy between a diagonal multivariate normal, and a standard normal distribution (with zero mean and unit variance):


Relation to metricsEdit

While relative entropy is a statistical distance, it is not a metric on the space of probability distributions, but instead it is a divergence.[3] While metrics are symmetric and generalize linear distance, satisfying the triangle inequality, divergences are asymmetric in general and generalize squared distance, in some cases satisfying a generalized Pythagorean theorem. In general   does not equal  , and while this can be symmetrized (see § Symmetrised divergence), the asymmetry is an important part of the geometry.[3]

It generates a topology on the space of probability distributions. More concretely, if   is a sequence of distributions such that


then it is said that


Pinsker's inequality entails that


where the latter stands for the usual convergence in total variation.

Fisher information metricEdit

Relative entropy is directly related to the Fisher information metric. This can be made explicit as follows. Assume that the probability distributions   and   are both parameterized by some (possibly multi-dimensional) parameter  . Consider then two close by values of   and   so that the parameter   differs by only a small amount from the parameter value  . Specifically, up to first order one has (using the Einstein summation convention)


with   a small change of   in the   direction, and   the corresponding rate of change in the probability distribution. Since relative entropy has an absolute minimum 0 for  , i.e.  , it changes only to second order in the small parameters  . More formally, as for any minimum, the first derivatives of the divergence vanish


and by the Taylor expansion one has up to second order


where the Hessian matrix of the divergence


must be positive semidefinite. Letting   vary (and dropping the subindex 0) the Hessian   defines a (possibly degenerate) Riemannian metric on the θ parameter space, called the Fisher information metric.

Fisher information metric theoremEdit

When   satisfies the following regularity conditions:


where ξ is independent of ρ




Variation of informationEdit

Another information-theoretic metric is Variation of information, which is roughly a symmetrization of conditional entropy. It is a metric on the set of partitions of a discrete probability space.

Relation to other quantities of information theoryEdit

Many of the other quantities of information theory can be interpreted as applications of relative entropy to specific cases.


The self-information, also known as the information content of a signal, random variable, or event is defined as the negative logarithm of the probability of the given outcome occurring.

When applied to a discrete random variable, the self-information can be represented as[citation needed]


is the relative entropy of the probability distribution   from a Kronecker delta representing certainty that   — i.e. the number of extra bits that must be transmitted to identify   if only the probability distribution   is available to the receiver, not the fact that  .

Mutual informationEdit

The mutual information,


is the relative entropy of the product   of the two marginal probability distributions from the joint probability distribution   — i.e. the expected number of extra bits that must be transmitted to identify   and   if they are coded using only their marginal distributions instead of the joint distribution. Equivalently, if the joint probability   is known, it is the expected number of extra bits that must on average be sent to identify   if the value of   is not already known to the receiver.

Shannon entropyEdit

The Shannon entropy,


is the number of bits which would have to be transmitted to identify   from   equally likely possibilities, less the relative entropy of the uniform distribution on the random variates of  ,  , from the true distribution   — i.e. less the expected number of bits saved, which would have had to be sent if the value of   were coded according to the uniform distribution   rather than the true distribution  .

Conditional entropyEdit

The conditional entropy[20],


is the number of bits which would have to be transmitted to identify   from   equally likely possibilities, less the relative entropy of the product distribution   from the true joint distribution   — i.e. less the expected number of bits saved which would have had to be sent if the value of   were coded according to the uniform distribution   rather than the conditional distribution   of   given  .

Cross entropyEdit

When we have a set of possible events, coming from the distribution p, we can encode them (with a lossless data compression) using entropy encoding. This compresses the data by replacing each fixed-length input symbol with a corresponding unique, variable-length, prefix-free code (e.g.: the events (A, B, C) with probabilities p = (1/2, 1/4, 1/4) can be encoded as the bits (0, 10, 11)). If we know the distribution p in advance, we can devise an encoding that would be optimal (e.g.: using Huffman coding). Meaning the messages we encode will have the shortest length on average (assuming the encoded events are sampled from p), which will be equal to Shannon's Entropy of p (denoted as  ). However, if we use a different probability distribution (q) when creating the entropy encoding scheme, then a larger number of bits will be used (on average) to identify an event from a set of possibilities. This new (larger) number is measured by the cross entropy between p and q.

The cross entropy between two probability distributions (p and q) measures the average number of bits needed to identify an event from a set of possibilities, if a coding scheme is used based on a given probability distribution q, rather than the "true" distribution p. The cross entropy for two distributions p and q over the same probability space is thus defined as follows.


For explicit derivation of this, see the Motivation section above.

Under this scenario, relative entropies (kl-divergence) can be interpreted as the extra number of bits, on average, that are needed (beyond  ) for encoding the events because of using q for constructing the encoding scheme instead of p.

Bayesian updatingEdit

In Bayesian statistics, relative entropy can be used as a measure of the information gain in moving from a prior distribution to a posterior distribution:  . If some new fact   is discovered, it can be used to update the posterior distribution for   from   to a new posterior distribution   using Bayes' theorem:


This distribution has a new entropy:


which may be less than or greater than the original entropy  . However, from the standpoint of the new probability distribution one can estimate that to have used the original code based on   instead of a new code based on   would have added an expected number of bits:


to the message length. This therefore represents the amount of useful information, or information gain, about  , that has been learned by discovering  .

If a further piece of data,  , subsequently comes in, the probability distribution for   can be updated further, to give a new best guess  . If one reinvestigates the information gain for using   rather than  , it turns out that it may be either greater or less than previously estimated:

  may be ≤ or > than  

and so the combined information gain does not obey the triangle inequality:

  may be <, = or > than  

All one can say is that on average, averaging using  , the two sides will average out.

Bayesian experimental designEdit

A common goal in Bayesian experimental design is to maximise the expected relative entropy between the prior and the posterior.[21] When posteriors are approximated to be Gaussian distributions, a design maximising the expected relative entropy is called Bayes d-optimal.

Discrimination informationEdit

Relative entropy   can also be interpreted as the expected discrimination information for   over  : the mean information per sample for discriminating in favor of a hypothesis   against a hypothesis  , when hypothesis   is true.[22] Another name for this quantity, given to it by I. J. Good, is the expected weight of evidence for   over   to be expected from each sample.

The expected weight of evidence for   over   is not the same as the information gain expected per sample about the probability distribution   of the hypotheses,


Either of the two quantities can be used as a utility function in Bayesian experimental design, to choose an optimal next question to investigate: but they will in general lead to rather different experimental strategies.

On the entropy scale of information gain there is very little difference between near certainty and absolute certainty—coding according to a near certainty requires hardly any more bits than coding according to an absolute certainty. On the other hand, on the logit scale implied by weight of evidence, the difference between the two is enormous – infinite perhaps; this might reflect the difference between being almost sure (on a probabilistic level) that, say, the Riemann hypothesis is correct, compared to being certain that it is correct because one has a mathematical proof. These two different scales of loss function for uncertainty are both useful, according to how well each reflects the particular circumstances of the problem in question.

Principle of minimum discrimination informationEdit

The idea of relative entropy as discrimination information led Kullback to propose the Principle of Minimum Discrimination Information (MDI): given new facts, a new distribution   should be chosen which is as hard to discriminate from the original distribution   as possible; so that the new data produces as small an information gain   as possible.

For example, if one had a prior distribution   over   and  , and subsequently learnt the true distribution of   was  , then the relative entropy between the new joint distribution for   and  ,  , and the earlier prior distribution would be:


i.e. the sum of the relative entropy of   the prior distribution for   from the updated distribution  , plus the expected value (using the probability distribution  ) of the relative entropy of the prior conditional distribution   from the new conditional distribution  . (Note that often the later expected value is called the conditional relative entropy (or conditional Kullback-Leibler divergence) and denoted by   [2][20]: p. 22 ) This is minimized if   over the whole support of  ; and we note that this result incorporates Bayes' theorem, if the new distribution   is in fact a δ function representing certainty that   has one particular value.

MDI can be seen as an extension of Laplace's Principle of Insufficient Reason, and the Principle of Maximum Entropy of E.T. Jaynes. In particular, it is the natural extension of the principle of maximum entropy from discrete to continuous distributions, for which Shannon entropy ceases to be so useful (see differential entropy), but the relative entropy continues to be just as relevant.

In the engineering literature, MDI is sometimes called the Principle of Minimum Cross-Entropy (MCE) or Minxent for short. Minimising relative entropy from   to   with respect to   is equivalent to minimizing the cross-entropy of   and  , since


which is appropriate if one is trying to choose an adequate approximation to  . However, this is just as often not the task one is trying to achieve. Instead, just as often it is   that is some fixed prior reference measure, and   that one is attempting to optimise by minimising   subject to some constraint. This has led to some ambiguity in the literature, with some authors attempting to resolve the inconsistency by redefining cross-entropy to be  , rather than  .

Relationship to available workEdit

Pressure versus volume plot of available work from a mole of argon gas relative to ambient, calculated as   times the Kullback–Leibler divergence.

Surprisals[23] add where probabilities multiply. The surprisal for an event of probability   is defined as  . If   is   then surprisal is in  nats, bits, or   so that, for instance, there are   bits of surprisal for landing all "heads" on a toss of   coins.

Best-guess states (e.g. for atoms in a gas) are inferred by maximizing the average surprisal   (entropy) for a given set of control parameters (like pressure   or volume  ). This constrained entropy maximization, both classically[24] and quantum mechanically,[25] minimizes Gibbs availability in entropy units[26]   where   is a constrained multiplicity or partition function.

When temperature   is fixed, free energy ( ) is also minimized. Thus if   and number of molecules   are constant, the Helmholtz free energy   (where   is energy and   is entropy) is minimized as a system "equilibrates." If   and   are held constant (say during processes in your body), the Gibbs free energy   is minimized instead. The change in free energy under these conditions is a measure of available work that might be done in the process. Thus available work for an ideal gas at constant temperature   and pressure   is   where   and   (see also Gibbs inequality).

More generally[27] the work available relative to some ambient is obtained by multiplying ambient temperature   by relative entropy or net surprisal   defined as the average value of   where   is the probability of a given state under ambient conditions. For instance, the work available in equilibrating a monatomic ideal gas to ambient values of   and   is thus  , where relative entropy


The resulting contours of constant relative entropy, shown at right for a mole of Argon at standard temperature and pressure, for example put limits on the conversion of hot to cold as in flame-powered air-conditioning or in the unpowered device to convert boiling-water to ice-water discussed here.[28] Thus relative entropy measures thermodynamic availability in bits.

Quantum information theoryEdit

For density matrices   and   on a Hilbert space, the quantum relative entropy from   to   is defined to be


In quantum information science the minimum of   over all separable states   can also be used as a measure of entanglement in the state  .

Relationship between models and realityEdit

Just as relative entropy of "actual from ambient" measures thermodynamic availability, relative entropy of "reality from a model" is also useful even if the only clues we have about reality are some experimental measurements. In the former case relative entropy describes distance to equilibrium or (when multiplied by ambient temperature) the amount of available work, while in the latter case it tells you about surprises that reality has up its sleeve or, in other words, how much the model has yet to learn.

Although this tool for evaluating models against systems that are accessible experimentally may be applied in any field, its application to selecting a statistical model via Akaike information criterion are particularly well described in papers[29] and a book[30] by Burnham and Anderson. In a nutshell the relative entropy of reality from a model may be estimated, to within a constant additive term, by a function of the deviations observed between data and the model's predictions (like the mean squared deviation) . Estimates of such divergence for models that share the same additive term can in turn be used to select among models.

When trying to fit parametrized models to data there are various estimators which attempt to minimize relative entropy, such as maximum likelihood and maximum spacing estimators.[citation needed]

Symmetrised divergenceEdit

Kullback & Leibler (1951) also considered the symmetrized function:[5]


which they referred to as the "divergence", though today the "KL divergence" refers to the asymmetric function (see § Etymology for the evolution of the term). This function is symmetric and nonnegative, and had already been defined and used by Harold Jeffreys in 1948;[6] it is accordingly called the Jeffreys divergence.

This quantity has sometimes been used for feature selection in classification problems, where   and   are the conditional pdfs of a feature under two different classes. In the Banking and Finance industries, this quantity is referred to as Population Stability Index (PSI), and is used to assess distributional shifts in model features through time.

An alternative is given via the   divergence,


which can be interpreted as the expected information gain about   from discovering which probability distribution   is drawn from,   or  , if they currently have probabilities   and   respectively.[clarification needed][citation needed]

The value   gives the Jensen–Shannon divergence, defined by


where   is the average of the two distributions,


  can also be interpreted as the capacity of a noisy information channel with two inputs giving the output distributions   and  . The Jensen–Shannon divergence, like all f-divergences, is locally proportional to the Fisher information metric. It is similar to the Hellinger metric (in the sense that it induces the same affine connection on a statistical manifold).

Furthermore, the Jensen-Shannon divergence can be generalized using abstract statistical M-mixtures relying on an abstract mean M. [31][32]

Relationship to other probability-distance measuresEdit

There are many other important measures of probability distance. Some of these are particularly connected with relative entropy. For example:

  • The total variation distance,  . This is connected to the divergence through Pinsker's inequality:  
  • The family of Rényi divergences generalize relative entropy. Depending on the value of a certain parameter,  , various inequalities may be deduced.

Other notable measures of distance include the Hellinger distance, histogram intersection, Chi-squared statistic, quadratic form distance, match distance, Kolmogorov–Smirnov distance, and earth mover's distance.[33]

Data differencingEdit

Just as absolute entropy serves as theoretical background for data compression, relative entropy serves as theoretical background for data differencing – the absolute entropy of a set of data in this sense being the data required to reconstruct it (minimum compressed size), while the relative entropy of a target set of data, given a source set of data, is the data required to reconstruct the target given the source (minimum size of a patch).

See alsoEdit


  1. ^ Kullback, S.; Leibler, R.A. (1951). "On information and sufficiency". Annals of Mathematical Statistics. 22 (1): 79–86. doi:10.1214/aoms/1177729694. JSTOR 2236703. MR 0039968.
  2. ^ a b c Kullback, S. (1959), Information Theory and Statistics, John Wiley & Sons. Republished by Dover Publications in 1968; reprinted in 1978: ISBN 0-8446-5625-9.
  3. ^ a b c d e Amari 2016, p. 11.
  4. ^ a b Amari 2016, p. 28.
  5. ^ a b Kullback & Leibler 1951, p. 80.
  6. ^ a b Jeffreys 1948, p. 158.
  7. ^ Kullback 1959, p. 7.
  8. ^ Kullback, S. (1987). "Letter to the Editor: The Kullback–Leibler distance". The American Statistician. 41 (4): 340–341. doi:10.1080/00031305.1987.10475510. JSTOR 2684769.
  9. ^ Kullback 1959, p. 6.
  10. ^ MacKay, David J.C. (2003). Information Theory, Inference, and Learning Algorithms (First ed.). Cambridge University Press. p. 34. ISBN 9780521642989.
  11. ^ Bishop C. (2006). Pattern Recognition and Machine Learning
  12. ^ Burnham, K. P.; Anderson, D. R. (2002). Model Selection and Multi-Model Inference (2nd ed.). Springer. p. 51. ISBN 9780387953649.
  13. ^ Hobson, Arthur (1971). Concepts in statistical mechanics. New York: Gordon and Breach. ISBN 978-0677032405.
  14. ^ Sanov, I.N. (1957). "On the probability of large deviations of random magnitudes". Mat. Sbornik. 42 (84): 11–44.
  15. ^ Novak S.Y. (2011), Extreme Value Methods with Applications to Finance ch. 14.5 (Chapman & Hall). ISBN 978-1-4398-3574-6.
  16. ^ Bonnici, V. (2020). "Kullback-Leibler divergence between quantum distributions, and its upper-bound". arXiv:2008.05932 [cs.LG].
  17. ^ See the section "differential entropy – 4" in Relative Entropy video lecture by Sergio Verdú NIPS 2009
  18. ^ Donsker, Monroe D.; Varadhan, SR Srinivasa (1983). "Asymptotic evaluation of certain Markov process expectations for large time. IV". Communications on Pure and Applied Mathematics: 183–212.
  19. ^ Duchi J., "Derivations for Linear Algebra and Optimization".
  20. ^ a b Cover, Thomas M.; Thomas, Joy A. (1991), Elements of Information Theory, John Wiley & Sons
  21. ^ Chaloner, K.; Verdinelli, I. (1995). "Bayesian experimental design: a review". Statistical Science. 10 (3): 273–304. doi:10.1214/ss/1177009939.
  22. ^ Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. (2007). "Section 14.7.2. Kullback–Leibler Distance". Numerical Recipes: The Art of Scientific Computing (3rd ed.). Cambridge University Press. ISBN 978-0-521-88068-8.
  23. ^ Myron Tribus (1961), Thermodynamics and Thermostatics (D. Van Nostrand, New York)
  24. ^ Jaynes, E. T. (1957). "Information theory and statistical mechanics" (PDF). Physical Review. 106 (4): 620–630. Bibcode:1957PhRv..106..620J. doi:10.1103/physrev.106.620.
  25. ^ Jaynes, E. T. (1957). "Information theory and statistical mechanics II" (PDF). Physical Review. 108 (2): 171–190. Bibcode:1957PhRv..108..171J. doi:10.1103/physrev.108.171.
  26. ^ J.W. Gibbs (1873), "A method of geometrical representation of thermodynamic properties of substances by means of surfaces", reprinted in The Collected Works of J. W. Gibbs, Volume I Thermodynamics, ed. W. R. Longley and R. G. Van Name (New York: Longmans, Green, 1931) footnote page 52.
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