Kullback–Leibler divergence
In probability theory and information theory, the Kullback–Leibler divergence,^{[1]}^{[2]} also called discrimination information (the name preferred by Kullback^{[3]}), information divergence, information gain, relative entropy, KLIC, KL divergence, is a measure of the difference between two probability distributions P and Q. It is not symmetric in P and Q. In applications, P typically represents the "true" distribution of data, observations, or a precisely calculated theoretical distribution, while Q typically represents a theory, model, description, or approximation of P.
Specifically, the Kullback–Leibler divergence from Q to P, denoted D_{KL}(P‖Q), is a measure of the information gained when one revises one's beliefs from the prior probability distribution Q to the posterior probability distribution P. In other words, it is the amount of information lost when Q is used to approximate P.^{[4]} Kullback–Leibler divergence also measures the expected number of extra bits required to code samples from P using a code optimized for Q rather than the code optimized for P.
Although it is often intuited as a way of measuring the distance between probability distributions, the Kullback–Leibler divergence is not a true metric. It does not obey the triangle inequality, and in general D_{KL}(P‖Q) does not equal D_{KL}(Q‖P). However, its infinitesimal form, specifically its Hessian, gives a metric tensor known as the Fisher information metric.
The Kullback–Leibler divergence is a special case of a broader class of divergences called fdivergences as well as the class of Bregman divergences. It is the only such divergence over probabilities that is a member of both classes.
The Kullback–Leibler divergence was originally introduced by Solomon Kullback and Richard Leibler in 1951 as the directed divergence between two distributions. It is discussed in Kullback's historic text, Information Theory and Statistics.^{[2]}
The Kullback–Leibler divergence is sometimes also called the information gain achieved if P is used instead of Q. It is also called the relative entropy of P with respect to Q.
Definition
For discrete probability distributions P and Q, the Kullback–Leibler divergence from Q to P is defined^{[5]} to be
In words, it is the expectation of the logarithmic difference between the probabilities P and Q, where the expectation is taken using the probabilities P. The Kullback–Leibler divergence is defined only if Q(i)=0 implies P(i)=0, for all i (absolute continuity). Whenever P(i) is zero the contribution of the ith term is interpreted as zero because .
For distributions P and Q of a continuous random variable, the Kullback–Leibler divergence is defined to be the integral:^{[6]}
where p and q denote the densities of P and Q.
More generally, if P and Q are probability measures over a set X, and P is absolutely continuous with respect to Q, then the Kullback–Leibler divergence from Q to P is defined as
where is the Radon–Nikodym derivative of P with respect to Q, and provided the expression on the righthand side exists. Equivalently, this can be written as
which we recognize as the entropy of P relative to Q. Continuing in this case, if is any measure on X for which and exist (meaning that p and q are absolutely continuous with respect to ), then the Kullback–Leibler divergence from Q to P is given as
The logarithms in these formulae are taken to base 2 if information is measured in units of bits, or to base e if information is measured in nats. Most formulas involving the Kullback–Leibler divergence hold regardless of the base of the logarithm.
Various conventions exist for referring to D_{KL}(P‖Q) in words. Often it is referred to as the divergence between P and Q; however this fails to convey the fundamental asymmetry in the relation. Sometimes, as in this article, it may be found described as the divergence of P from, or with respect to Q. This reflects the asymmetry in Bayesian inference, which starts from a prior Q and updates to the posterior P.
Characterization
Arthur Hobson proved that the Kullback–Leibler divergence is the only measure of difference between probability distributions that satisfies some desiderata, which are the canonical extension to those appearing in a commonly used characterization of entropy.^{[7]} Consequently, mutual information is the only measure of mutual dependence that obeys certain related conditions, since it can be defined in terms of KullbackLeibler divergence.
There is also a Bayesian characterization of the Kullback–Leibler divergence.^{[8]}
Motivation
In information theory, the Kraft–McMillan theorem establishes that any directly decodable coding scheme for coding a message to identify one value x_{i} out of a set of possibilities X can be seen as representing an implicit probability distribution q(x_{i})=2^{−li} over X, where l_{i} is the length of the code for x_{i} in bits. Therefore, the Kullback–Leibler divergence can be interpreted as the expected extra messagelength per datum that must be communicated if a code that is optimal for a given (wrong) distribution Q is used, compared to using a code based on the true distribution P.
where H(P,Q) is the cross entropy of P and Q, and H(P) is the entropy of P.
Note also that there is a relation between the Kullback–Leibler divergence and the "rate function" in the theory of large deviations.^{[9]}^{[10]}
Properties
 The Kullback–Leibler divergence is always nonnegative,
 a result known as Gibbs' inequality, with D_{KL}(P‖Q) zero if and only if P = Q almost everywhere. The entropy H(P) thus sets a minimum value for the crossentropy H(P,Q), the expected number of bits required when using a code based on Q rather than P; and the Kullback–Leibler divergence therefore represents the expected number of extra bits that must be transmitted to identify a value x drawn from X, if a code is used corresponding to the probability distribution Q, rather than the "true" distribution P.
 The Kullback–Leibler divergence remains welldefined for continuous distributions, and furthermore is invariant under parameter transformations. For example, if a transformation is made from variable x to variable y(x), then, since P(x) dx = P(y) dy and Q(x) dx = Q(y) dy the Kullback–Leibler divergence 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 Kullback–Leibler divergence produces a dimensionally consistent quantity, since if x is a dimensioned variable, P(x) and Q(x) are also dimensioned, since e.g. P(x) dx 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^{[11]} (such as selfinformation or Shannon entropy), which can become undefined or negative for nondiscrete probabilities.
 The Kullback–Leibler divergence is additive for independent distributions in much the same way as Shannon entropy. If are independent distributions, with the joint distribution , and likewise, then
Kullback–Leibler divergence for multivariate normal distributions
Suppose that we have two multivariate normal distributions, with means and with (nonsingular) covariance matrices . If the two distributions have the same dimension, k, then the Kullback–Leibler divergence between the distributions is as follows.^{[12]}
The logarithm in the last term must be taken to base e since all terms apart from the last are basee 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 log_{e} 2 yields the divergence in bits.
Relation to metrics
One might be tempted to call the Kullback–Leibler divergence a "distance metric" on the space of probability distributions, but this would not be correct as it is not symmetric – that is, , – nor does it satisfy the triangle inequality. Even so, being a premetric, it generates a topology on the space of probability distributions. More concretely, if is a sequence of distributions such that

(1)
then it is said that

.
(2)
Pinsker's inequality entails that

,
(3)
where the latter stands for the usual convergence in total variation.
Following Rényi (1970, 1961)^{[13]}^{[14]}
Fisher information metric
However, the Kullback–Leibler divergence is rather directly related to a metric, specifically, the Fisher information metric. This can be made explicit as follows. Assume that the probability distributions P and Q are both parameterized by some (possibly multidimensional) 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 j direction, and the corresponding rate of change in the probability distribution. Since the Kullback–Leibler divergence has an absolute minimum 0 for P = Q, 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 parameter space, called the Fisher information metric.
Relation to other quantities of information theory
Many of the other quantities of information theory can be interpreted as applications of the Kullback–Leibler divergence to specific cases.
The selfinformation,
is the Kullback–Leibler divergence of the probability distribution P(i) from a Kronecker delta representing certainty that i = m — i.e. the number of extra bits that must be transmitted to identify i if only the probability distribution P(i) is available to the receiver, not the fact that i = m.
The mutual information,
is the Kullback–Leibler divergence of the product P(X)P(Y) of the two marginal probability distributions from the joint probability distribution P(X,Y) — i.e. the expected number of extra bits that must be transmitted to identify X and Y if they are coded using only their marginal distributions instead of the joint distribution. Equivalently, if the joint probability P(X,Y) is known, it is the expected number of extra bits that must on average be sent to identify Y if the value of X is not already known to the receiver.
The Shannon entropy,
is the number of bits which would have to be transmitted to identify X from N equally likely possibilities, less the Kullback–Leibler divergence of the uniform distribution P_{U}(X) from the true distribution P(X) — i.e. less the expected number of bits saved, which would have had to be sent if the value of X were coded according to the uniform distribution P_{U}(X) rather than the true distribution P(X).
The conditional entropy,
is the number of bits which would have to be transmitted to identify X from N equally likely possibilities, less the Kullback–Leibler divergence of the product distribution P_{U}(X) P(Y) from the true joint distribution P(X,Y) — i.e. less the expected number of bits saved which would have had to be sent if the value of X were coded according to the uniform distribution P_{U}(X) rather than the conditional distribution P(X  Y) of X given Y.
The cross entropy between two probability distributions 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 , rather than the "true" distribution . The cross entropy for two distributions and over the same probability space is thus defined as follows:
Kullback–Leibler divergence and Bayesian updating
In Bayesian statistics the Kullback–Leibler divergence 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 we can estimate 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 design
A common goal in Bayesian experimental design is to maximise the expected Kullback–Leibler divergence between the prior and the posterior.^{[15]} When posteriors are approximated to be Gaussian distributions, a design maximising the expected Kullback–Leibler divergence is called Bayes doptimal.
Discrimination information
The Kullback–Leibler divergence D_{KL}( p(xH_{1}) ‖ p(xH_{0}) ) can also be interpreted as the expected discrimination information for H_{1} over H_{0}: the mean information per sample for discriminating in favor of a hypothesis H_{1} against a hypothesis H_{0}, when hypothesis H_{1} is true.^{[16]} Another name for this quantity, given to it by I.J. Good, is the expected weight of evidence for H_{1} over H_{0} to be expected from each sample.
The expected weight of evidence for H_{1} over H_{0} is not the same as the information gain expected per sample about the probability distribution p(H) 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 information
The idea of Kullback–Leibler divergence as discrimination information led Kullback to propose the Principle of Minimum Discrimination Information (MDI): given new facts, a new distribution f should be chosen which is as hard to discriminate from the original distribution f_{0} as possible; so that the new data produces as small an information gain D_{KL}( f ‖ f_{0} ) as possible.
For example, if one had a prior distribution p(x,a) over x and a, and subsequently learnt the true distribution of a was u(a), the Kullback–Leibler divergence between the new joint distribution for x and a, q(xa) u(a), and the earlier prior distribution would be:
i.e. the sum of the Kullback–Leibler divergence of p(a) the prior distribution for a from the updated distribution u(a), plus the expected value (using the probability distribution u(a)) of the Kullback–Leibler divergence of the prior conditional distribution p(xa) from the new conditional distribution q(xa). (Note that often the later expected value is called the conditional Kullback–Leibler divergence (or conditional relative entropy) and denoted by D_{KL}(q(xa)‖p(xa))^{[17]}) This is minimized if q(xa) = p(xa) over the whole support of u(a); and we note that this result incorporates Bayes' theorem, if the new distribution u(a) is in fact a δ function representing certainty that a 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 Kullback–Leibler divergence continues to be just as relevant.
In the engineering literature, MDI is sometimes called the Principle of Minimum CrossEntropy (MCE) or Minxent for short. Minimising the Kullback–Leibler divergence from m to p with respect to m is equivalent to minimizing the crossentropy of p and m, since
which is appropriate if one is trying to choose an adequate approximation to p. However, this is just as often not the task one is trying to achieve. Instead, just as often it is m that is some fixed prior reference measure, and p that one is attempting to optimise by minimising D_{KL}(p‖m) subject to some constraint. This has led to some ambiguity in the literature, with some authors attempting to resolve the inconsistency by redefining crossentropy to be D_{KL}(p‖m), rather than H(p,m).
Relationship to available work
Surprisals^{[18]} 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.
Bestguess 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^{[19]} and quantum mechanically,^{[20]} minimizes Gibbs availability in entropy units^{[21]} 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) 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^{[22]} the work available relative to some ambient is obtained by multiplying ambient temperature by Kullback–Leibler divergence 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 Kullback–Leibler divergence . The resulting contours of constant Kullback–Leibler divergence, 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 flamepowered airconditioning or in the unpowered device to convert boilingwater to icewater discussed here.^{[23]} Thus Kullback–Leibler divergence measures thermodynamic availability in bits.
Quantum information theory
For density matrices P and Q on a Hilbert space, the K–L divergence (or quantum relative entropy as it is often called in this case) from Q to P is defined to be
In quantum information science the minimum of over all separable states Q can also be used as a measure of entanglement in the state P.
Relationship between models and reality
Just as Kullback–Leibler divergence of "actual from ambient" measures thermodynamic availability, Kullback–Leibler divergence 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 Kullback–Leibler divergence 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^{[24]} and a book^{[25]} by Burnham and Anderson. In a nutshell the Kullback–Leibler divergence of reality from a model may be estimated, to within a constant additive term, by a function (like the squares summed) of the deviations observed between data and the model's predictions. 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 Kullback–Leibler divergence, such as maximum likelihood and maximum spacing estimators.
Symmetrised divergence
Kullback and Leibler themselves actually defined the divergence as:
which is symmetric and nonnegative. This quantity has sometimes been used for feature selection in classification problems, where P and Q are the conditional pdfs of a feature under two different classes.
An alternative is given via the λ divergence,
which can be interpreted as the expected information gain about X from discovering which probability distribution X is drawn from, P or Q, if they currently have probabilities λ and (1 − λ) respectively.
The value λ = 0.5 gives the Jensen–Shannon divergence, defined by
where M is the average of the two distributions,
D_{JS} can also be interpreted as the capacity of a noisy information channel with two inputs giving the output distributions p and q. The Jensen–Shannon divergence, like all fdivergences, is locally proportional to the Fisher information metric. It is similar to the Hellinger metric (in the sense that induces the same affine connection on a statistical manifold), and equal to onehalf the socalled Jeffreys divergence.^{[26]}^{[27]}
Relationship to other probabilitydistance measures
There are many other important measures of probability distance. Some of these are particularly connected with the Kullback–Leibler divergence. For example:
 The total variation distance, This is connected to the divergence through Pinsker's inequality:
 The family of Rényi divergences provide generalizations of the Kullback–Leibler divergence. Depending on the value of a certain parameter, , various inequalities may be deduced.
Other notable measures of distance include the Hellinger distance, histogram intersection, Chisquared statistic, quadratic form distance, match distance, Kolmogorov–Smirnov distance, and earth mover's distance.^{[26]}
Data differencing
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 also
 Akaike Information Criterion
 Bayesian information criterion
 Bregman divergence
 Crossentropy
 Deviance information criterion
 Entropic value at risk
 Entropy power inequality
 Information gain in decision trees
 Information gain ratio
 Information theory and measure theory
 Jensen–Shannon divergence
 Quantum relative entropy
 Solomon Kullback and Richard Leibler
References
 ↑ Kullback, S.; Leibler, R.A. (1951). "On information and sufficiency". Annals of Mathematical Statistics. 22 (1): 79–86. doi:10.1214/aoms/1177729694. MR 39968.
 1 2 Kullback S. (1959), Information Theory and Statistics (John Wiley & Sons).
 ↑ 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.
 ↑ Burnham K.P., Anderson D.R. (2002), Model Selection and MultiModel Inference (Springer). (2nd edition), p.51
 ↑ MacKay, David J.C. (2003). Information Theory, Inference, and Learning Algorithms (First ed.). Cambridge University Press. p. 34.
 ↑ Bishop C. (2006). Pattern Recognition and Machine Learning p. 55.
 ↑ Hobson, Arthur (1971). Concepts in statistical mechanics. New York: Gordon and Breach. ISBN 0677032404.
 ↑ Baez, John; Fritz, Tobias (2014). "A Bayesian characterization of relative entropy". Theory and Application of Categories. 29: 421–456. arXiv:1402.3067.
 ↑ Sanov, I.N. (1957). "On the probability of large deviations of random magnitudes". Matem. Sbornik. 42 (84): 11–44.
 ↑ Novak S.Y. (2011), Extreme Value Methods with Applications to Finance ch. 14.5 (Chapman & Hall). ISBN 9781439835746.
 ↑ See the section "differential entropy  4" in Relative Entropy video lecture by Sergio Verdú NIPS 2009
 ↑ Duchi J., "Derivations for Linear Algebra and Optimization", p. 13.
 ↑ Rényi A. (1970). Probability Theory. Elsevier. Appendix, Sec.4. ISBN 0486458679.
 ↑ Rényi, A. (1961), "On measures of entropy and information" (PDF), Proceedings of the 4th Berkeley Symposium on Mathematics, Statistics and Probability 1960, pp. 547–561
 ↑ Chaloner, K.; Verdinelli, I. (1995). "Bayesian experimental design: a review". Statistical Science. 10 (3): 273–304. doi:10.1214/ss/1177009939.
 ↑ 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 9780521880688
 ↑ Thomas M. Cover, Joy A. Thomas (1991) Elements of Information Theory (John Wiley & Sons), p.22
 ↑ Myron Tribus (1961), Thermodynamics and Thermostatics (D. Van Nostrand, New York)
 ↑ Jaynes, E. T. (1957). "Information theory and statistical mechanics" (PDF). Physical Review. 106: 620–630. Bibcode:1957PhRv..106..620J. doi:10.1103/physrev.106.620.
 ↑ Jaynes, E. T. (1957). "Information theory and statistical mechanics II" (PDF). Physical Review. 108: 171–190. Bibcode:1957PhRv..108..171J. doi:10.1103/physrev.108.171.
 ↑ 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.
 ↑ Tribus, M.; McIrvine, E. C. (1971). "Energy and information". Scientific American. 224: 179–186. doi:10.1038/scientificamerican0971179.
 ↑ Fraundorf, P. (2007). "Thermal roots of correlationbased complexity". Complexity. 13 (3): 18–26. doi:10.1002/cplx.20195.
 ↑ Burnham, K.P.; Anderson, D.R. (2001). "Kullback–Leibler information as a basis for strong inference in ecological studies". Wildlife Research. 28: 111–119. doi:10.1071/WR99107.
 ↑ Burnham, K. P. and Anderson D. R. (2002), Model Selection and Multimodel Inference: A Practical InformationTheoretic Approach, Second Edition (Springer Science) ISBN 9780387953649.
 1 2 Rubner, Y.; Tomasi, C.; Guibas, L. J. (2000). "The earth mover's distance as a metric for image retrieval". International Journal of Computer Vision. 40 (2): 99–121.
 ↑ Jeffreys, H. (1946). "An invariant form for the prior probability in estimation problems". Proceedings of the Royal Society of London, Series A. 186: 453–461. Bibcode:1946RSPSA.186..453J. doi:10.1098/rspa.1946.0056. JSTOR 97883.
External links
 Information Theoretical Estimators Toolbox
 Ruby gem for calculating Kullback–Leibler divergence
 Jon Shlens' tutorial on Kullback–Leibler divergence and likelihood theory
 Matlab code for calculating Kullback–Leibler divergence for discrete distributions
 Sergio Verdú, Relative Entropy, NIPS 2009. Onehour video lecture.
 A modern summary of infotheoretic divergence measures