The expectationmaximization em algorithm is a convenient tool for approximating maximum likelihood estimators in situations when available data are incomplete, as is the case f. The difference between the two quantities is the kullbackleibler divergence or relative entropy. Vision researchers have created an incredible range of algorithms and systems to detect, track, recognize, and contextualize objects in a scene, using a myriad of internal models to represent their problem and solution. A new refinement of jensens inequality in linear spaces with applications article pdf available in mathematical and computer modelling 52910. If we use the generalized jensens inequality instead, we can get the weighted amgm inequality. Document advertisement m athematical l inguistics a ndr a s k ornai final draft version 1. The bound 0 can be hence merely understood as a reliability bound imposed on the conveyed information 4. Kullbackleibler divergence, also known as kl divergence, relative entropy, or information divergence. Shannon entropy and kullbackleibler divergence cmu statistics. The kullbackleibler divergence is a measure of the dissimilarity between two probability distributions. For discrete probability distributions and defined on the same probability.
Ayres, jensens inequality predicts effects of environmental variation, trends ecol. A companion volume bishop and nabney, 2008 will deal with practical aspects of pattern recognition and machine learning, and will be accompanied by matlab software implementing most of the algorithms discussed in this book. Consider a pair of jointly distributed random variables and. Comparison of the normalized jensen functionals of two. We are going to give two separate definitions of kullbackleibler kl divergence, one for discrete random variables and one for continuous variables. Gms are a flexible inference framework based on factorizing a highdimensional multivariate joint distribution into lowerdimensional conditionals and marginals 15. Graphical models for inferring single molecule dynamics. The first kernel selected has the greatest weight and hence it contains the majority of the information about the free energy curve. Free energy computations by minimization of kullback.
Methods for applied macroeconomic research pdf free download. Kullback preferred the term discrimination information. A family of statistical symmetric divergences based on jensens. We can look at the minus kl divergence which equals to the expected value of the logarithm of the ratio. Statistical inference based on divergence measures. Full text of recent advances in applied probability.
We also provide new bounds for bhattacharyya divergence. Verejones an introduction to the theory of point processes volume 12002 springer. Pricing nondiversifiable credit risk in the corporate eurobond market. In probability theory and information theory, the kullbackleibler divergence. Assume that the marginal distribution of is uniform over a finite set of size i. Expectation propagation for poisson data iopscience. Pdf a new refinement of jensens inequality in linear. Saying that f of the expected value of t is greater than or equal to the expected value of f of t. And thats kind of easy to prove because you can use minus kl divergence.
In a gm, the nodes of the graph represent either observable variables data, denoted by filled circles, latent variables hidden states, denoted by open circles, or fixed parameters denoted by dots. Why do we optimize kl divergence in addition to the optimization of matching a binomial distribution example given in the blog, i will give another case where optimization of divergence may occur. And finally, the kl divergence is nonnegative for any of the distributions. The former reasonings may, to a certain extent, vindicate the use of 0 in usual information theory. The jensenshannon divergence can be interpreted as the total kl. Jensens inequality states that, if a function fx is convex, then. We refer to the book of deza and deza 2 and the survey of. The next result shows that convergence in the lq norm obtains when we know that convergence in the lq norm occurs, q q 0. Given its generality, the inequality appears in many forms depending on the context, some of which are presented below. In information theory, gibbs inequality is a statement about the mathematical entropy of a. Renyi divergence and kullbackleibler divergence arxiv. The second connection which we intend to advocate and progress here is the connection with the maximal entropy principle maxent. The task at step a is to construct the ith tilted distribution.
However in order to effectively make use of these algorithms sophisticated expert knowledge is required to understand and properly utilize the internal models used. So kullbackleibler divergence between a distribution and itself is 0. For continuous variables, jensens inequality takes the form f xpxdx fxpxdx. Its primary aim is to encourage and publish scholarship in digital history that offers disciplinespecific arguments and.
By jensens inequality, the divergence is always nonnegative, and it vanishes if and only if almost everywhere. Byrnes1 and anders lindquist2 1 2 washington university, st. It is straightforward to show that the kl divergence is never negative using jensens inequality and the concavity of the \log function. The expectationmaximization algorithm for illposed. New probabilistic inference algorithms that harness the. However, in spite of the fact that divergence statistics have become a very good alternative to the classical likelihood ratio test and the pearsontype statistic in discrete models, many statisticians remain unaware of this powerful approach. In mathematics, jensens inequality, named after the danish mathematician johan jensen, relates the value of a convex function of an integral to the integral of the convex function. We think of as the unknown parameter and as the data from which we have to draw inference about. On the other hand, if an equality sign is meant to convey the exactness of expressions on both sides, then the bigo actually is not the right tool, since it destroys some information in return for the convenience of simpler expressions. Top american libraries canadian libraries universal library community texts project gutenberg biodiversity heritage library childrens library.
This fact provided kullback and leibler 1951 the opportunity of introducing a measure of divergence, as a generalization of shannons entropy, called the kullbackleibler divergence. In particular, inspired by the idea of kullbackleibler divergence, we aim to propose a new method to compute the dynamic reliability of the component under stressstrength model. Em never decreases the likelihood the e and m steps together never decrease the log likelihood. Proof of nonnegativity of kl divergence using jensens.
The four main technical contributions of this thesis are. Maximum likelihood estimation and complexity regularization. The central objective of this thesis is to develop new algorithms for inference in probabilistic graphical models that improve upon the stateoftheart and lend new insight into the computational nature of probabilistic inference. Taken from the book elements of information theory by thomas m.
As applications, we give some valuable upper and lower bounds for the agm inequality, which lead to some comparison results regarding kullbackleibler divergence and shannons entropy. The result can alternatively be proved using jensens inequality, the log sum. The kullbackleibler divergence quantifies the average number of extra bits required to represent a distribution \p\ when using an arbitrary distribution \q\. In other words, the kl divergence is not symmetric. So the average with weights being the probabilities. Exponentiating both sides, we get the amgm inequality. The idea of using functionals of information theory, such as entropies or divergences, in statistical inference is not new. In this paper, we compare normalized jensen functionals of two given convex functions defined on an interval of the real line. Since logis strictly concave, jensens inequality holds in equality only if qxpx 1 for all xvalues. The kullbackleibler kl divergence is a measure of the di erence between two.
Pattern recognition and machine learning pdf free download. Lower bounds using information theory tools lecturer. In the article, we establish an inequality for csiszar divergence associated with sconvex functions, present several inequalities for kullbackleibler, renyi, hellinger, chisquare, jefferys, and variational distance divergences by using particular sconvex functions in the csiszar divergence. Pdf we introduce a novel parametric family of symmetric.
Mean, mode and median relationship if 1 0 with equality if, and only if, the two distributions are equal. Quantitative and qualitative approaches to the development. Lby jensen or, equivalently, from the nonnegativity of kl if the mstep is executed so that. Mexas omnitab a simple language for statistical computations. If p and q are probability distributions on the real line, such that p is absolutely continuous with respect to q, i. Video created by national research university higher school of economics for the course bayesian methods for machine learning. Association of jensens inequality for s convex function. Journal of statistical computation and simulation volume 1, number 2, 1972 nils aall barricell numerical testing of evolution theories 97127 d.
In information theory and statistics, kullbacks inequality is a lower bound on the kullbackleibler divergence expressed in terms of the large deviations rate function. Statistical inference based on divergence measures pdf. It follows that the relative entropy, or kullbackleibler divergence, between these two beta distributions is measured in nats the relative entropy, or kullbackleibler divergence, is always nonnegative. The divergence is discussed in kullbacks 1959 book, information theory and statistics. The uncertain generalized moment problem with complexity constraint christopher i. In information theory and statistics, kullbacks inequality is a lower bound on the kullback leibler divergence expressed in terms of the large deviations rate function. Expansionofthekullback4leiblerdivergence, andanewclassofinformation metrics david.