A Maximization Technique Occurring in the Statistical Analysis of Probabilistic Functions of Markov Chains
01 Feb 1970-Annals of Mathematical Statistics (Institute of Mathematical Statistics)-Vol. 41, Iss: 1, pp 164-171
About: This article is published in Annals of Mathematical Statistics.The article was published on 1970-02-01 and is currently open access. It has received 4618 citations till now. The article focuses on the topics: Examples of Markov chains & Markov chain.
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01 Feb 1989TL;DR: In this paper, the authors provide an overview of the basic theory of hidden Markov models (HMMs) as originated by L.E. Baum and T. Petrie (1966) and give practical details on methods of implementation of the theory along with a description of selected applications of HMMs to distinct problems in speech recognition.
Abstract: This tutorial provides an overview of the basic theory of hidden Markov models (HMMs) as originated by L.E. Baum and T. Petrie (1966) and gives practical details on methods of implementation of the theory along with a description of selected applications of the theory to distinct problems in speech recognition. Results from a number of original sources are combined to provide a single source of acquiring the background required to pursue further this area of research. The author first reviews the theory of discrete Markov chains and shows how the concept of hidden states, where the observation is a probabilistic function of the state, can be used effectively. The theory is illustrated with two simple examples, namely coin-tossing, and the classic balls-in-urns system. Three fundamental problems of HMMs are noted and several practical techniques for solving these problems are given. The various types of HMMs that have been studied, including ergodic as well as left-right models, are described. >
21,819 citations
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TL;DR: The purpose of this tutorial paper is to give an introduction to the theory of Markov models, and to illustrate how they have been applied to problems in speech recognition.
Abstract: The basic theory of Markov chains has been known to mathematicians and engineers for close to 80 years, but it is only in the past decade that it has been applied explicitly to problems in speech processing. One of the major reasons why speech models, based on Markov chains, have not been developed until recently was the lack of a method for optimizing the parameters of the Markov model to match observed signal patterns. Such a method was proposed in the late 1960's and was immediately applied to speech processing in several research institutions. Continued refinements in the theory and implementation of Markov modelling techniques have greatly enhanced the method, leading to a wide range of applications of these models. It is the purpose of this tutorial paper to give an introduction to the theory of Markov models, and to illustrate how they have been applied to problems in speech recognition.
4,546 citations
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TL;DR: In this paper, the authors proposed an iterative method for scene reconstruction based on a non-degenerate Markov Random Field (MRF) model, where the local characteristics of the original scene can be represented by a nondegenerate MRF and the reconstruction can be estimated according to standard criteria.
Abstract: may 7th, 1986, Professor A. F. M. Smith in the Chair] SUMMARY A continuous two-dimensional region is partitioned into a fine rectangular array of sites or "pixels", each pixel having a particular "colour" belonging to a prescribed finite set. The true colouring of the region is unknown but, associated with each pixel, there is a possibly multivariate record which conveys imperfect information about its colour according to a known statistical model. The aim is to reconstruct the true scene, with the additional knowledge that pixels close together tend to have the same or similar colours. In this paper, it is assumed that the local characteristics of the true scene can be represented by a nondegenerate Markov random field. Such information can be combined with the records by Bayes' theorem and the true scene can be estimated according to standard criteria. However, the computational burden is enormous and the reconstruction may reflect undesirable largescale properties of the random field. Thus, a simple, iterative method of reconstruction is proposed, which does not depend on these large-scale characteristics. The method is illustrated by computer simulations in which the original scene is not directly related to the assumed random field. Some complications, including parameter estimation, are discussed. Potential applications are mentioned briefly.
4,490 citations
Cites background from "A Maximization Technique Occurring ..."
...…regard the y as mixture data with the complication that the underlying (unobservable) classification variables making up x are not independent; see Baum et al. (1970) for a version of E M for the one-dimensional case, in which the components of x follow a Markov chain, and Geman and McClure…...
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TL;DR: This paper presents a tutorial introduction to the use of variational methods for inference and learning in graphical models (Bayesian networks and Markov random fields), and describes a general framework for generating variational transformations based on convex duality.
Abstract: This paper presents a tutorial introduction to the use of variational methods for inference and learning in graphical models (Bayesian networks and Markov random fields). We present a number of examples of graphical models, including the QMR-DT database, the sigmoid belief network, the Boltzmann machine, and several variants of hidden Markov models, in which it is infeasible to run exact inference algorithms. We then introduce variational methods, which exploit laws of large numbers to transform the original graphical model into a simplified graphical model in which inference is efficient. Inference in the simpified model provides bounds on probabilities of interest in the original model. We describe a general framework for generating variational transformations based on convex duality. Finally we return to the examples and demonstrate how variational algorithms can be formulated in each case.
4,093 citations
Cites background from "A Maximization Technique Occurring ..."
...The cost is that we have obtained a free parameter λ that must be set, once for each x....
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References
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2,919 citations
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TL;DR: In this paper, a polynomial with nonnegative coefficients homogeneous of degree d in its variables is shown to be polynomially homogeneous unless 3(3(x))>P(x), where 3(x)=x.
Abstract: 1. Summary. The object of this note is to prove the theorem below and sketch two applications, one to statistical estimation for (proba-bilistic) functions of Markov processes [l] and one to Blakley's model for ecology [4]. 2. Result. THEOREM. Let P(x)=P({xij}) be a polynomial with nonnegative coefficients homogeneous of degree d in its variables {##}. Let x= {##} be any point of the domain D: ## §:(), ]pLi ## = 1, i = l, • • • , p, j=l, • • • , q%. For x= {xij} ££> let 3(#) = 3{##} denote the point of D whose i, j coordinate is (dP\\ \\ f « dP 3(*) Then P(3(x))>P(x) unless 3(x)=x. Notation, fi will denote a doubly indexed array of nonnegative integers: fx= {M#}> i = l> • • • >
1,145 citations
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TL;DR: The Gamma function as discussed by the authors is a generalized factorial function that can be used to estimate the probability distribution of a probability distribution, and it has been used in many applications, e.g., as part of probability distributions.
Abstract: In what follows, we introduce the classical Gamma function in Sect. 2.1. It is essentially understood to be a generalized factorial. However, there are many further applications, e.g., as part of probability distributions (see, e.g., Evans et al. 2000). The main properties of the Gamma function are explained in this chapter (for a more detailed discussion the reader is referred to, e.g., Artin (1964), Lebedev (1973), Muller (1998), Nielsen (1906), and Whittaker and Watson (1948) and the references therein).
267 citations