Above are papers related to the following paper: [Paper_ID] Paper #4335. 	[Title] A VIEW OF THE EM ALGORITHM THAT JUSTIFIES INCREMENTAL, SPARSE, AND OTHER VARIANTS [Abstract] The EM algorithm performs maximum likelihood estimation for data in which some variables are unobserved. We present a function that resembles negative free energy and show that the M step maximizes this function with respect to the model parameters and the E step maximizes it with respect to the distribution over the unobserved variables. From this perspective, it is easy to justify an incremental variant of the EM algorithm in which the distribution for only one of the unobserved variables is recalculated in each E step. This variant is shown empirically to give faster convergence in a mixture estimation problem. A variant of the algorithm that exploits sparse conditional distributions is also described, and a wide range of other variant algorithms are also seen to be possible. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #41216. 	[Title] A simple algorithm that discovers efficient perceptual codes [Abstract] We describe the "wake-sleep" algorithm that allows a multilayer, unsupervised, neural network to build a hierarchy of representations of sensory input. The network has bottom-up "recognition" connections that are used to convert sensory input into underlying representations. Unlike most artificial neural networks, it also has top-down "generative" connections that can be used to reconstruct the sensory input from the representations. In the "wake" phase of the learning algorithm, the network is driven by the bottom-up recognition connections and the top-down generative connections are trained to be better at reconstructing the sensory input from the representation chosen by the recognition process. In the "sleep" phase, the network is driven top-down by the generative connections to produce a fantasized representation and a fantasized sensory input. The recognition connections are then trained to be better at recovering the fantasized representation from the fantasized sensory input. In both phases, the synaptic learning rule is simple and local. The combined effect of the two phases is to create representations of the sensory input that are efficient in the following sense: On average, it takes more bits to describe each sensory input vector directly than to first describe the representation of the sensory input chosen by the recognition process and then describe the difference between the sensory input and its reconstruction from the chosen representation. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #6913. 	[Title] On Convergence of the EM Algorithm and the Gibbs Sampler  SUMMARY [Abstract] In this article we investigate the relationship between the two popular algorithms, the EM algorithm and the Gibbs sampler. We show that the approximate rate of convergence of the Gibbs sampler by Gaussian approximation is equal to that of the corresponding EM type algorithm. This helps in implementing either of the algorithms as improvement strategies for one algorithm can be directly transported to the other. In particular, by running the EM algorithm we know approximately how many iterations are needed for convergence of the Gibbs sampler. We also obtain a result that under conditions, the EM algorithm used for finding the maximum likelihood estimates can be slower to converge than the corresponding Gibbs sampler for Bayesian inference which uses proper prior distributions. We illustrate our results in a number of realistic examples all based on the generalized linear mixed models. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #46468. 	[Title] A Statistical Approach to Decision Tree Modeling [Abstract] A statistical approach to decision tree modeling is described. In this approach, each decision in the tree is modeled parametrically as is the process by which an output is generated from an input and a sequence of decisions. The resulting model yields a likelihood measure of goodness of fit, allowing ML and MAP estimation techniques to be utilized. An efficient algorithm is presented to estimate the parameters in the tree. The model selection problem is presented and several alternative proposals are considered. A hidden Markov version of the tree is described for data sequences that have temporal dependencies. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #32260. 	[Title] On-Line Portfolio Selection Using Multiplicative Updates [Abstract] We present an on-line investment algorithm which achieves almost the same wealth as the best constant-rebalanced portfolio determined in hindsight from the actual market outcomes. The algorithm employs a multiplicative update rule derived using a framework introduced by Kivinen and Warmuth. Our algorithm is very simple to implement and requires only constant storage and computing time per stock in each trading period. We tested the performance of our algorithm on real stock data from the New York Stock Exchange accumulated during a 22-year period. On this data, our algorithm clearly outperforms the best single stock as well as Cover's universal portfolio selection algorithm. We also present results for the situation in which the investor has access to additional "side information." Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1135108. 	[Title] A Gentle Tutorial of the EM Algorithm and its Application to Parameter Estimation for Gaussian [Abstract] We describe the maximum-likelihood parameter estimation problem and how the Expectation-Maximization (EM) algorithm can be used for its solution. We first describe the abstract form of the EM algorithm as it is often given in the literature. We then develop the EM parameter estimation procedure for two applications: 1) finding the parameters of a mixture of Gaussian densities, and 2) finding the parameters of a hidden Markov model (HMM) (i.e., the Baum-Welch algorithm) for both discrete and Gaussian mixture observation models. We derive the update equations in fairly explicit detail but we do not prove any convergence properties. We try to emphasize intuition rather than mathematical rigor. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #24966. 	[Title] A Practical Bayesian Framework for Backprop Networks [Abstract] A quantitative and practical Bayesian framework is described for learning of mappings in feedforward networks. The framework makes possible: (1) objective comparisons between solutions using alternative network architectures; (2) objective stopping rules for network pruning or growing procedures; (3) objective choice of magnitude and type of weight decay terms or additive regularisers (for penalising large weights, etc.); (4) a measure of the effective number of well-determined parameters in a model; (5) quantified estimates of the error bars on network parameters and on network output; (6) objective comparisons with alternative learning and interpolation models such as splines and radial basis functions. The Bayesian `evidence' automatically embodies `Occam's razor,' penalising over-flexible and over-complex models. The Bayesian approach helps detect poor underlying assumptions in learning models. For learning models well matched to a problem, a good correlation between generalisation ability and the Bayesian evidence is obtained. This paper makes use of the Bayesian framework for regularisation and model comparison described in the companion paper `Bayesian interpolation' (MacKay, 1991a). This framework is due to Gull and Skilling (Gull, 1989a). Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1152904. 	[Title] GIBBS-MARKOV MODELS [Abstract] In this paper we present a framework for building probabilistic automata parameterized by context-dependent probabilities. Gibbs distributions are used to model state transitions and output generation, and parameter estimation is carried out using an EM algorithm where the M-step uses a generalized iterative scaling procedure. We discuss relations with certain classes of stochastic feedforward neural networks, a geometric interpretation for parameter estimation, and a simple example of a statistical language model constructed using this methodology. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1136393. 	[Title] Data-driven Modeling and Synthesis of Acoustical Instruments [Abstract] We present a framework for the analysis and synthesis of acoustical instruments based on data-driven probabilistic inference modeling. Audio time series and boundary conditions of a played instrument are recorded and the non-linear mapping from the control data into the audio space is inferred using the general inference framework of Cluster-Weighted Modeling. The resulting model is used for real-time synthesis of audio sequences from new input data. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #390922. 	[Title] A MEAN FIELD LEARNING ALGORITHM FOR UNSUPERVISED NEURAL NETWORKS [Abstract] We introduce a learning algorithm for unsupervised neural networks based on ideas from statistical mechanics. The algorithm is derived from a mean field approximation for large, layered sigmoid belief networks. We show how to (approximately) infer the statistics of these networks without resort to sampling. This is done by solving the mean field equations, which relate the statistics of each unit to those of its Markov blanket. Using these statistics as target values, the weights in the network are adapted by a local delta rule. We evaluate the strengths and weaknesses of these networks for problems in statistical pattern recognition. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #906. 	[Title] Regularization Theory and Neural Networks Architectures [Abstract] We had previously shown that regularization principles lead to approximation schemes which are equivalent to networks with one layer of hidden units, called Regularization Networks. In particular, standard smoothness functionals lead to a subclass of regularization networks, the well known Radial Basis Functions approximation schemes. This paper shows that regularization networks encompass a much broader range of approximation schemes, including many of the popular general additive models and some of the neural networks. In particular, we introduce new classes of smoothness functionals that lead to different classes of basis functions. Additive splines as well as some tensor product splines can be obtained from appropriate classes of smoothness functionals. Furthermore, the same generalization that extends Radial Basis Functions (RBF) to Hyper Basis Functions (HBF) also leads from additive models to ridge approximation models, containing as special cases Breiman's hinge functions, some forms of Projection Pursuit Regression and several types of neural networks. We propose to use the term Generalized Regularization Networks for this broad class of approximation schemes that follow from an extension of regularization. In the probabilistic interpretation of regularization, the different classes of basis functions correspond to different classes of prior probabilities on the approximating function spaces, and therefore to different types of smoothness assumptions. In summary, different multilayer networks with one hidden layer, which we collectively call Generalized Regularization Networks, correspond to different classes of priors and associated smoothness functionals in a classical regularization principle. Three broad classes are a) Radial Basis Functions that can be generalized to Hyper Basis Functions, b) some tensor product splines, and c) additive splines that can be generalized to schemes of the type of ridge approximation, hinge functions and several perceptron-like neural networks with one-hidden layer. 1 This paper will appear on Neural Computation, vol. 7, pages 219-269, 1995. An earlier version of Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #755082. 	[Title] DRAFT Cluster-Weighted Modeling for Time Series Prediction and Characterization [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1136397. 	[Title] State Reconstruction for Determining Predictability in Driven Nonlinear Acoustical Systems [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1152272. 	[Title] GIBBS-MARKOV MODELS [Abstract] In this paper we present a framework for building probabilistic automata parameterized by context-dependent probabilities. Gibbs distributions are used to model state transitions and output generation, and parameter estimation is carried out using an EM algorithm where the M-step uses a generalized iterative scaling procedure. We discuss relations with certain classes of stochastic feedforward neural networks, a geometric interpretation for parameter estimation, and a simple example of a statistical language model constructed using this methodology. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1115291. 	[Title] How Many Clusters? Which Clustering Method? Answers Via Model-Based Cluster Analysis 1 [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #28957. 	[Title] Operations for Learning with Graphical Models decomposition techniques and the demonstration that graphical models provide [Abstract] This paper is a multidisciplinary review of empirical, statistical learning from a graphical model perspective. Well-known examples of graphical models include Bayesian networks, directed graphs representing a Markov chain, and undirected networks representing a Markov field. These graphical models are extended to model data analysis and empirical learning using the notation of plates. Graphical operations for simplifying and manipulating a problem are provided including decomposition, differentiation, and the manipulation of probability models from the exponential family. Two standard algorithm schemas for learning are reviewed in a graphical framework: Gibbs sampling and the expectation maximization algorithm. Using these operations and schemas, some popular algorithms can be synthesized from their graphical specification. This includes versions of linear regression, techniques for feed-forward networks, and learning Gaussian and discrete Bayesian networks from data. The paper concludes by sketching some implications for data analysis and summarizing how some popular algorithms fall within the framework presented. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #3229. 	[Title] Book Review  Introduction to the Theory of Neural Computation Reviewed by: 2 [Abstract] Neural computation, also called connectionism, parallel distributed processing, neural network modeling or brain-style computation, has grown rapidly in the last decade. Despite this explosion, and ultimately because of impressive applications, there has been a dire need for a concise introduction from a theoretical perspective, analyzing the strengths and weaknesses of connectionist approaches and establishing links to other disciplines, such as statistics or control theory. The Introduction to the Theory of Neural Computation by Hertz, Krogh and Palmer (subsequently referred to as HKP) is written from the perspective of physics, the home discipline of the authors. The book fulfills its mission as an introduction for neural network novices, provided that they have some background in calculus, linear algebra, and statistics. It covers a number of models that are often viewed as disjoint. Critical analyses and fruitful comparisons between these models Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #50336. 	[Title] Computing upper and lower bounds on likelihoods in intractable networks [Abstract] We present deterministic techniques for computing upper and lower bounds on marginal probabilities in sigmoid and noisy-OR networks. These techniques become useful when the size of the network (or clique size) precludes exact computations. We illustrate the tightness of the bounds by numerical experi ments. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #50337. 	[Title] Recursive algorithms for approximating probabilities in graphical models [Abstract] MIT Computational Cognitive Science Technical Report 9604 Abstract We develop a recursive node-elimination formalism for efficiently approximating large probabilistic networks. No constraints are set on the network topologies. Yet the formalism can be straightforwardly integrated with exact methods whenever they are/become applicable. The approximations we use are controlled: they maintain consistently upper and lower bounds on the desired quantities at all times. We show that Boltzmann machines, sigmoid belief networks, or any combination (i.e., chain graphs) can be handled within the same framework. The accuracy of the methods is verified exper imentally. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1152162. 	[Title] GIBBS-MARKOV MODELS [Abstract] In this paper we present a framework for building probabilistic automata parameterized by context-dependent probabilities. Gibbs distributions are used to model state transitions and output generation, and parameter estimation is carried out using an EM algorithm where the M-step uses a generalized iterative scaling procedure. We discuss relations with certain classes of stochastic feedforward neural networks, a geometric interpretation for parameter estimation, and a simple example of a statistical language model constructed using this methodology. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #6818. 	[Title] Exploiting Tractable Substructures in Intractable Networks [Abstract] We develop a refined mean field approximation for inference and learning in probabilistic neural networks. Our mean field theory, unlike most, does not assume that the units behave as independent degrees of freedom; instead, it exploits in a principled way the existence of large substructures that are computationally tractable. To illustrate the advantages of this framework, we show how to incorporate weak higher order interactions into a first-order hidden Markov model, treating the corrections (but not the first order structure) within mean field theory. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #66982. 	[Title] Learning Generative Models with the Up-Propagation Algorithm [Abstract] Up-propagation is an algorithm for inverting and learning neural network generative models. Sensory input is processed by inverting a model that generates patterns from hidden variables using top-down connections. The inversion process is iterative, utilizing a negative feedback loop that depends on an error signal propagated by bottom-up connections. The error signal is also used to learn the generative model from examples. The algorithm is benchmarked against principal component analysis in In his doctrine of unconscious inference, Helmholtz argued that perceptions are formed by the interaction of bottom-up sensory data with top-down expectations. According to one interpretation of this doctrine, perception is a procedure of sequential hypothesis testing. We propose a new algorithm, called up-propagation, that realizes this interpretation in layered neural networks. It uses top-down connections to generate hypotheses, and bottom-up connections to revise them. It is important to understand the difference between up-propagation and its ancestor, the backpropagation algorithm[1]. Backpropagation is a learning algorithm for recognition models. As shown in Figure 1a, bottom-up connections recognize patterns, while top-down connections propagate an error signal that is used to learn the recognition model. In contrast, up-propagation is an algorithm for inverting and learning generative models, as shown in Figure 1b. Top-down connections generate patterns from a set of hidden variables. Sensory input is processed by inverting the generative model, recovering hidden variables that could have generated the sensory data. This operation is called either pattern recognition or pattern analysis, depending on the meaning of the hidden variables. Inversion of the generative model is done iteratively, through a negative feedback loop driven by an error signal from the bottom-up connections. The error signal is also used for learning the connections experiments on images of handwritten digits. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #173863. 	[Title] Learning Belief Networks in the Presence of Missing Values and Hidden Variables [Abstract] In recent years there has been a flurry of works on learning probabilistic belief networks. Current state of the art methods have been shown to be successful for two learning scenarios: learning both network structure and parameters from complete data, and learning parameters for a fixed network from incomplete datathat is, in the presence of missing values or hidden variables. However, no method has yet been demonstrated to effectively learn network structure from incomplete data. In this paper, we propose a new method for learning network structure from incomplete data. This method is based on an extension of the Expectation-Maximization (EM) algorithm for model selection problems that performs search for the best structure inside the EM procedure. We prove the convergence of this algorithm, and adapt it for learning belief networks. We then describe how to learn networks in two scenarios: when the data contains missing values, and in the presence of hidden variables. We provide experimental results that show the effectiveness of our procedure in both scenarios. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1104300. 	[Title] Efficient Stochastic Source Coding and an Application to a Bayesian Network Source Model [Abstract] Brendan J. Frey and Geoffrey E. Hinton 1997. Efficient stochastic source coding and an application to a Bayesian network source model. The Computer Journal 40, 157-165. In this paper, we introduce a new algorithm called "bits-back coding" that makes stochastic source codes efficient. For a given one-to-many source code, we show that this algorithm can actually be more efficient than the algorithm that always picks the shortest codeword. Optimal efficiency is achieved when codewords are chosen according to the Boltzmann distribution based on the codeword lengths. It turns out that a commonly used technique for determining parameters | maximum likelihood estimation | actually minimizes the bits-back coding cost when codewords are chosen according to the Boltzmann distribution. A tractable approximation to maximum likelihood estimation | the generalized expectation maximization algorithm | minimizes the bits-back coding cost. After presenting a binary Bayesian network model that assigns exponentially many codewords to each symbol, we show how a tractable approximation to the Boltzmann distribution can be used for bits-back coding. We illustrate the performance of bits-back coding using using nonsynthetic data with a binary Bayesian network source model that produces 2 60 possible codewords for each input symbol. The rate for bits-back coding is nearly one half of that obtained by picking the shortest codeword for each symbol. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #66990. 	[Title] Generative Models for Discovering Sparse Distributed Representations [Abstract] We describe a hierarchical, generative model that can be viewed as a non-linear generalization of factor analysis and can be implemented in a neural network. The model uses bottom-up, top-down and lateral connections to perform Bayesian perceptual inference correctly. Once perceptual inference has been performed the connection strengths can be updated using a very simple learning rule that only requires locally available information. We demon strate that the network learns to extract sparse, distributed, hierarchical representations. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #194223. 	[Title] Bayesian Unsupervised Learning of Higher Order Structure [Abstract] Multilayer architectures such as those used in Bayesian belief networks and Helmholtz machines provide a powerful framework for representing and learning higher order statistical relations among inputs. Because exact probability calculations with these models are often intractable, there is much interest in finding approximate algorithms. We present an algorithm that efficiently discovers higher order structure using EM and Gibbs sampling. The model can be interpreted as a stochastic recurrent network in which ambiguity in lower-level states is resolved through feedback from higher levels. We demonstrate the performance of the algorithm on bench mark problems. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1116336. 	[Title] On Computing the Largest Fraction of Missing Information for the EM Algorithm and the Worst [Abstract] We address the problem of computing the largest fraction of missing information for the EM algorithm and the worst linear function for data augmentation. These are the largest eigenvalue and its associated eigenvector for the Jacobian of the EM operator at a maximum likelihood estimate, which are important for assessing convergence in iterative simulation. An estimate of the largest fraction of missing information is available from the EM iterates; this is often adequate since only a few figures of accuracy are needed. In some instances the EM iteration also gives an estimate of the worst linear function. We show that the power method for eigencomputation can be used to compute efficient and accurate estimates of both quantities. Unlike eigenvalue decomposition, the power method computes only the largest eigenvalue and eigenvector of a matrix, it can take advantage of a good eigenvector estimate as an initial value and it can be terminated after only a few figures of accuracy are obtained. Moreover, the matrix products needed in the power method can be computed by extrapolation, obviating the need to form the Jacobian of the EM operator. We give results of simultation studies on multivariate normal data showing this approach becomes more efficient as the data dimension increases than methods that use a finite-difference approximation to the Jacobian, which is the only general-purpose alternative available. fl Funded by National Institutes of Health Small Business Innovation Reseach Grant 5R44CA65147-03, and by Office of Naval Research contracts N00014-96-1-0192 and N00014-96-1-0330. We are indebted to Tim Hesterberg, Jim Schimert, Doug Clarkson, Anne Greenbaum, and Adrian Raftery for comments and discussion that helped advance this research and improve this paper. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #32688. 	[Title] CLASSIFICATION USING HIERARCHICAL MIXTURES OF EXPERTS [Abstract] There has recently been widespread interest in the use of multiple models for classification and regression in the statistics and neural networks communities. The Hierarchical Mixture of Experts (HME) [1] has been successful in a number of regression problems, yielding significantly faster training through the use of the Expectation Maximisation algorithm. In this paper we extend the HME to classification and results are reported for three common classification benchmark tests: Exclusive-Or, N-input Parity and Two Spirals. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #156977. 	[Title] The EM Algorithm for Mixtures of Factor Analyzers [Abstract] Technical Report CRG-TR-96-1 May 21, 1996 (revised Feb 27, 1997) Abstract Factor analysis, a statistical method for modeling the covariance structure of high dimensional data using a small number of latent variables, can be extended by allowing different local factor models in different regions of the input space. This results in a model which concurrently performs clustering and dimensionality reduction, and can be thought of as a reduced dimension mixture of Gaussians. We present an exact Expectation-Maximization algorithm for fitting the parameters of this mixture of factor analyzers. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1717. 	[Title] On Convergence Properties of the EM Algorithm for Gaussian Mixtures [Abstract] We build up the mathematical connection between the "Expectation-Maximization" (EM) algorithm and gradient-based approaches for maximum likelihood learning of finite Gaussian mixtures. We show that the EM step in parameter space is obtained from the gradient via a projection matrix P , and we provide an explicit expression for the matrix. We then analyze the convergence of EM in terms of special properties of P and provide new results analyzing the effect that P has on the likelihood surface. Based on these mathematical results, we present a comparative discussion of the advantages and disadvantages of EM and other algorithms for the learning of Gaussian mixture models. This report describes research done at the Center for Biological and Computational Learning and the Artificial Intelligence Laboratory of the Massachusetts Institute of Technology. Support for the Center is provided in part by a grant from the National Science Foundation under contract ASC-9217041. Support for the laboratory's artificial intelligence research is provided in part by the Advanced Research Projects Agency of the Department of Defense under Office of Naval Research contract N00000-00-A-0000. The authors were also supported by the HK RGC Earmarked Grant CUHK250/94E, by a grant from the McDonnell-Pew Foundation, by a grant from ATR Human Information Processing Research Laboratories, by a grant from Siemens Corporation, and by grant N00014-90-1-0777 from the Office of Naval Research. Michael I. Jordan is an NSF Presidential Young Investigator. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1136310. 	[Title] The Expectation-Maximization Algorithm for MAP Estimation [Abstract] The Expectation-Maximization algorithm given by Dempster et al (1977) has enjoyed considerable popularity for solving MAP estimation problems. This note gives a simple derivation of the algorithm, due to Luttrell (1994), that better illustrates the convergence properties of the algorithm and its variants. The algorithm is illustrated with two examples: pooling data from multiple noisy sources and fitting a mixture density. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #116790. 	[Title] Sequential Update of Bayesian Network Structure [Abstract] There is an obvious need for improving the performance and accuracy of a Bayesian network as new data is observed. Because of errors in model construction and changes in the dynamics of the domains, we cannot afford to ignore the information in new data. While sequential update of parameters for a fixed structure can be accomplished using standard techniques, sequential update of network structure is still an open problem. In this paper, we investigate sequential update of Bayesian networks were both parameters and structure are expected to change. We introduce a new approach that allows for the flexible manipulation of the tradeoff between the quality of the learned networks and the amount of information that is maintained about past observations. We formally describe our approach including the necessary modifications to the scoring functions for learning Bayesian networks, evaluate its effectiveness through and empirical study, and extend it to the case of missing data. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #239800. 	[Title] Ensemble Learning and Evidence Maximization [Abstract] Ensemble learning by variational free energy minimization is a tool introduced to neural networks by Hinton and van Camp in which learning is described in terms of the optimization of an ensemble of parameter vectors. The optimized ensemble is an approximation to the posterior probability distribution of the parameters. This tool has now been applied to a variety of statistical inference problems. In this paper I study a linear regression model with both parameters and hyper-parameters. I demonstrate that the evidence approximation for the optimization of regularization constants can be derived in detail from a free energy minimization viewpoint. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #32698. 	[Title] Ensemble Learning for Hidden Markov Models [Abstract] The standard method for training Hidden Markov Models optimizes a point estimate of the model parameters. This estimate, which can be viewed as the maximum of a posterior probability density over the model parameters, may be susceptible to over-fitting, and contains no indication of parameter uncertainty. Also, this maximum may be unrepresentative of the posterior probability distribution. In this paper we study a method in which we optimize an ensemble which approximates the entire posterior probability distribution. The ensemble learning algorithm requires the same resources as the traditional Baum-Welch algorithm. The traditional training algorithm for hidden Markov models is an expectation-maximization (EM) algorithm (Dempster et al. 1977) known as the Baum-Welch algorithm. It is a maximum likelihood method, or, with a simple modification, a penalized maximum likelihood method, which can be viewed as maximizing a posterior probability density over the model parameters. Recently, Hinton and van Camp (1993) developed a technique known as ensemble learning (see also MacKay (1995) for a review). Whereas maximum a posteriori methods optimize a point estimate of the parameters, in ensemble learning an ensemble is optimized, so that it approximates the entire posterior probability distribution over the parameters. The objective function that is optimized is a variational free energy (Feynman 1972) which measures the relative entropy between the approximating ensemble and the true distribution. In this paper we derive and test an ensemble learning algorithm for hidden Markov models, building on Neal Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #73146. 	[Title] LEARNING BAYESIAN NETWORKS WITH LOCAL STRUCTURE [Abstract] We examine a novel addition to the known methods for learning Bayesian networks from data that improves the quality of the learned networks. Our approach explicitly represents and learns the local structure in the conditional probability distributions (CPDs) that quantify these networks. This increases the space of possible models, enabling the representation of CPDs with a variable number of parameters. The resulting learning procedure induces models that better emulate the interactions present in the data. We describe the theoretical foundations and practical aspects of learning local structures and provide an empirical evaluation of the proposed learning procedure. This evaluation indicates that learning curves characterizing this procedure converge faster, in the number of training instances, than those of the standard procedure, which ignores the local structure of the CPDs. Our results also show that networks learned with local structures tend to be more complex (in terms of arcs), yet require fewer parameters. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #240321. 	[Title] Nonlinear gated experts for time series: discovering regimes and avoiding overfitting [Abstract] In: International Journal of Neural Systems 6 (1995) p. 373 - 399. URL of this paper: ftp://ftp.cs.colorado.edu/pub/Time-Series/MyPapers/experts.ps.Z, or http://www.cs.colorado.edu/~andreas/Time-Series/MyPapers/experts.ps.Z University of Colorado Computer Science Technical Report CU-CS-798-95. In the analysis and prediction of real-world systems, two of the key problems are nonstationarity(often in the form of switching between regimes), and overfitting (particularly serious for noisy processes). This article addresses these problems using gated experts, consisting of a (nonlinear) gating network, and several (also nonlinear) competing experts. Each expert learns to predict the conditional mean, and each expert adapts its width to match the noise level in its regime. The gating network learns to predict the probability of each expert, given the input. This article focuses on the case where the gating network bases its decision on information from the inputs. This can be contrasted to hidden Markov models where the decision is based on the previous state(s) (i.e., on the output of the gating network at the previous time step), as well as to averaging over several predictors. In contrast, gated experts soft-partition the input space. This article discusses the underlying statistical assumptions, derives the weight update rules, and compares the performance of gated experts to standard methods on three time series: (1) a computer-generated series, obtained by randomly switching between two nonlinear processes, (2) a time series from the Santa Fe Time Series Competition (the light intensity of a laser in chaotic state), and (3) the daily electricity demand of France, a real-world multivariate problem with structure on several time scales. The main results are (1) the gating network correctly discovers the different regimes of the process, (2) the widths associated with each expert are important for the segmentation task (and they can be used to characterize the sub-processes), and (3) there is less overfitting compared to single networks (homogeneous multi-layer perceptrons), since the experts learn to match their variances to the (local) noise levels. This can be viewed as matching the local complexity of the model to the local complexity of the data. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #62274. 	[Title] A Comparison of New and Old Algorithms for A Mixture Estimation Problem [Abstract] We investigate the problem of estimating the proportion vector which maximizes the likelihood of a given sample for a mixture of given densities. We adapt a framework developed for supervised learning and give simple derivations for many of the standard iterative algorithms like gradient projection and EM. In this framework, the distance between the new and old proportion vectors is used as a penalty term. The square distance leads to the gradient projection update, and the relative entropy to a new update which we call the exponentiated gradient update (EG ). Curiously, when a second order Taylor expansion of the relative entropy is used, we arrive at an update EM which, for = 1, gives the usual EM update. Experimentally, both the EM -update and the EG -update for &gt; 1 outperform the EM algorithm and its variants. We also prove a polynomial bound on the rate of convergence of the EG algorithm. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1140547. 	[Title] Localist Attractor Networks [Abstract] Attractor networks, which map a continuous input space to a discrete output space, are useful for pattern completion, cleaning up noisy or missing features in an input. However, designing a net to have a given set of attractors is notoriously tricky; training procedures are CPU intensive and often produce spurious attractors and ill-conditioned attractor basins. These difficulties occur because each connection in the network participates in the encoding of multiple attractors. We describe an alternative formulation of attractor networks in which the encoding of knowledge is local, not distributed. Although localist attractor nets have similar dynamics to their distributed counterparts, they are much easier to work with and interpret. We propose a statistical formulation of localist attractor net dynamics, which yields a convergence proof and a mathematical interpretation of model parameters. We present simulation experiments that explore the behavior of lo-calist attractor nets, showing that they produce a gang effectthe presence of an attractor enhances the attractor basins of neighboring attractorsand that spurious attractors occur only at points of symmetry in state space. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #50381. 	[Title] Training Algorithms for Hidden Markov Models Using Entropy Based Distance Functions [Abstract] We present new algorithms for parameter estimation of HMMs. By adapting a framework used for supervised learning, we construct iterative algorithms that maximize the likelihood of the observations while also attempting to stay close to the current estimated parameters. We use a bound on the relative entropy between the two HMMs as a distance measure between them. The result is new iterative training algorithms which are similar to the EM (Baum-Welch) algorithm for training HMMs. The proposed algorithms are composed of a step similar to the expectation step of Baum-Welch and a new update of the parameters which replaces the maximization (re-estimation) step. The algorithm takes only negligibly more time per iteration and an approximated version uses the same expectation step as Baum-Welch. We evaluate experimentally the new algorithms on synthetic and natural speech pronunciation data. For sparse models, i.e. models with relatively small number of non-zero parameters, the proposed algorithms require significantly fewer iterations. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #33231. 	[Title] Probabilistic Independence Networks for Hidden Markov Probability Models [Abstract] Graphical techniques for modeling the dependencies of random variables have been explored in a variety of different areas including statistics, statistical physics, artificial intelligence, speech recognition, image processing, and genetics. Formalisms for manipulating these models have been developed relatively independently in these research communities. In this paper we explore hidden Markov models (HMMs) and related structures within the general framework of probabilistic independence networks (PINs). The paper contains a self-contained review of the basic principles of PINs. It is shown that the well-known forward-backward (F-B) and Viterbi algorithms for HMMs are special cases of more general inference algorithms for arbitrary PINs. Furthermore, the existence of inference and estimation algorithms for more general graphical models provides a set of analysis tools for HMM practitioners who wish to explore a richer class of HMM structures. Examples of relatively complex models to handle sensor fusion and coarticulation in speech recognition are introduced and treated within the graphical model framework to illustrate the advantages of the general approach. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #396412. 	[Title] Boltzmann Machine learning using mean field theory and linear response correction [Abstract] We present a new approximate learning algorithm for Boltzmann Machines, using a systematic expansion of the Gibbs free energy to second order in the weights. The linear response correction to the correlations is given by the Hessian of the Gibbs free energy. The computational complexity of the algorithm is cubic in the number of neurons. We compare the performance of the exact BM learning algorithm with first order (Weiss) mean field theory and second order (TAP) mean field theory. The learning task consists of a fully connected Ising spin glass model on 10 neurons. We conclude that 1) the method works well for paramagnetic problems 2) the TAP correction gives a significant improvement over the Weiss mean field theory, both for paramagnetic and spin glass problems and 3) that the inclusion of diagonal weights improves the Weiss approximation for paramagnetic problems, but not for spin glass problems. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #62417. 	[Title] Free energy coding  In [Abstract] In this paper, we introduce a new approach to the problem of optimal compression when a source code produces multiple codewords for a given symbol. It may seem that the most sensible codeword to use in this case is the shortest one. However, in the proposed free energy approach, random codeword selection yields an effective codeword length that can be less than the shortest codeword length. If the random choices are Boltzmann distributed, the effective length is optimal for the given source code. The expectation-maximization parameter estimation algorithms minimize this effective codeword length. We illustrate the performance of free energy coding on a simple problem where a compression factor of two is gained by using the new method. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #35922. 	[Title] Mean Field Theory for Sigmoid Belief Networks [Abstract] We develop a mean field theory for sigmoid belief networks based on ideas from statistical mechanics. Our mean field theory provides a tractable approximation to the true probability distribution in these networks; it also yields a lower bound on the likelihood of evidence. We demonstrate the utility of this framework on a benchmark problem in statistical pattern recognition|the classification of handwritten digits. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #18777. 	[Title] A Tutorial on Learning With Bayesian Networks [Abstract] Technical Report MSR-TR-95-06 Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1105116. 	[Title] Large Deviation Methods for Approximate Probabilistic Inference, with Rates of Convergence a free parameter. The [Abstract] We study layered belief networks of binary random variables in which the conditional probabilities Pr[childjparents] depend monotonically on weighted sums of the parents. For these networks, we give efficient algorithms for computing rigorous bounds on the marginal probabilities of evidence at the output layer. Our methods apply generally to the computation of both upper and lower bounds, as well as to generic transfer function parameterizations of the conditional probability tables (such as sigmoid and noisy-OR). We also prove rates of convergence of the accuracy of our bounds as a function of network size. Our results are derived by applying the theory of large deviations to the weighted sums of parents at each node in the network. Bounds on the marginal probabilities are computed from two contributions: one assuming that these weighted sums fall near their mean values, and the other assuming that they do not. This gives rise to an interesting trade-off between probable explanations of the evidence and improbable deviations from the mean. In networks where each child has N parents, the gap between our upper and lower bounds behaves as a sum of two terms, one of order p In addition to providing such rates of convergence for large networks, our methods also yield efficient algorithms for approximate inference in fixed networks. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #71904. 	[Title] Free Energy Minimization Algorithm for Decoding and Cryptanalysis three binary vectors: s of length N [Abstract] where A is a binary matrix. Our task is to infer s given z and A, and given assumptions about the statistical properties of s and n. This problem arises in the decoding of a noisy signal transmitted using a linear code A, and in the inference of the sequence of a linear feedback shift register (LFSR) from noisy observations [1, 2]. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1107558. 	[Title] Bits-back coding software guide [Abstract] Abstract | In this document, I first review the theory behind bits-back coding (aka. free energy coding) (Frey and Hinton 1996) and then describe the interface to C-language software that can be used for bits-back coding. This method is a new approach to the problem of optimal compression when a source code produces multiple codewords for a given symbol. It may seem that the most sensible codeword to use in this case is the shortest one. However, in the proposed bits-back approach, random codeword selection yields an effective codeword length that can be less than the shortest codeword length. If the random choices are Boltzmann distributed, the effective length is optimal for the given source code. The software which I describe in this guide is easy to use and the source code is only a few pages long. I illustrate the bits-back coding software on a simple quantized Gaussian mixture problem. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #95719. 	[Title] Choice of Basis for Laplace Approximation [Abstract] Maximum a posteriori optimization of parameters and the Laplace approximation for the marginal likelihood are both basis-dependent methods. This note compares two choices of basis for models parameterized by probabilities, showing that it is possible to improve on the traditional choice, the probability simplex, by transforming to the `softmax' basis. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1121254. 	[Title] On Structured Variational Approximations [Abstract] The problem of approximating a probability distribution occurs frequently in many areas of applied mathematics, including statistics, communication theory, machine learning, and the theoretical analysis of complex systems such as neural networks. Saul and Jordan (1996) have recently proposed a powerful method for efficiently approximating probability distributions known as structured variational approximations. In structured variational approximations, exact algorithms for probability computation on tractable substructures are combined with variational methods to handle the interactions between the substructures which make the system as a whole intractable. In this note, I present a mathematical result which can simplify the derivation of struc tured variational approximations in the exponential family of distributions. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #390889. 	[Title] A HIERARCHICAL COMMUNITY OF EXPERTS [Abstract] We describe a directed acyclic graphical model that contains a hierarchy of linear units and a mechanism for dynamically selecting an appropriate subset of these units to model each observation. The non-linear selection mechanism is a hierarchy of binary units each of which gates the output of one of the linear units. There are no connections from linear units to binary units, so the generative model can be viewed as a logistic belief net (Neal 1992) which selects a skeleton linear model from among the available linear units. We show that Gibbs sampling can be used to learn the parameters of the linear and binary units even when the sampling is so brief that the Markov chain is far from equilibrium. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #4330. 	[Title] Hierarchical Mixtures of Experts and the EM Algorithm [Abstract] We present a tree-structured architecture for supervised learning. The statistical model underlying the architecture is a hierarchical mixture model in which both the mixture coefficients and the mixture components are generalized linear models (GLIM's). Learning is treated as a maximum likelihood problem; in particular, we present an Expectation-Maximization (EM) algorithm for adjusting the parameters of the architecture. We also develop an on-line learning algorithm in which the parameters are updated incrementally. Comparative simulation results are presented in the robot dynamics domain. This report describes research done at the Dept. of Brain and Cognitive Sciences, the Center for Biological and Computational Learning, and the Artificial Intelligence Laboratory of the Massachusetts Institute of Technology. Support for CBCL is provided in part by a grant from the NSF (ASC-9217041). Support for the laboratory's artificial intelligence research is provided in part by the Advanced Research Projects Agency of the Dept. of Defense. The authors were supported by a grant from the McDonnell-Pew Foundation, by a grant from ATR Human Information Processing Research Laboratories, by a grant from Siemens Corporation, by by grant IRI-9013991 from the National Science Foundation, by grant N00014-90-J-1942 from the Office of Naval Research, and by NSF grant ECS-9216531 to support an Initiative in Intelligent Control at MIT. Michael I. Jordan is a NSF Presidential Young Investigator. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #4329. 	[Title] Supervised learning from incomplete data via an EM approach [Abstract] Real-world learning tasks may involve high-dimensional data sets with arbitrary patterns of missing data. In this paper we present a framework based on maximum likelihood density estimation for learning from such data sets. We use mixture models for the density estimates and make two distinct appeals to the Expectation-Maximization (EM) principle (Dempster et al., 1977) in deriving a learning algorithm|EM is used both for the estimation of mixture components and for coping with missing data. The resulting algorithm is applicable to a wide range of supervised as well as unsupervised learning problems. Results from a classification benchmark|the iris data set|are presented. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #25702. 	[Title] Nonlinear Prediction of Chaotic Time Series Using Support Vector Machines [Abstract] A novel method for regression has been recently proposed by V. Vapnik et al. [8, 9]. The technique, called Support Vector Machine (SVM), is very well founded from the mathematical point of view and seems to provide a new insight in function approximation. We implemented the SVM and tested it on the same data base of chaotic time series that was used in [1] to compare the performances of different approximation techniques, including polynomial and rational approximation, local polynomial techniques, Radial Basis Functions, and Neural Networks. The SVM performs better than the approaches presented in [1]. We also study, for a particular time series, the variability in performance with respect to the few free parameters of SVM. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #95718. 	[Title] Developments in Probabilistic Modelling with Neural Networks|Ensemble Learning [Abstract] In this paper I give a review of ensemble learning using a simple example. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #390894. 	[Title] A variational approach to Bayesian logistic regression models and their extensions [Abstract] We consider a logistic regression model with a Gaussian prior distribution over the parameters. We show that accurate variational techniques can be used to obtain a closed form posterior distribution over the parameters given the data thereby yielding a posterior predictive model. The results are readily extended to (binary) belief networks. For belief networks we also derive closed form posteriors in the presence of missing values. Finally, we show that the dual of the regression problem gives a latent variable density model, the variational formulation of which leads to exactly solvable EM updates. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #733167. 	[Title] Soft Vector Quantization and the EM Algorithm Running Title: Soft Vector Quantization and EM Section: [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #19697. 	[Title] Keeping Neural Networks Simple by Minimizing the Description Length of the Weights [Abstract] Supervised neural networks generalize well if there is much less information in the weights than there is in the output vectors of the training cases. So during learning, it is important to keep the weights simple by penalizing the amount of information they contain. The amount of information in a weight can be controlled by adding Gaussian noise and the noise level can be adapted during learning to optimize the trade-off between the expected squared error of the network and the amount of information in the weights. We describe a method of computing the derivatives of the expected squared error and of the amount of information in the noisy weights in a network that contains a layer of non-linear hidden units. Provided the output units are linear, the exact derivatives can be computed efficiently without time-consuming Monte Carlo simulations. The idea of minimizing the amount of information that is required to communicate the weights of a neural network leads to a number of interesting schemes for encoding the weights. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #390896. 	[Title] IMPROVING THE MEAN FIELD APPROXIMATION VIA THE USE OF MIXTURE DISTRIBUTIONS [Abstract] Mean field methods provide computationally efficient approximations to posterior probability distributions for graphical models. Simple mean field methods make a completely factorized approximation to the posterior, which is unlikely to be accurate when the posterior is multimodal. Indeed, if the posterior is multi-modal, only one of the modes can be captured. To improve the mean field approximation in such cases, we employ mixture models as posterior approximations, where each mixture component is a factorized distribution. We describe efficient methods for optimizing the parameters in these models. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #27249. 	[Title] Machine Learning Bias, Statistical Bias, and Statistical Variance of Decision Tree Algorithms [Abstract] The term "bias" is widely used|and with different meanings|in the fields of machine learning and statistics. This paper clarifies the uses of this term and shows how to measure and visualize the statistical bias and variance of learning algorithms. Statistical bias and variance can be applied to diagnose problems with machine learning bias, and the paper shows four examples of this. Finally, the paper discusses methods of reducing bias and variance. Methods based on voting can reduce variance, and the paper compares Breiman's bagging method and our own tree randomization method for voting decision trees. Both methods uniformly improve performance on data sets from the Irvine repository. Tree randomization yields perfect performance on the Letter Recognition task. A weighted nearest neighbor algorithm based on the infinite bootstrap is also introduced. In general, decision tree algorithms have moderate-to-high variance, so an important implication of this work is that variance|rather than appropriate or inappropriate machine learning bias|is an important cause of poor performance for decision tree algorithms. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #94713. 	[Title] Update rules for parameter estimation in Bayesian networks [Abstract] This paper re-examines the problem of parameter estimation in Bayesian networks with missing values and hidden variables from the perspective of recent work in on-line learning [12]. We provide a unified framework for parameter estimation that encompasses both on-line learning, where the model is continuously adapted to new data cases as they arrive, and the more traditional batch learning, where a pre-accumulated set of samples is used in a one-time model selection process. In the batch case, our framework encompasses both the gradient projection algorithm [2, 3] and the EM algorithm [14] for Bayesian networks. The framework also leads to new on-line and batch parameter update schemes, including a parameterized version of EM. We provide both empirical and theoretical results indicating that parameterized EM allows faster convergence to the maximum likelihood parame ters than does standard EM. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #345340. 	[Title] Generalized Prioritized Sweeping [Abstract] Prioritized sweeping is a model-based reinforcement learning method that attempts to focus an agent's limited computational resources to achieve a good estimate of the value of environment states. To choose effectively where to spend a costly planning step, classic prioritized sweeping uses a simple heuristic to focus computation on the states that are likely to have the largest errors. In this paper, we introduce generalized prioritized sweeping, a principled method for generating such estimates in a representation-specific manner. This allows us to extend prioritized sweeping beyond an explicit, state-based representation to deal with compact representations that are necessary for dealing with large state spaces. We apply this method for generalized model approximators (such as Bayesian networks), and describe preliminary experiments that compare our approach with classical prioritized sweeping. Please summarise all the given information.

Paper #4335 has the title `A VIEW OF THE EM ALGORITHM THAT JUSTIFIES INCREMENTAL, SPARSE, AND OTHER VARIANTS  
` and the abstract `The EM algorithm performs maximum likelihood estimation for data in which some variables are unobserved. We present a function that resembles negative free energy and show that the M step maximizes this function with respect to the model parameters and the E step maximizes it with respect to the distribution over the unobserved variables. From this perspective, it is easy to justify an incremental variant of the EM algorithm in which the distribution for only one of the unobserved variables is recalculated in each E step. This variant is shown empirically to give faster convergence in a mixture estimation problem. A variant of the algorithm that exploits sparse conditional distributions is also described, and a wide range of other variant algorithms are also seen to be possible. 
`.
Question: Which category should Paper #4335 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].


Paper #4335 has the title `A VIEW OF THE EM ALGORITHM THAT JUSTIFIES INCREMENTAL, SPARSE, AND OTHER VARIANTS  
` and the abstract `The EM algorithm performs maximum likelihood estimation for data in which some variables are unobserved. We present a function that resembles negative free energy and show that the M step maximizes this function with respect to the model parameters and the E step maximizes it with respect to the distribution over the unobserved variables. From this perspective, it is easy to justify an incremental variant of the EM algorithm in which the distribution for only one of the unobserved variables is recalculated in each E step. This variant is shown empirically to give faster convergence in a mixture estimation problem. A variant of the algorithm that exploits sparse conditional distributions is also described, and a wide range of other variant algorithms are also seen to be possible. 
`.
Question: Which category should Paper #4335 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].

This paper should be classified into: Probabilistic_Methods