The Q-function used in the EM algorithm is based on the log likelihood. Therefore, it is regarded as the log-EM algorithm. Obtaining this Q-function is a generalized E step.
 Estimation of a Two-component Mixture Model with Applications to Multiple Testing
Its maximization is a generalized M step. No computation of gradient or Hessian matrix is needed. EM is a partially non-Bayesian, maximum likelihood method. In this paradigm, the distinction between the E and M steps disappears. Now, k steps per iteration are needed, where k is the number of latent variables. For graphical models this is easy to do as each variable's new Q depends only on its Markov blanket , so local message passing can be used for efficient inference.
In information geometry , the E step and the M step are interpreted as projections under dual affine connections , called the e-connection and the m-connection; the Kullback—Leibler divergence can also be understood in these terms. The aim is to estimate the unknown parameters representing the mixing value between the Gaussians and the means and covariances of each:.
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The inner sum thus reduces to one term. These are called the "membership probabilities" which are normally considered the output of the E step although this is not the Q function of below. This has the same form as the MLE for the binomial distribution , so. The algorithm illustrated above can be generalized for mixtures of more than two multivariate normal distributions. The EM algorithm has been implemented in the case where an underlying linear regression model exists explaining the variation of some quantity, but where the values actually observed are censored or truncated versions of those represented in the model.
EM typically converges to a local optimum, not necessarily the global optimum, with no bound on the convergence rate in general. It is possible that it can be arbitrarily poor in high dimensions and there can be an exponential number of local optima. Hence, a need exists for alternative methods for guaranteed learning, especially in the high-dimensional setting. Alternatives to EM exist with better guarantees for consistency, which are termed moment-based approaches  or the so-called spectral techniques   [ citation needed ].
Moment-based approaches to learning the parameters of a probabilistic model are of increasing interest recently since they enjoy guarantees such as global convergence under certain conditions unlike EM which is often plagued by the issue of getting stuck in local optima. Algorithms with guarantees for learning can be derived for a number of important models such as mixture models, HMMs etc. For these spectral methods, no spurious local optima occur, and the true parameters can be consistently estimated under some regularity conditions [ citation needed ].
From Wikipedia, the free encyclopedia. Machine learning and data mining Problems. Dimensionality reduction.
Structured prediction. Graphical models Bayes net Conditional random field Hidden Markov. Anomaly detection. Artificial neural networks. Reinforcement learning.
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Machine-learning venues. Glossary of artificial intelligence. Related articles. List of datasets for machine-learning research Outline of machine learning. EM clustering of Old Faithful eruption data. The random initial model which, due to the different scales of the axes, appears to be two very flat and wide spheres is fit to the observed data.
In the first iterations, the model changes substantially, but then converges to the two modes of the geyser. Visualized using ELKI. Further information: Information geometry. Scandinavian Journal of Statistics. Maximum likelihood theory and applications for distributions generated when observing a function of an exponential family variable.
Communications in Statistics — Simulation and Computation. Contributions to the theory of estimation from grouped and partially grouped samples. Statistics from the point of view of statistical mechanics. Lecture notes, Mathematical Institute, Aarhus University. Stockholm University. The notion of redundancy and its use as a quantitative measure of the deviation between a statistical hypothesis and a set of observational data. Affiliations 1. Find all citations in this journal default.
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Or filter your current search. Abstract Mixtures of beta distributions are a flexible tool for modeling data with values on the unit interval, such as methylation levels. While ad-hoc corrections have been proposed to mitigate this problem, we propose a different approach to parameter estimation for beta mixtures where such problems do not arise in the first place. Our algorithm combines latent variables with the method of moments instead of maximum likelihood, which has computational advantages over the popular EM algorithm.
As an application, we demonstrate that methylation state classification is more accurate when using adaptive thresholds from beta mixtures than non-adaptive thresholds on observed methylation levels. We also demonstrate that we can accurately infer the number of mixture components. It is challenging to express the uncertainty around the MAP partition as each partition is composed of assignments of states to clusters.
The extent of the distance or spread between concentrations of low and high co-clustering probabilities provides an indication of the degree of concentration around the MAP point clustering estimate. A larger spread between low and high values for pairwise assignment probabilities indicates lower variability in cluster assignments. The focus is on the co-clustering probability for state i with state j , and not the assignment to clusters.
Our task was to estimate latent, state-indexed functions that improve the efficiency and interpretability of county-level employment statistics constructed from the state CPS estimates. We developed nonparametric mixture formulations that simultaneously estimate the latent, state-indexed functions and allow the data to discover a general dependence structure among them that borrows estimation strength to improve precision.
Our simulation study results demonstrated that failing to account for a dependence structure among states in the estimation model lessens the ability to uncover the true latent functions. Our DP mixture of GPs or iGMRFs, outlined in 3 and 5 , employ an unsupervised approach for discovering dependence among the state functions based on similarities in the time-indexed patterns expressed in the data. They perform well on our CPS employment count application, uncovering a differentiation among states based on their employment sensitivities to the Great Recession.
The simulation study revealed a greater robustness, both in estimation of accuracy of the latent functions and their clustering properties, of the GP mixture model as compared to the iGMRF mixture, due to the regulated smoothness properties of the rational quadratic covariance formulation and its inclusion of more parameters than iGMRF to reflect the trend, scale and frequency properties of the estimated latent functions. The iGMRF computes much faster, however, so it may still be useful, particularly in the case where the clusters are differentiated based primarily on vertical magnitude of the functions.
The authors wish to thank colleagues at the Bureau of Labor Statistics who supported this project and provided the data for this project. We thank the following important contributors: Sean B.
Wilson, Senior Economist, who formulated the project and provided feedback from the states on our results; Garrett T. Schmitt, Senior Economist, who helped us think through alternative approaches; Bradley A. Jensen, Senior Economist, who provided us multiple data slices that allowed us to perform our estimation.
We first outline our sequential scan in the case where we marginalize out the latent functions and conclude this section by highlighting changes required in the case we co-sample them. The mixtures of GPs model specification are highly non-conjugate. We carefully configure blocks of parameters to permit application of posterior sampling approaches designed to produce robust chain mixing under non-conjugate specifications. This posterior representation is a relatively straightforward Gaussian kernel of a non-conjugate probability model. If our lower dimensional approximations are relatively good, this approach will speed chain convergence by producing draws of lower autocorrelation since each proposal includes a sequence of moves generated in the temporary space.
Since the moves in the temporary space are executed with fast approximations, Wang and Neal show that this algorithm has the potential to substantially reduce computation time, as compared to the usual Metropolis-Hastings algorithm, for drawing an equivalent effective sample size. The probability of move formulation evaluates the proposals on the full space of size T , however, so that the resulting sampled draws are from the exact posterior distribution, rather than from a sparse approximation.
Unlike the mixtures of GPs, we specify this model in a conjugate formulation that allows for fast sampling. Relatively high and low values provide indication of higher concentration of the posterior distribution over the space of partitions. We observe relatively high and low values bounded away from 0. Table 2. Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide.
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