Fitting Directed Graphical Gaussian Models with One Hidden Variable
نویسنده
چکیده
We discuss directed acyclic graph (DAG) models to represent the independence structure of linear Gaussian systems with continuous variables: such models can be interpreted as a set of recursive univariate regressions. Then we consider Gaussian models in which one of the variables is not observed and we show how the incomplete log-likelihood of the observed data can be maximized using the EM. As the EM algorithm does not provide the matrix of the second derivatives we show how to get an explicit formula for the observed information matrix. We illustrate the utility of the models with two examples.
منابع مشابه
The Hidden Life of Latent Variables: Bayesian Learning with Mixed Graph Models
Directed acyclic graphs (DAGs) have been widely used as a representation of conditional independence in machine learning and statistics. Moreover, hidden or latent variables are often an important component of graphical models. However, DAG models suffer from an important limitation: the family of DAGs is not closed under marginalization of hidden variables. This means that in general we cannot...
متن کاملGraphical Models and Exponential Families
We provide a classi cation of graphical models according to their representation as subfamilies of exponential families. Undirected graphical models with no hidden variables are linear exponential families (LEFs), directed acyclic graphical models and chain graphs with no hidden variables, including Bayesian networks with several families of local distributions, are curved exponential families ...
متن کاملGraphical Answers to Questions about Likelihood Inference for Gaussian Covariance Models
In graphical modelling, a bi-directed graph encodes marginal independences among random variables that are identified with the vertices of the graph (alternatively graphs with dashed edges have been used for this purpose). Bi-directed graphs are special instances of ancestral graphs, which are mixed graphs with undirected, directed, and bi-directed edges. In this paper, we show how simplicial s...
متن کاملSpectral Methods for Learning Multivariate Latent Tree Structure
This work considers the problem of learning the structure of multivariate linear tree models, whichinclude a variety of directed tree graphical models with continuous, discrete, and mixed latent variablessuch as linear-Gaussian models, hidden Markov models, Gaussian mixture models, and Markov evolu-tionary trees. The setting is one where we only have samples from certain observe...
متن کاملImage Segmentation using Gaussian Mixture Model
Abstract: Stochastic models such as mixture models, graphical models, Markov random fields and hidden Markov models have key role in probabilistic data analysis. In this paper, we used Gaussian mixture model to the pixels of an image. The parameters of the model were estimated by EM-algorithm. In addition pixel labeling corresponded to each pixel of true image was made by Bayes rule. In fact,...
متن کامل