This tutorial was basically written for students/researchers who want to get into first touch with the Expectation Maximization (EM) Algorithm. The main motivation for writing this tutorial was the fact that I did not find any text that fitted my needs. I started with the great book “Artificial Intelligence: A modern Approach”of Russel and Norvig [6], which provides lots of intuition, but I was...

This tutorial attempts to provide a gentle introduction to EM by way of simple examples involving maximum-likelihood estimation of mixture-model parameters. Readers familiar with ML paramter estimation and clustering may want to skip directly to Sections 5.2 and 5.3.

Expectation-maximization (EM) is an iterative algorithm for finding the maximum likelihood or maximum a posteriori estimate of the parameters of a statistical model with latent variables or when we have missing data. In this work, we view EM in a generalized surrogate optimization framework and analyze its convergence rate under commonly-used assumptions. We show a lower bound on the decrease i...

We present a noise-injected version of the Expectation-Maximization (EM) algorithm: the Noisy Expectation Maximization (NEM) algorithm. The NEM algorithm uses noise to speed up the convergence of the EM algorithm. The NEM theorem shows that additive noise speeds up the average convergence of the EM algorithm to a local maximum of the likelihood surface if a positivity condition holds. Corollary...

In the previous class we already mentioned that many of the most powerful probabilistic models contain hidden variables. We will denote these variables with y. It is usually also the case that these models are most easily written in terms of their joint density, p(d,y,θ) = p(d|y,θ) p(y|θ) p(θ) (1) Remember also that the objective function we want to maximize is the log-likelihood (possibly incl...

The expectation maximization (EM) algorithm is a widely used maximum likelihood estimation procedure for statistical models when the values of some of the variables in the model are not observed. Very often, however, our aim is primarily to find a model that assigns values to the latent variables that have intended meaning for our data and maximizing expected likelihood only sometimes accomplis...

Let $S$ be an ordered semigroup. A fuzzy subset of $S$ is anarbitrary mapping   from $S$ into $[0,1]$, where $[0,1]$ is theusual interval of real numbers. In this paper,  the concept of fuzzygeneralized bi-ideals of an ordered semigroup $S$ is introduced.Regular ordered semigroups are characterized by means of fuzzy leftideals, fuzzy right ideals and fuzzy (generalized) bi-ideals.Finally, two m...

UNLABELLED The aim of this study was to compare the performance of filtered backprojection (FBP) and ordered-subset expectation maximization (OSEM) reconstruction algorithms available in several types of commercial SPECT software. METHODS Numeric simulations of SPECT acquisitions of 2 phantoms were used: the National Electrical Manufacturers Association line phantom used for the assessment of...

OBJECTIVES We observed whether clearer tumor delineation and greater tumor to non-tumor (T/N) count ratios could be obtained using an iterative ordered-subsets expectation maximization (OSEM) algorithm than conventional filtered-back projection algorithm (FBP) in the image reconstruction of thallium-201 (201Tl) lung scintigraphy. METHODS In 29 patients with lung cancer and phantom studies, to...