A Tutorial on the Expectation-Maximization Algorithm Including Maximum-Likelihood Estimation and EM Training of Probabilistic Context-Free Grammars

The paper gives a brief review of the expectation-maximization algorithm (Dempster 1977) in the comprehensible framework of discrete mathematics. In Section 2, two prominent estimation methods, the relative-frequency estimation and the maximum-likelihood estimation are presented. Section 3 is dedicated to the expectation-maximization algorithm and a simpler variant, the generalized expectation-maximization algorithm. In Section 4, two loaded dice are rolled. A more interesting example is presented in Section 5: The estimation of probabilistic context-free grammars.Comment: Presented at the 15th European Summer School in Logic, Language and Information (ESSLLI 2003). Example 5 extended (and partially corrected

Foreword

This report reviews the Expectation Maximization EM algorithm and applies it to the data segmentation problem yielding the Expectation Maximization Segmentation EMS algorithm The EMS algorithm requires batch processing of the data and can be applied to mode switching or jumping linear dynamical state space models The EMS algorithm consists of an optimal fusion of fixed interval Kalman smoothing and discrete optimization. The next section gives a short introduction to the EM algorithm with some background and convergence results In Section the data segmentation problem is dened and in Section the EM algorithm is applied to this problem Section contains simulation results and Section some conclusive remarks

Alternative EM algorithms for nonlinear state-space models

This is the author accepted manuscript. The final version is available from the publisher via the DOI in this recordThe expectation-maximization algorithm is a commonly employed tool for system identification. However, for a large set of state-space models, the maximization step cannot be solved analytically. In these situations, a natural remedy is to make use of the expectation-maximization gradient algorithm, i.e., to replace the maximization step by a single iteration of Newton’s method. We propose alternative expectationmaximization algorithms that replace the maximization step with a single iteration of some other well-known optimization method. These algorithms parallel the expectation-maximization gradient algorithm while relaxing the assumption of a concave objective function. The benefit of the proposed expectation-maximization algorithms is demonstrated with examples based on standard observation models in tracking and localization

Expectation-Maximization Algorithm

EM (Expectation-Maximization) algoritmus je iterativní metoda sloužící k nalezení odhadu maximální věrohodnosti v případech, kdy buď data obsahují chybějící hodnoty, nebo předpokladem existence dalších skrytých proměnných může dojít ke zjednodušení modelu. Každá jeho iterace se skládá ze dvou částí. V kroku E (expectation) vytváříme očekávání logaritmované věrohodnosti úplných dat, která je podmíněna daty pozorovanými a také současným odhadem zkoumaného parametru. Krok M (maximization) následně hledá nový odhad, který bude maximalizovat funkci získanou v předchozí části a který se následně použije v další iteraci v kroku E. EM algoritmus má významné využití např. v oceňování a řízení rizik portfolia.EM (Expectation-Maximization) algorithm is an iterative method for finding maximum likelihood estimates in cases, when either complete data include missing values or assuming the existence of additional unobserved data points can lead to more simple formulation of the model. Each of its iterations consists of two parts. During the E step (expectation) we calculate the expected value of the log-likelihood function of the complete data, with respect to the observed data and the current estimate of the parameter. The M step (maximization) then finds new estimate, which will maximize the function obtained in the previous step and which will be used in the next iteration in step E. EM algorithm has important use in e.g. price and manage risk of the portfolio.Department of Probability and Mathematical StatisticsKatedra pravděpodobnosti a matematické statistikyFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult