scholarly journals Tutorial on EM Algorithm

2018 ◽  
Author(s):  
Loc Nguyen
Keyword(s):  
Em Algorithm ◽  
Likelihood Function ◽  
Joint Probability ◽  
Likelihood Estimation ◽  
Mathematical Tool ◽  
Probability And Statistics ◽  
Hidden Data ◽  
Parameter Estimators ◽  
The Relationship ◽  
Powerful Mathematical Tool

Maximum likelihood estimation (MLE) is a popular method for parameter estimation in both applied probability and statistics but MLE cannot solve the problem of incomplete data or hidden data because it is impossible to maximize likelihood function from hidden data. Expectation maximum (EM) algorithm is a powerful mathematical tool for solving this problem if there is a relationship between hidden data and observed data. Such hinting relationship is specified by a mapping from hidden data to observed data or by a joint probability between hidden data and observed data. In other words, the relationship helps us know hidden data by surveying observed data. The essential ideology of EM is to maximize the expectation of likelihood function over observed data based on the hinting relationship instead of maximizing directly the likelihood function of hidden data. Pioneers in EM algorithm proved its convergence. As a result, EM algorithm produces parameter estimators as well as MLE does. This tutorial aims to provide explanations of EM algorithm in order to help researchers comprehend it.

scholarly journals Tutorial on EM Algorithm

2018 ◽  
Author(s):  
Loc Nguyen
Keyword(s):  
Em Algorithm ◽  
Likelihood Function ◽  
Joint Probability ◽  
Likelihood Estimation ◽  
Mathematical Tool ◽  
Probability And Statistics ◽  
Hidden Data ◽  
Parameter Estimators ◽  
The Relationship ◽  
Powerful Mathematical Tool

Maximum likelihood estimation (MLE) is a popular method for parameter estimation in both applied probability and statistics but MLE cannot solve the problem of incomplete data or hidden data because it is impossible to maximize likelihood function from hidden data. Expectation maximum (EM) algorithm is a powerful mathematical tool for solving this problem if there is a relationship between hidden data and observed data. Such hinting relationship is specified by a mapping from hidden data to observed data or by a joint probability between hidden data and observed data. In other words, the relationship helps us know hidden data by surveying observed data. The essential ideology of EM is to maximize the expectation of likelihood function over observed data based on the hinting relationship instead of maximizing directly the likelihood function of hidden data. Pioneers in EM algorithm proved its convergence. As a result, EM algorithm produces parameter estimators as well as MLE does. This tutorial aims to provide explanations of EM algorithm in order to help researchers comprehend it.

scholarly journals Tutorial on EM Algorithm

2018 ◽  
Author(s):  
Loc Nguyen
Keyword(s):  
Em Algorithm ◽  
Likelihood Function ◽  
Joint Probability ◽  
Likelihood Estimation ◽  
Mathematical Tool ◽  
Probability And Statistics ◽  
Hidden Data ◽  
Parameter Estimators ◽  
The Relationship ◽  
Powerful Mathematical Tool

Maximum likelihood estimation (MLE) is a popular method for parameter estimation in both applied probability and statistics but MLE cannot solve the problem of incomplete data or hidden data because it is impossible to maximize likelihood function from hidden data. Expectation maximum (EM) algorithm is a powerful mathematical tool for solving this problem if there is a relationship between hidden data and observed data. Such hinting relationship is specified by a mapping from hidden data to observed data or by a joint probability between hidden data and observed data. In other words, the relationship helps us know hidden data by surveying observed data. The essential ideology of EM is to maximize the expectation of likelihood function over observed data based on the hinting relationship instead of maximizing directly the likelihood function of hidden data. Pioneers in EM algorithm proved its convergence. As a result, EM algorithm produces parameter estimators as well as MLE does. This tutorial aims to provide explanations of EM algorithm in order to help researchers comprehend it.

scholarly journals Tutorial on EM Algorithm

2020 ◽  
Author(s):  
Loc Nguyen
Keyword(s):  
Em Algorithm ◽  
Likelihood Function ◽  
Joint Probability ◽  
Likelihood Estimation ◽  
Pso Algorithm ◽  
Descent Method ◽  
Gradient Descent Method ◽  
Mathematical Tool ◽  
Hidden Data ◽  
Process Combination

Maximum likelihood estimation (MLE) is a popular method for parameter estimation in both applied probability and statistics but MLE cannot solve the problem of incomplete data or hidden data because it is impossible to maximize likelihood function from hidden data. Expectation maximum (EM) algorithm is a powerful mathematical tool for solving this problem if there is a relationship between hidden data and observed data. Such hinting relationship is specified by a mapping from hidden data to observed data or by a joint probability between hidden data and observed data. In other words, the relationship helps us know hidden data by surveying observed data. The essential ideology of EM is to maximize the expectation of likelihood function over observed data based on the hinting relationship instead of maximizing directly the likelihood function of hidden data. Pioneers in EM algorithm proved its convergence. As a result, EM algorithm produces parameter estimators as well as MLE does. This tutorial aims to provide explanations of EM algorithm in order to help researchers comprehend it. Moreover some improvements of EM algorithm are also proposed in the tutorial such as combination of EM and third-order convergence Newton-Raphson process, combination of EM and gradient descent method, and combination of EM and particle swarm optimization (PSO) algorithm.

scholarly journals Tutorial on EM Algorithm

2020 ◽  
Author(s):  
Loc Nguyen
Keyword(s):  
Em Algorithm ◽  
Likelihood Function ◽  
Joint Probability ◽  
Likelihood Estimation ◽  
Pso Algorithm ◽  
Descent Method ◽  
Gradient Descent Method ◽  
Mathematical Tool ◽  
Hidden Data ◽  
Process Combination

Maximum likelihood estimation (MLE) is a popular method for parameter estimation in both applied probability and statistics but MLE cannot solve the problem of incomplete data or hidden data because it is impossible to maximize likelihood function from hidden data. Expectation maximum (EM) algorithm is a powerful mathematical tool for solving this problem if there is a relationship between hidden data and observed data. Such hinting relationship is specified by a mapping from hidden data to observed data or by a joint probability between hidden data and observed data. In other words, the relationship helps us know hidden data by surveying observed data. The essential ideology of EM is to maximize the expectation of likelihood function over observed data based on the hinting relationship instead of maximizing directly the likelihood function of hidden data. Pioneers in EM algorithm proved its convergence. As a result, EM algorithm produces parameter estimators as well as MLE does. This tutorial aims to provide explanations of EM algorithm in order to help researchers comprehend it. Moreover some improvements of EM algorithm are also proposed in the tutorial such as combination of EM and third-order convergence Newton-Raphson process, combination of EM and gradient descent method, and combination of EM and particle swarm optimization (PSO) algorithm.

scholarly journals Tutorial on EM Algorithm

2020 ◽  
Author(s):  
Loc Nguyen
Keyword(s):  
Em Algorithm ◽  
Likelihood Function ◽  
Joint Probability ◽  
Likelihood Estimation ◽  
Pso Algorithm ◽  
Descent Method ◽  
Gradient Descent Method ◽  
Mathematical Tool ◽  
Hidden Data ◽  
Process Combination

Maximum likelihood estimation (MLE) is a popular method for parameter estimation in both applied probability and statistics but MLE cannot solve the problem of incomplete data or hidden data because it is impossible to maximize likelihood function from hidden data. Expectation maximum (EM) algorithm is a powerful mathematical tool for solving this problem if there is a relationship between hidden data and observed data. Such hinting relationship is specified by a mapping from hidden data to observed data or by a joint probability between hidden data and observed data. In other words, the relationship helps us know hidden data by surveying observed data. The essential ideology of EM is to maximize the expectation of likelihood function over observed data based on the hinting relationship instead of maximizing directly the likelihood function of hidden data. Pioneers in EM algorithm proved its convergence. As a result, EM algorithm produces parameter estimators as well as MLE does. This tutorial aims to provide explanations of EM algorithm in order to help researchers comprehend it. Moreover some improvements of EM algorithm are also proposed in the tutorial such as combination of EM and third-order convergence Newton-Raphson process, combination of EM and gradient descent method, and combination of EM and particle swarm optimization (PSO) algorithm.

scholarly journals Tutorial on EM Algorithm

2020 ◽  
Author(s):  
Loc Nguyen
Keyword(s):  
Em Algorithm ◽  
Likelihood Function ◽  
Joint Probability ◽  
Likelihood Estimation ◽  
Pso Algorithm ◽  
Descent Method ◽  
Gradient Descent Method ◽  
Mathematical Tool ◽  
Hidden Data ◽  
Process Combination

Maximum likelihood estimation (MLE) is a popular method for parameter estimation in both applied probability and statistics but MLE cannot solve the problem of incomplete data or hidden data because it is impossible to maximize likelihood function from hidden data. Expectation maximum (EM) algorithm is a powerful mathematical tool for solving this problem if there is a relationship between hidden data and observed data. Such hinting relationship is specified by a mapping from hidden data to observed data or by a joint probability between hidden data and observed data. In other words, the relationship helps us know hidden data by surveying observed data. The essential ideology of EM is to maximize the expectation of likelihood function over observed data based on the hinting relationship instead of maximizing directly the likelihood function of hidden data. Pioneers in EM algorithm proved its convergence. As a result, EM algorithm produces parameter estimators as well as MLE does. This tutorial aims to provide explanations of EM algorithm in order to help researchers comprehend it. Moreover some improvements of EM algorithm are also proposed in the tutorial such as combination of EM and third-order convergence Newton-Raphson process, combination of EM and gradient descent method, and combination of EM and particle swarm optimization (PSO) algorithm.

scholarly journals Tutorial on EM Algorithm

2020 ◽  
Author(s):  
Loc Nguyen
Keyword(s):  
Em Algorithm ◽  
Likelihood Function ◽  
Joint Probability ◽  
Likelihood Estimation ◽  
Pso Algorithm ◽  
Descent Method ◽  
Gradient Descent Method ◽  
Mathematical Tool ◽  
Hidden Data ◽  
Process Combination

Maximum likelihood estimation (MLE) is a popular method for parameter estimation in both applied probability and statistics but MLE cannot solve the problem of incomplete data or hidden data because it is impossible to maximize likelihood function from hidden data. Expectation maximum (EM) algorithm is a powerful mathematical tool for solving this problem if there is a relationship between hidden data and observed data. Such hinting relationship is specified by a mapping from hidden data to observed data or by a joint probability between hidden data and observed data. In other words, the relationship helps us know hidden data by surveying observed data. The essential ideology of EM is to maximize the expectation of likelihood function over observed data based on the hinting relationship instead of maximizing directly the likelihood function of hidden data. Pioneers in EM algorithm proved its convergence. As a result, EM algorithm produces parameter estimators as well as MLE does. This tutorial aims to provide explanations of EM algorithm in order to help researchers comprehend it. Moreover some improvements of EM algorithm are also proposed in the tutorial such as combination of EM and third-order convergence Newton-Raphson process, combination of EM and gradient descent method, and combination of EM and particle swarm optimization (PSO) algorithm.

scholarly journals Tutorial on EM Algorithm

2020 ◽  
Author(s):  
Loc Nguyen
Keyword(s):  
Em Algorithm ◽  
Likelihood Function ◽  
Joint Probability ◽  
Likelihood Estimation ◽  
Pso Algorithm ◽  
Descent Method ◽  
Gradient Descent Method ◽  
Mathematical Tool ◽  
Hidden Data ◽  
Process Combination

Maximum likelihood estimation (MLE) is a popular method for parameter estimation in both applied probability and statistics but MLE cannot solve the problem of incomplete data or hidden data because it is impossible to maximize likelihood function from hidden data. Expectation maximum (EM) algorithm is a powerful mathematical tool for solving this problem if there is a relationship between hidden data and observed data. Such hinting relationship is specified by a mapping from hidden data to observed data or by a joint probability between hidden data and observed data. In other words, the relationship helps us know hidden data by surveying observed data. The essential ideology of EM is to maximize the expectation of likelihood function over observed data based on the hinting relationship instead of maximizing directly the likelihood function of hidden data. Pioneers in EM algorithm proved its convergence. As a result, EM algorithm produces parameter estimators as well as MLE does. This tutorial aims to provide explanations of EM algorithm in order to help researchers comprehend it. Moreover some improvements of EM algorithm are also proposed in the tutorial such as combination of EM and third-order convergence Newton-Raphson process, combination of EM and gradient descent method, and combination of EM and particle swarm optimization (PSO) algorithm.

A Block Successive Lower-Bound Maximization Algorithm for the Maximum Pseudo-Likelihood Estimation of Fully Visible Boltzmann Machines

2016 ◽  
Vol 28 (3) ◽  
pp. 485-492 ◽  
Author(s):  
Hien D. Nguyen ◽  
Ian A. Wood
Keyword(s):  
Lower Bound ◽  
Likelihood Function ◽  
Likelihood Estimation ◽  
Convergence Criterion ◽  
Boltzmann Machines ◽  
Pseudo Likelihood ◽  
Global Maximizer ◽  
Ga Algorithm ◽  
Pseudo Likelihood Estimation ◽  
The Relationship

Maximum pseudo-likelihood estimation (MPLE) is an attractive method for training fully visible Boltzmann machines (FVBMs) due to its computational scalability and the desirable statistical properties of the MPLE. No published algorithms for MPLE have been proven to be convergent or monotonic. In this note, we present an algorithm for the MPLE of FVBMs based on the block successive lower-bound maximization (BSLM) principle. We show that the BSLM algorithm monotonically increases the pseudo-likelihood values and that the sequence of BSLM estimates converges to the unique global maximizer of the pseudo-likelihood function. The relationship between the BSLM algorithm and the gradient ascent (GA) algorithm for MPLE of FVBMs is also discussed, and a convergence criterion for the GA algorithm is given.

scholarly journals Evaluating the Observed Log-Likelihood Function in Two-Level Structural Equation Modeling with Missing Data: From Formulas to R Code

Psych ◽  
2021 ◽  
Vol 3 (2) ◽  
pp. 197-232
Author(s):  
Yves Rosseel
Keyword(s):  
Structural Equation ◽  
Structural Equation Models ◽  
Likelihood Function ◽  
Likelihood Estimation ◽  
Missing At Random ◽  
Equation Modeling ◽  
Computationally Efficient ◽  
Log Likelihood ◽  
Efficient Expression ◽  
Special Case

This paper discusses maximum likelihood estimation for two-level structural equation models when data are missing at random at both levels. Building on existing literature, a computationally efficient expression is derived to evaluate the observed log-likelihood. Unlike previous work, the expression is valid for the special case where the model implied variance–covariance matrix at the between level is singular. Next, the log-likelihood function is translated to R code. A sequence of R scripts is presented, starting from a naive implementation and ending at the final implementation as found in the lavaan package. Along the way, various computational tips and tricks are given.

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