# Questions tagged [expectation-maximization]

An optimization algorithm often used for maximum-likelihood estimation in the presence of missing data.

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### Example of Parameter-Expanded EM (PX-EM) algorithm

Can someone provide me some paper or book with and example of the PX-EM algorithm? I'm kind of understanding the theory, but having big trouble understanding how to put it in practice... In "The EM ...
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### Fitting mixture model on data with duplicate values

What is the correct procedure to fit finite mixture models on data with many duplicate values using EM? Let's say I have N(0,1) distributed data and try to fit a 2 component mixture using EM. There ...
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### Confusion in Sampling using the IP algorithm (Bishop PRML)

I'm reading Bishop's PRML p. 537 and I don't understand one piece of the IP (data augmentation) algorithm. Namely, the part that says "we use the samples $\{\mathbf{Z}^{(l)}\}$ obtained in the I step ...
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### Newton-Raphson method to solve for dof when performing MLE of a multivariate Student-t distribution using EM

I am reading the derivation of EM algorithm to estimate the maximum likelihood of a multivariate Student-t distribution $\mathcal{T}(\mathbf{x} \vert \pmb{\mu}, \pmb{\Sigma}, \nu)$ in Kevin Murphy's ...
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### Why Expectation and Maximization algorithm not used in Machine Learning while Gradient Descent algorithm used in Machine Learning?

I know that Newton Raphson, Expectation & Maximization, and Gradient Descent are all known to be optimization methods. Somehow, I wonder why Gradient Descent is chosen to be used in most of ...
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### Do I impute missing values with the response?

I have a dataset with missing values in both predictors and the response. As far as I know, the data are missing not at random, so I cannot simply use listwise deletion. Instead, I employed the EM ...
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### Can someone verify if the following Bayesian Information Criterion (BIC) model selection algorithm is correct for Gaussian mixture models?

I am trying to find an automated way of picking the number of clusters $K \in \mathbb{N}$ for unsupervised learning scenarios, specifically for GMM. I was suggested to use something called the "...
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### Expectation Maximisation (EM) Algorithm

Some of my parameters do not have a closed form solution. Thus, for these parameters the M-step is implemented via a one-step Newton-Raphson update, i.e., \begin{equation} \theta^{t+1} = \theta^t - \...
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### Outlier detection with EM

I am interested in using expectation maximization for outlier detection. In the literature this is usually done assuming that the data of interest are normally distributed while the outliers are ...
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### Using expectation maximization for robust regression

What are the advantages/disadvantages of using EM for robust estimation vs. the robust estimation with Huber or Tukey loss functions?
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### assessing the stability of importance (sampling) weights

I have read that when importance weights are used, the stability (variability) of the weights should be assessed (Levine and Casella, 2001) -- however, I wonder how this might be accomplished. For ...
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### EM-algorithm for two clusters (when one of the distributions is uniform)

I am having a hard time with the EM-algorithm. Here's the problem that I am trying to solve. Dealing with noisy annotations is a common problem in computer vision, especially when using ...
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### Accounting for uncertain information (few observations) in a prior (empirial Bayes)

I did not really know how to choose an adequate title for this question, so please feel free to change it. I have a weird case wherein frequentist and Bayesian philosophies come together. I am ...
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### Expectation Maximisation vs Expectation Propagation in the context of Bayesian Networks

I am confused about Expectation Maximisation and Expectation Propagation algorithms in the context of Bayesian Networks, especially whether one comprise another. What is the difference between ...
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### Consistency of EM for missing data in non-parametric setting

When we have missing data, a parametric model, and an expectation-maximization procedure, and we want to show that our procedure leads to consistent estimators, we can sometimes set up score functions ...
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### Calculation of AIC in finite mixture modeling

I have a question about calculation the AIC to find my optimal amount of clusters. I am applying mixture modeling with the EM algorithm. I know the formula AIC = -2ln(log-lik) + 2k. These are my log-...
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### Analysing faithful dataset in R using GMM

I have got a project on analysing the faithful data in R found in the package "datasets" and called using data(faithful) which is the data set off eruption time and waiting time of the Old Faithful ...
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### Are my data a good candidate for EM imputation followed by exploratory factor analysis?

I am doing Exploratory Factor Analysis (EFA) in R, using principal axis factoring in the psych package. I have missing data that prevent me getting factor scores, so I am imputing data. I am using ...
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### Cosine Similarity for Classification to EM Cluster?

Perhaps my question sounds naive, uncovering the very little knowledge that I have in the field of Statistics, but is very urgent to get a solid answer or trigger for further insights for my concerns. ...
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### A trivial question about EM algorithm theory

In "The EM Algorithm and Extensions", second edition, from Geoffrey J. McLachlan and Thriyambakam Krishnan, X is the latent variable, and Y is de observed (incomplete) variable I'm little confuse ...
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### Using AIC/BIC within cross-validation for likelihood based loss functions

For a course I am teaching, I am having my students fit a Gaussian mixture model using MLEs via the EM algorithm to a bivariate dataset. I have asked the students to use use cross-validation to choose ...
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### Compound distribution

I'm trying to compute a maximum likelihood of compound Poisson exp distribution in R by using EM-algorithm method. The distribution is defined by ∑Nj=1Yj where Yn is i.i.d sequence independent exp(θ) ...
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### EM algorithm, Elements of Statistical Learning, expectation of log-likelihood

While I am reading ESL, I have some questions in chapter 8 (Model inference and averaging). Specifically,8.5.2 The EM algorithm in general. This part explains how EM works, in general, referring to ...
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### Expectation Maximization (EM) stopping criterion

In several EM description (e.g., Theory and Use of the EM Algorithm By Maya R. Gupta and Yihua Chen) I read that two tipical stopping criterions for EM are defined on the difference between log-...
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### Visualized Imputation of 2D Sample Space With Saturation Corresponding to Confidence?

I want to visualize a 2D sample space imputed from a 2D scatter of measurements. Something like a 2D hexbin color histogram would work if it were enhanced in the ...
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### Is there a way to run constrained EM?

I would like to run an EM (expectation maximization) algorithm to estimate hidden Markov model parameters except, in my case, I have an extra constraint on my start and final states in my HMM. How do ...
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### State Space models: rewriting the Likelihood to estimate the covariance matrix

I have a State Space model $\begin{matrix} Y_t & = & FX_t +R_{t}^{1/2}\epsilon_t \\ X_{t+1} & = & GX_{t}+Du_t \end{matrix}$ where $\epsilon_t$ and $u_{t}$ distributed ...
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### Convergence of Modified Expectation-Maximisation Algorithm - interpreting language of question

We're going to consider a modified E-M algorithm and its convergence properties. To do so, we will first need to review the convergence of the standard E-M algorithm as I'll need to refer back to it. ...
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### Convergence of a Expectation Maximisation Algorithm

Consider using standard expectation maximisation to learn the parameters of a Hidden Markov Model. We can show the effect of standard expectation-maximisation on the log-likelihood by first writing ...
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### What is an appropriate threshold for the EM algorithm?

I am implementing the Baum-Welch algorithm (special case of the EM algorithm) on a hidden Markov model and I now have to pick an appropriate stopping criteria $\epsilon$ so that the algorithm ...
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### Figueiredo and Jain's Gaussian mixture EM convergence criterion

I have implemented and been playing around Figueiredo & Jain 's trainer in this paper http://www.lx.it.pt/~mtf/IEEE_TPAMI_2002.pdf for Gaussian mixture. Fig. 2 in the paper depicts the full ...
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### Convergence of EM algorithm

I am aware that EM eventually converges. However, I still have some confusions regarding this property: 1: As far as I am aware, HMM, Gaussian mixture model and MCMC can converge and all of them use ...
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### Conceptual explanation of Maximum Likelihood Estimation

Given a generic time-series $$y_{t+1}= \alpha y_{t} + \Sigma_{t+1}^{1/2}\varepsilon_{t+1} \quad \text{with} \quad \varepsilon_{t+1}=N(0, I)$$ where $\Sigma_{t+1}^{1/2}$ indicates the conditional ...
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### MLE when variance of residuals is null (y is a linear combination of x)

Suppose I have the following model to be estimated via MLE assuming normal errors $y_{t}=x_{t} \beta +e_{t}$ with $e=N(0,\sigma^{2})$, where $y, x$ are matrixes and $\beta$ is a vector, so $\sigma^{2}$...
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### MLE vs Expectation Maximization to estimate time-changing parameters in state space model

Suppose I have a generic model in state-space form described as $$x_{t+1}=\phi_{t} x_{t}+w_{t+1}\epsilon_{t+1}$$ $$y_{t}=H_{t}x_{t}+v_{t}e_{t+1}$$ where both $e_{t+1}$ and $\epsilon_{t+1}$ are iid ...
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