# Questions tagged [expectation-maximization]

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

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### Compute membership probabilities in E-step of EM algorithm with log-densities instead of densities

As an exercise I have implemented the EM algorithm for Gaussian mixtures, however, I have the problem that in high dimensions the densities of data points become so small that I get a numerical ...
22 views

### Clarification of a passage of Bishop about EM algorithm

I am trying to get a very good grasp of the EM algorithm, including the MCEM variant. There is one little passage of the famous Bishop book (pag 440, Mixture Models and EM) where it says: Now ...
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### Derivation of EM in bishop

I'm working through chapter 9 in Bishop (Mixture models and EM) and I'm stuck on equation 9.29. For those without the book: Bishop states that the log likelihood for a latent variable model with ...
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### Random forest (or regression) when predictors are intervals

I'm trying to fit a Random Forest model where the predictors are reported as an interval rather than a point estimate. The structure of each data point is a triplet $(\bar{y_i}, x_{i1}, x_{i2})$ where ...
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### Evaluating goodness of fit of a model estimated with EM-algorithm (with AIC or BIC)

I am learning a Hidden Markov Model with time varying transition probabilities depending on different features. I do this by estimating the model parameters with the EM-algorithm. Now I would like to ...
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### E-step of E-M algorithm with missing data

I am learning expectation-maximization (E-M) algorithm on Coursera and during the course the teacher says that it can be used to handle missing data when fitting Gaussian mixtures (GM) but did not ...
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### EM algorithm for mixture of categorical distributions instantly stabilizes

Brief Summary of Question I'm trying to fit a mixture model of categorical distributions (see https://en.wikipedia.org/wiki/Categorical_distribution). The expectation at the second time step is ...
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### Use Expectation-Maximization algorithm for obtaining maximal likelihood estimator

For $X = {(Z_{i}, Y_{i}) : i = 1, ... ,n}$, consider the model: $Y_{i} = \beta_{1} + \beta_{2}Z_{i} + \epsilon_{i}$ where $\epsilon_{1}, ... ,\epsilon_{n}$ are i.i.d $N(0,\sigma^2)$, $Z_{i},...Z_{i}$ ...
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### Applying the EM algorithm for model with 'fixed' coefficients

My problem is that I want to apply the EM algorithm on a stochastic model (knowing full well that the model is misspecified). \begin{align} log(y_{t}^2) &= h_t + log(\epsilon_t ^2) \\ h_{t+1} = \...
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### Derivation of maximum likelihood for a Gaussian mixture model

I'm working my way through the derivation of EM in Bishop (p. 435). I'm stuck trying to derive to MLE for $\mu_k$ for the gaussian mixture model. Basically I get an extra sum in the numerator. For ...
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### Bayesian PCA, problem understanding Expectation Maximization scheme

I'm reading the following article https://papers.nips.cc/paper/1549-bayesian-pca.pdf of Christophe M. Bishop. I've understood the general method, however, I have trouble understanding the EM scheme ...
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### Why don't we treat the mean and variances in EM algorithm as latent variables

I know how the Expectation Maximization works. What I fail to understand is why only the mixture components are treated as latent variables and why not the mean and variances values of the K gaussians?...
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### What is the Q distribution in expectation maximization in the following explanation?

I am reading a blog on expectation maximization - http://krasserm.github.io/2019/11/21/latent-variable-models-part-1/ Here, I encounter the following expression: When you look at the above ...
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### In expectation maximization, why do we have a latent variable distribution for every sample of the data

I am reading this blog on expectation maximization - http://krasserm.github.io/2019/11/21/latent-variable-models-part-1/ Starting the section where the author starts explaining how EM is done in the ...
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### Does EM algorithm require us to know the joint (predictive) distribution of the latent variables $Z$ when $Z$ is two-dimensional?

In its general form the E-step of the EM algorithm finds the expectation $$Q(\theta|\theta') =\int \log[ p(Y,Z | \theta)] p(Z|Y,\theta') d Z$$ where $Y$ the data, $Z$ the latent variables, $\theta'$...
142 views

### ELBO maximization with SGD

In cases such as Gaussian mixture models, there's is no closed-term solution for the original likelihood maximization. Maximizing the ELBO, however, does have analytical update formulas (i.e. formulas ...
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### interpretation of the estimated parameters of a gaussian mixture model

I need to find/fit a model for the color of an object. Suppose its color is generally yellow and we have 10000-by-3 data which are pixel values for R, G, B channels. Firstly I choose a Multivariate ...
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### Gaussian Mixture model - Penalized log-likelihood in EM algorithm not monotone increasing

I am working on a multivariate Gaussian Mixture Model in R. The goal is to do regularized clustering on the data, where each component represents a cluster. I wrote an EM algorithm to maximize a ...
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### How to calculate weights in EM-algorithm?

Assume that we have 2 clusters with some initial model: Can you please explain how weights are calculated?
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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., \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 exponential distribution in R by using EM-algorithm method. The distribution is defined by $āN_j=1 Y_j$ where $Y_n$ is i.i.d sequence ...
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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 ...