Questions tagged [expectation-maximization]

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

Filter by
Sorted by
Tagged with
0
votes
0answers
7 views

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 ...
0
votes
0answers
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 ...
2
votes
1answer
31 views

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 ...
1
vote
0answers
28 views

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 ...
0
votes
0answers
17 views

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 ...
0
votes
0answers
22 views

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 ...
0
votes
0answers
33 views

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 ...
1
vote
1answer
47 views

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 $N(0,\sigma^2)$, $Z_{i},...Z_{i}$ ...
0
votes
0answers
9 views

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} = \...
4
votes
1answer
48 views

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 ...
0
votes
0answers
14 views

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 ...
2
votes
1answer
27 views

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?...
0
votes
1answer
21 views

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 ...
0
votes
1answer
16 views

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 ...
5
votes
0answers
98 views

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, $\theta'$...
4
votes
1answer
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 ...
2
votes
2answers
75 views

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 ...
1
vote
1answer
21 views

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 ...
0
votes
0answers
17 views

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?
1
vote
1answer
13 views

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 ...
1
vote
0answers
20 views

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 ...
0
votes
0answers
19 views

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 ...
0
votes
0answers
42 views

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 ...
0
votes
0answers
22 views

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 ...
2
votes
0answers
31 views

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 "...
4
votes
0answers
39 views

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 - \...
0
votes
0answers
12 views

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 ...
0
votes
0answers
12 views

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?
0
votes
0answers
32 views

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 ...
2
votes
2answers
51 views

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 ...
3
votes
2answers
79 views

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 ...
1
vote
0answers
31 views

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 ...
1
vote
0answers
9 views

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 ...
1
vote
1answer
84 views

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-...
0
votes
0answers
18 views

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 ...
1
vote
0answers
44 views

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 ...
0
votes
0answers
19 views

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. ...
1
vote
0answers
39 views

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 ...
1
vote
1answer
97 views

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 ...
0
votes
0answers
35 views

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 $∑N_j=1 Y_j$ where $Y_n$ is i.i.d sequence ...
1
vote
0answers
32 views

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 ...
0
votes
0answers
98 views

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-...
2
votes
1answer
16 views

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 ...
1
vote
0answers
21 views

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 ...
1
vote
1answer
46 views

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 ...
0
votes
0answers
98 views

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. ...
0
votes
0answers
123 views

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 ...
0
votes
0answers
29 views

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 ...
2
votes
1answer
115 views

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 ...
0
votes
2answers
61 views

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 ...

1
2 3 4 5
10