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

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Why should one use EM vs. say, Gradient Descent with MLE?

Mathematically, it's often seen that expressions and algorithms for Expectation Maximization (EM) are often simpler for mixed models, yet it seems that almost everything (if not everything) that can ...
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47 views

Help in the expression for log-likelihood and maximization for PRBS signal

I am trying to do blind system identification of a univariate linear FIR model : $$y(n)=\mathbf{h^Tz(n)} + w(n)$$ Where the input $z(n)$ is a PRBS sequence of $\pm1$. Assuming, the input to be from ...
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35 views

Real time example: Estimation for incomplete data

Following is from Csiszar and Shields' FnT monograph "Information Theory and Statistics": The expectation–maximization or EM algorithm is an iterative method frequently used in statistics to ...
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9 views

How to reestimate GMMs in a HMM-GMM

Context: Automatic Speech Recognition I understand the training of a pure HMM with Baum-Welch: Expectation step compute $\gamma_t(i) = P(q_t=i |O,\lambda)$ //p(passing state $i$ at frame ...
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85 views

MCEM algorithm in normal distribution

Consider $z_1,\ldots,z_n$ as a sample of observations of $Z$ and $y_1,\ldots,y_n$ the missing data, where $Z\sim N(\mu,\sigma^2+1)$ and $Y\sim N(0,1)$. i)Find the expression of ...
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21 views

EM algorithm to impute missing value for one variable

This is from Robert Hogg's Introduction to Mathematical Statistics 6th, exercise 6.6.5. p366, It says, Suppose $X_1$, $X_2$, $X_{n1}$, are a random sample from a $N(\theta,1)$ distribution. Suppose ...
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11 views

Calculate parameter standard errors when using E-M algorithm?

When using the E-M algorithm, how can we get standard errors for estimated parameters? If we were just maximizing log-likelihood, then hessian=-observed information, then we can get variance from ...
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53 views

I am having trouble understanding EM gradient algorithm can someone help?

I understand the formula for the EM gradient algorithm but I don't understand what it means by taking one step of Newton's method. Does it mean that it just take one iteration to converge to the ...
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66 views

Convergence of EM for Mixture of Gaussians

Is the Mixture of Gaussians model (an example of latent class analysis) gauranteed to converge on a viable solution even on Unimodal data using the Expectation Maximization algorithm to estimate the ...
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35 views

Numerical Approxmation of standard errors for parameter estimation in the EM algorithm

Generally, when you want to compute standard errors for estimated parameters within the ML framework, one uses the diagonal elements of the observed information matrix. In for instance ...
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21 views

How do you differentiate a function in the em-algorithm?

I am conducting the EM algorithm. I understand the algorithm and my question is more related to the differentiation procedure within the algorithm more than the algorithm itself. Through using the ...
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27 views

Is there any alternative to the EM algorithm?

I am working on biomedical signal analysis and the most used method for parameters estimation is the EM algorithm. My question is : what are the most powerful alternatives to this algorithm?
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30 views

Regularizing soft kmeans with entropy

So in classical fuzzy k-means clustering, the objective function is $\sum_i \sum_j u_{ij} \|x_i - c_j\|^2$ Now, we want to regularize this objective function using the entropy: $\sum_i^n H(U_i) = - ...
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24 views

log multivariate normal differentiation with VAR process

I am trying to estimate a regime switching model with an autoregressive component using the EM algorithm. The process itself can be presented this way: $$ r_{t}= A_{n \times (n+1)} \boldsymbol ...
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1answer
42 views

Prove the loglikelihood is strictly concave for ABO allele frequency blood type data

I am working through the problems in Kenn Lange's book Numerical Analysis for Statisticians. I am going to try and do all of the problems in the book, though none of them are specifically assigned for ...
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25 views

Reference request: EM algorithm and hidden Markov model books with solutions

I am studying missing data problems and the applications of the EM algorithm to missing data problems, like mixture models and hidden Markov models. We have been using Schafer's book Analysis of ...
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37 views

Quasi-Newton Accelerator (QN1) for EM Algorithm

I am trying to implement what is called a "very simple to implement" accelerator for the EM algorithm. Specifically I am talking about the QN1 algorithm, described here, and am using a multivariate ...
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15 views

Self-study (Expectation Maximization on Bivariate Normal Distribution)

I see this example is also "classic", and I am attempting to understand how to approach it. I have an iid sample drawn from a bivariate normal distribution with mean vector ($\mu_1, \mu_2$) and ...
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12 views

How to fix co variance matrix going singular in Gaussian mixture model implementation?

I am implementing GMM in Matlab without using any machine learning library. I am able to initialize the parameters, perform expectation and maximization for one iteration; but when I put expectation ...
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73 views

Expectation-Maximization Algorithm for Binomial

I have a multinomial distribution with four outcomes, with a pdf: $$p(x_1,x_2,x_3,x_4)=\frac{n!}{x_1!x_2!x_3!x_4!}p_1^{x_1}p_2^{x_2}p_3^{x_3}p_4^{x_4}, \sum_{i=1}^4x_i=n, \sum_{i=1}^4p_i=1$$ The ...
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18 views

Fast soft-assign to a large high-dimensional GMM

We wish to perform a fast (possibly approximate) soft-assign of high-dimensional data points to a large Gaussian mixture with diagonal covariance (this is related to the E step of the EM algorithm). ...
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37 views

EM algorithm: With prior vs. not prior

I have a working EM algorithm without prior. I am asking for some advice on how to add prior on latent variables. Define: $t_i \in \{ +1, -1 \} $: variables of interest to be predicted $p_j \in ...
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121 views

What will be the estimator for these parameters

Question: $y_0 = z^d$ is computed from the sum of some recordings by a sensor. Let, there be $k$ sensor nodes. This parameter is calculated by each sensor node and then transmitted to the base ...
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38 views

When does the EM for Gaussian mixture model has one of the Gaussian diminish to exactly one point and have zero variance?

I had implemented the EM algorithm for mixture models as follows: For the E-step I compute the soft-counts of assigning each point $x^{(t)} \in Data_n$ to an individual cluster $j \in \{1, ..., K \}$ ...
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How do you do EM algorithm for a factored model for a recommender system?

Let $X$ be a $n \times d$ matrix with users as rows and movies as columns. Each user is a single row $x^{(u)} \in \mathbb{R}^d$ (i.e. for user u there are at most d ratings for the d movies). Also ...
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167 views

Convergence from the EM Algorithm with bivariate mixture distribution

I have a mixture model which I want to find the maximum likelihood estimator of given a set of data $x$ and a set of partially observed data $z$. I have implemented both the E-step (calculating the ...
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51 views

Expectation Maximisation

I'm currently reading Thomas Hofmamms paper on Probabilistic Latent Semantic Analysis. He includes a formula for the E step in Expectation Maximisation, but has proposed an alternative to this step ...
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121 views

Kalman Filter Expectation Maximization

I'm not very familiar with the EM algorithm for the Kalman Filter. I've been using pykalman to do my analysis in Python. The package comes with a simple EM algo: ...
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Convergence of EM in Mixture Models w.r.t unlikely events $(f(\cdot)=0)$ in either distribution

To maximize the likelihood of a mixture model with unobserved latent variables, the Expectation Maximization is conventionally applied. Assuming we have data $x_1,\dots,x_n$ from a fixed number of ...
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24 views

Non-Causal time-series filtering techniques for standard noise with unkown variance. (EM vs. weiner vs. kalman)

This is a quick question about filtering stored time-series data using kalman/weiner filtering techniques or expectation maximization. I'm just hoping to fix some confusion about questioning ...
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25 views

How to evaluate the goodness of Fit of parameters obtained from EM algorithm

I have a set of observations $\mathcal{Y} = {Y_1, \cdots, Y_T}$. I am running EM algorithm to fit the observations to the following Hidden Markov Model $$A = [a_{ij}]_{N \times N}, a_{ij} = P(X_{k+1} ...
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Beginner level: How to plug in the smoothing equations into E step (Part 2)

Considering Gaussian Linear Dynamical system, $x_{t+1} = Ax_t + w_t$ $y_t = Cx_t + v_t$ $w_t = N(0,Q)$, $v_t = N(0,R)$ By Kalman Filter we are estimating the state variables and the state estimate ...
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255 views

Why is there a E in the name EM algorithm?

I understand where the E step happens in the algorithm (as explicated in the math section below). In my mind, the key ingenuity of the algorithm is the use of the Jensen's inequality to create a lower ...
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Expectation maxmisation algorithm increases true likelihood at each iteration

I've heard that the EM algorithm ensures that the true likelihood is non-decreasing at each iteration of the algorithm, but I'm not sure why this is the case. I've provided a basic plot which I ...
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65 views

Maximum likelihood estimate parameters estimation

In this tutorial on mixture models, page 2, how did the author arrive to the parameters for maximum likelihood in the fully observed case? This is the general setting (based on an excerpt from the ...
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Fitting multiple power laws, Zipf's law in the real-world

As a preface, the following questions are related: How to calculate Zipf's law coefficient from a set of top frequencies? How to estimate parameters for Zipf truncated distribution from a data ...
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pLSA using EM not converging

I have found a lot of questions related to EM but nothing specific to my question. I am using the EM algorithm to fit the pLSA model. As far as I can tell (multiple rounds of checking the code) I can ...
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Reason for using log-likelihood in EM algorithm

When I learned EM algorithm, I saw many literatures use (the expectation of ) the log-likelihood. Is there any reason other than that the log-likelihood may reduce computation? Thanks!
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Cross Validation: Which classifier to use in the end - more difficult setting with the EM algorithm

Referring to already discussed question, I solve something more difficult. During the cross validation, I obtain say $n$ models. The discussed question assumes that the best way is to train a new ...
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95 views

Likelihood maximization: MCEM algorithm versus MCMC algorithm

Hello Everyone this is my first question. I am a particle physicist and I am doing some empirical studiues on parameters estimation using different methods (this might give me some handle to study on ...
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why is the incomplete log-likelihood difficult to optimize

I am trying to teach myself the expectation-maximization algorithm and the texts say the EM is particularly useful when the incomplete log-likelihood i.e. $P(X|\theta)$ where $\theta$ are the ...
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Question on how to use EM to estimate parameters of this model

I am trying to understand EM and trying to infer parameters of this model using this technique but am having trouble understanding how to begin: So, I have a weighted linear regression model as ...
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80 views

Deriving K-means algorithm as a limit of Expectation Maximization for Gaussian Mixtures

Christopher Bishop defines the expected value of the complete-data log likelihood function (i.e. assuming that we are given both the observable data X as well as the latent data Z) as follows: $$ ...
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52 views

Update Rules in Expectation Maximization

I am emulating a certain PDF behaviour using a function. However, due to divergent improper integral, I don't have a closed form expression for the normalization constant. To get the PDF, I just ...
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229 views

“The EM algorithm failed to converge in 25 iterations”

When I Replace Missing Values - Expectation-Maximization in SPSS, I receive the following message: The EM algorithm failed to converge in 25 iterations. Should the algorithm be able to converge? Can ...
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Contribution to the components of a Gaussian mixture by data features

My question is about modelling data with a GMM using EM. One can split the mean and variance of each component into parts as well when working with data with multiple features. My question is what ...
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130 views

Self-study: Finding the maximum likelihood estimates of the parameters of a density function

Consider a random sample $x_1,x_2,...,x_n$ from a newly-generated distribution, whose probability density function is given below \begin{equation} f(x|\alpha,\beta,\sigma)=\frac{1}{\Gamma \left( ...
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Clusteriod questions

I would like to clear some things up because I'm confusing everything. A $clusteriod$ is a coordinate for the mean value of a cluster? So if I have a 2-d .csv file I wish to perform kmeans, the ...
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Help with an exponential function with an indicator and using the EM Algorithm

Two bulbs, Brand A and Brand B, in which their lifetimes are distributed exponentially with expectations $\lambda$ and $\mu$ respectively. They pair $X_i$ and $Y_i$. In the ith experiment, instead ...
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81 views

Questions revolving GMM & EM

I am currently reading about the guassian mixture model and the expectation–maximization algorithm. From what I am reading the two differences between the two here is what I've come up with so far, ...