Questions tagged [expectation-maximization]

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

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EM Algorithm For Bipolar Normal Distribution

Question: Let $x_1, \dots, x_m$ be an i.i.d. sample from a normal density with mean $\mu$ and variance $\sigma^2$. Suppose for each $x_i$ we observe $y_i = |x_i|$ . Formulate an EM algorithm for ...
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Nonparametric mixture estimation

Let's assume that we have two samples $\{X_i\}_{i=1..N}$ and $\{Y_i\}_{i=1..M}$ corresponding to random variables $X$ and $Y$. Let there also be a sample $\{Z_i\}_{i=1..K}$ corresponding to random ...
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Calculating ELBO in EM algorithm

In Andrew Ng's CS229 notes, a nice derivation of EM algorithm is given. With some minor notation modifications, the algorithm is written as follows: E-step: For each i, $Q_i(z_i)=p(z_i|x_i, \theta^{...
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Finding the Q function for the EM algorithm

I have a situation where $X_1,...X_n$ come from $N(\mu,1)$ and there is a realization of 10 $x$ values. I want to use the EM algorithm to work out the MLE. So, I am trying to compute the expected ...
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Why maximizing the expected value of log likelihood under the posterior distribution of latent variables maximize the observed data log-likelihood?

I am trying to understand the Expectation-Maximization algorithm and I am not able to get the intuition of a particular step. I am able to verify the mathematical derivation but I want to understand ...
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Fixing the parameters of the variational distribution in Expectation-Maximization

Consider directed graphical model $z \to x$ (with $z$ unobserved and $x$ observed). The evidence lower bound on the log-likelihood $\log p(x) = \log \sum_z p(x, z; \theta)$ for parameters $\theta$ (...
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28 views

How to get number of iterations in EM-algorithm using R mclust gaussian mixture model

I am clustering data using the mclust function from the R mclust package. I am struggling to get the number of iterations the EM ...
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1answer
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derivation of E step in EM algorithm for pLSA via Lagrangian

I have trouble deriving the EM algorithm for the Probabilistic latent semantic analysis (pLSA) model via Lagrange multipliers. I model the missing data $Q_{zij} \in \{0,1\}$ for word $w_j$ in document ...
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2answers
40 views

Understanding Bishop on EM for HMM's

I'm reading page 616 of Bishop's PRML (pdf), which introduces EM for hidden markov models with categorical hidden states and arbitrary emission distributions. Bishop defines $z_n$ as the hidden state ...
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EM, MCMC and VB, What are their similarity, difference and use scenorio?

I am recently learning MCMC, and realized some connections between MCMC, EM and VB (variational Bayesian). However, I did not find any discussion that can clear my mind on their similarity, difference ...
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1answer
23 views

Are EM and MCMC closed-form analytical solution

I came across the concept of closed-form analytical solution while learning MCMC and got a bit confused. At this point, my understanding is that the problem does not have closed-form solution if it ...
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The performance of EM algorithm at different local maximas

Setup: Say I have a problem (any problem that can be modeled by a probabilistic model) to solve, this problem involves some unknown parameters to estimate. I model this problem in a probabilistic ...
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What is the relationship between VAE and EM algorithm?

What's the relationship between Variational Autoencoders and the Expectation Maximization Algorithm? I know that the EM algorithm is used in latent variable models, specifically to do maximum ...
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Solving Markov Switching Autoregressiv (MS-AR) model

I am currently trying to estimate a MS-AR model of the form $$y_t - \mu_{s_t} = \sum_{l=1}^{p}\phi_l(y_{t-l}-\mu_{s_{t-l}}) + \epsilon_t$$ with $p=4$ and $n=3$ regimes. Further, the variance terms are ...
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EM Algorithm Derivation, Discrete Case

Just wanted to ask whether the following derivation is correct: Suppose $X$ is a vector of observed random variables, $Z$ is a vector of unobserved random variables and $\theta$ is a vector of ...
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Multi-layer regression (with non-linearity in between): Iterated regression algorithm?

Suppose we observe $z_1,\dots,z_n\in\mathbb{R}$ and $x_1,\dots,x_n\in\mathbb{R}^b$ generated from the following model, for $i=1,\dots,n$: $$y_i=x_i^\top \beta,$$ $$z_i={f(y_i)}^\top \gamma + \...
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What assumption need to hold for Monte Carlo EM algorithm?

I have a hierarchical(/multilevel) model from which I want to estimate its parameters. I do by the use of Maximimum Likelihood Estimation via the (Markov Chain) Monte Carlo EM algorithm. When solving ...
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Derivation of M step for Gaussian mixture model

Summary So to summarize my question, how can I take \begin{align} = \sum_{i=1}^{n}W_{i1} \left(log (1-\sum_{j=2}^{K}\pi_j) -\frac{1}{2} log(|\Sigma_1|) -\frac{d}{2} log(2\pi) -\frac{1}{2}(x_i-\mu_1)^{...
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Gaussian mixture model parameter updates derivation?

I've been following a helpful tutorial and I'm trying to understand the parameter updates. For example, the mu_k parameter update is below. I'm unsure why the sum(Bk) does not cancel out as it's in ...
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Expectation-Maximisation derivations [duplicate]

I've come across a few different sources on expectation-maximisation which I can't quite match up. The CS229 lecture 8 [1] states that the function we must write down and maximise is: $$ Q_1 = \sum_{...
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Is this the correct expression for variational distribution q(t) for expectation maximization algorithm?

Just hoping for a sanity check on my equations below. Ultimate goal: apply the EM algorithm to a GMM problem. First I need to come up with an expression for the variational distribution q(t) where ...
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The connection between the expectation in expectation maximization and the importance sampling?

The log-likelihood of the EM algorithm can be expressed as \begin{align} \ell(\theta, x) &= \log p(x|\theta) \\ &= \log \sum_z p(x, z|\theta) \\ &= \log \sum_z \frac{q(z|x)}{q(z|x)}p(x,z|...
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Fitting a mixture model distribution to kurtotic data

I need to fit a parametric distribution to data that has non-zero (unknown) kurtosis. First I tried to fit a Pearson type VII / Student's t, but the fitting is especially poor in the two tails, ...
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1answer
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Nonlinear filtering with unknown model parameters

I have a Markov chain with scalar states $x_t$ evolving according to $$x_t = x_{t-1} + \boldsymbol\vartheta^{\rm T} {\boldsymbol\varphi_{t-1}}(x_{t-1}) + w_t$$ with unknown but constant parameters $\...
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M step EM algorithm in Mixture Models. Expected value of the indicator variable under the posterior [closed]

I am not able to solve the following expectation. In the EM algorithm, the first step in the M step is to compute the expected value of $\log p(x,z)$ where $x$ are observations and $z$ indicator ...
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60 views

Why marginal likelihood is optimized in expectation maximization?

Suppose we would like maximize a likelihood function $p(\mathbf x, \mathbf z| \theta)$, where $\mathbf x$ is observed, $\mathbf z$ is a latent variable, and $\theta$ is the collection of model ...
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1answer
36 views

MCMC inside Expectation Maximization

I wish to optimize the following likelihood function for parameter $\Theta$: $$p(D|\Theta)=\int_X\int_Y p(x, y, D|\Theta)dydx$$ where $X$ and $Y$ are latent variables and only $D$ is observed. I ...
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Exponential Reestimation Formula in EM Algorithm

I'm trying to understand how to reestimate parameters, as part of the EM algorithm. As a simple example, I'm trying to derive the reestimation formula for an exponential distribution. Here's the setup:...
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How to fit a mixture of a normal and a half normal distribution?

I tried Expectation-Maximization (EM) based fitting using the mixfit function from the mixR package in the R environment. It yielded a normal mixture model with 2 components: 1) pi 0.21, mu: 0.47, sd: ...
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How to estimate parameters of a hierarchical model?

I have the following hierarchical model, where $t$ stands for time and $y_t, x_{1t}, x_{2t}, \dots, z_{1t}, z_{2t}, \dots$ are known. I want to estimate the parameters of the model using a ...
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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 ...
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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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1answer
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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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54 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 ...
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101 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 ...
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1answer
56 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 $N(0,\sigma^2)$, $Z_{i},...Z_{i}$ ...
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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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1answer
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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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1answer
46 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?...
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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, $\theta'$...
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165 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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2answers
119 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 ...
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1answer
50 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 ...
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24 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?
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21 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 ...

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