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 Negative Log Likelihood

I am fitting a shared parameter model where I have a binary $Y_2$ and a continuous $Y_1$, and I am using random effects in the mean structure to model them jointly $$f(Y_1,Y_2)=\int f(Y_1,Y_2,b)\,\...
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Expectation-Maximization (EM) algorithm with known means

I am trying to fit a mixture of 1D Gaussian distributions to some data. Can the EM algorithm be used in the case of known mean (all the mean values are equal to zero) and fit only the variances of the ...
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Why is the EM algorithm well suited for exponential families?

I've been brushing up on the EM algorithm, and while I feel like I understand the basics, I keep seeing the claim made (e.g. here, here, among several others) that EM works particularly well for ...
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28 views

Estimation of joint modeling of binary and continuous random variables via shared parameter

I am trying to fit a joint model of a binary and continuous outcome with repeated measures. I am trying to fit it using a shared parameter model that induces all the correlation between outcome and ...
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What calculation is implied by this Gaussian term?

I'd appreciate this community's help in understanding the calculation implied by the Gaussian term in this equation: $$w_{ik} = \alpha_{k} \mathcal{N}( \mathbf{y}_i | \mathbf{X}_i \mathbf{\beta}_k , \...
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Imputation methods for time series data (non-stationary)

I am looking for an impute method for non-stationary time series (financial indeces). From https://pypi.org/project/impyute/ (https://buildmedia.readthedocs.org/media/pdf/impyute/latest/impyute.pdf) I ...
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EM Algorithm For Distributions That Share Parameters

I've implemented an EM algorithm for a gaussian mixture where the mean is defined by an Ornstein–Uhlenbeck process. The pdf I'm trying to maximize is: $$f(r(s)|r(t)) = (1-q)\cdot e^{-(r(s) - r(t) - k(\...
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61 views

Find a lower bound for the minimax risk $\displaystyle \min_{\delta}\max_{\theta\in\Omega}EL(\theta,\delta(X))\ge1-\frac{1}{2^n}$

Consider a random binary vector $X\in\{0,1\}^n. $Let $\theta\in\Omega$ be a probability vector in $\mathbb R^{2^n}$ with $X\sim\theta$. Consider the loss function $\displaystyle L(\theta,a)=\max_{x\in\...
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Optimizing this log-likelihood

I have a HMM which emits an observation Z. The parameters of the HMM are $\boldsymbol\theta$. $$\boldsymbol\theta = {\boldsymbol{A},\boldsymbol{B},\pi}$$ Where $\boldsymbol{A}$ is the transition ...
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EM algorithm for MLE from a bivariate normal sample with missing data: Stuck on M-step

I'm trying to understand applying the EM algorithm to compute the MLE in a missing data problem. Specifically, suppose $(x_1,y_1),\ldots,(x_n,y_n)$ is a random sample from the bivariate normal ...
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Multiple imputation using proc mi EM method in SAS

I'd like to clarify how to use SAS for (multiple) imputation in SAS, specifically the EM method option for proc mi. Do I need to analyze multiple imputed sets using proc mianalyze as mentioned in ...
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Bayesian Regression- Expectation Maximization

In Bayesian regression, we have $y_i=x_i^{T}w+\epsilon_i$ where $w \sim \mathcal{N}(0,\alpha)$ and $\epsilon_i \sim \mathcal{N}(0,\frac{1}{\beta})$. Inference of $\alpha$ and $\beta$ is done by ...
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Statistical Interpretation of 2 data set measures

I was adviced to post here (initialy post on physics exchange but I am going to remove it). I have two independant experiments have measured $\tau_{1},\sigma_{1}$ and $\tau_{2},\sigma_{2}$ with $\...
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Likelihood of a latent graphical model

What is the approach to take when trying to find the likelihood of the observations on a latent graphical model, with intertwining conditional distributions? The model: Each vertex $X_i$ of a binary ...
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Derive expectation of the log determinant of a precision matrix from a Wishart distribution

I'm reading through section 21.6 of Murphy's Machine Learning: A probabilistic perspective where they derive the variational bayes algorithm for fitting a mixture of gaussians. One of the steps ...
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GMM for nonlinear mean

In conventional GMM, observations $\mathbf{X} = \left\lbrace \mathbf{x}_1,\mathbf{x}_2,\ldots\right\rbrace$ are draw from a distribution $$ \mathbf{x}_n \sim \sum_{k=1}^{K}\pi_k\mathcal{N}\left( \...
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Expectation Maximization Correctness of Problem Formulation

Suppose I draw $n$ iid samples from a Poisson$(\lambda)$ distribution, with $\lambda$ unknown. Now, I artificially turn every 3 I draw into a 1, so that the probability of observing any particular non-...
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Why do we say EM is a partially non-Bayesian method?

I have some difficulties understanding the phrase 'EM is a partially non-Bayesian method'. EM works in iterative fashion. Is it because the iterative nature of EM is somehow similar to prior - ...
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What is the nature of $r_i^t$?

Using Expectation Maximization (EM) algorithm, I want to vary the number of clusters used according to $K = [2,4, ... 50]$ for a normal distribution initialized randomly (...
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Conjectures regarding EM approximations of mixtures of multivariate normal distributions

Consider $X\in\mathbb{R}^{N\times d}$ containing data for $N$ points in $d$ dimensions drawn from a bimodal multivariate normal distribution, where any row $x$ of $X$ follows the mixed multivariate ...
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How does maximising ELBO for a Gaussian mixture model fit the model to data?

I am following along in Bishop's Pattern Recognition and ML chapters 9 and 10, and I understand that the EM algorithm works by iteratively updating model parameters using equations derived from ...
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Making sense of the belief propagation on graphs

I sort of understand when do I use variational Bayesian and when do I use expectation maximization. But now I want to know when do I use belief propagation in graphs to solve an estimation problem. ...
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51 views

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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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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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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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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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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22 views

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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112 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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51 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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81 views

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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