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Questions tagged [mixture]

A mixture distribution is one that is written as a convex combination of other distributions.

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How to estimate the probability distribution of two parameters given a discrete output?

I'm currently facing a new type of problem, and i have no idea how to solve it, so any suggestion will be really appreciated ! The problem is the following: I have a matrix of temperatures, depending ...
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Typo in the definition of Finite Mixed Model in Machine Learning a probabilistic Perspective

In subsection 25.2.1 it's stated, regarding finite mixture model: The usual representation (of a finite mixture model) is as follows: $p(x_i|z_i = k, \boldsymbol\theta) = p(x_i|\boldsymbol\...
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Fitting wrong copula type to a real data set

I have developed a new mixture copula model. This model overcomes some limitation of copula models. I tested my new model on a simulation data. The model shows a superior result. My supervisor asked ...
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Inferring GMM parameters with Gibbs Sampling

On my book, "Machine Learning A Probabilistic Approach". It's stated that is straightforward to derive a Gibbs sampling algorithm to fit a mixture model, especially if we use conjugate priors. So ...
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Understanding short animation about Dirichlet Process Mixture Model

On the wikipedia page of Dirichlet Process, there is the following video. I don't get the point of the video. My first impression was that the video was showing the fitting of one-dimensional data ...
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Fitting a Dirichlet Process Mixture Model [closed]

I have understood the Dirichlet Process used for Mixture Model. Now my question is how to fit this model with real data? More clearly if we a data set D, how do we fit the Mixture Model in order to ...
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Confusion in modelling finite mixture model

From the book "Machine Learning a probabilistic Perspective", I'm reading about finite/infinite mixture models. Particularly at paragraph 25.2.1 it's stated: The usual representation (of a finite ...
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46 views

Why is variational Bayesian mixture model an alternative to MCMC? What are the similarities?

Why do people say that a variational Bayesian mixture model could be an alternative to MCMC for clustering? For example see the details here: https://en.wikipedia.org/wiki/Variational_Bayesian_method. ...
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Error in node group[1] Failure to calculate log density [closed]

I am analyzing the data using Mixture model (Bayesian analysis of truncated Poisson regression). ============================R-Code===================== model{ for (i in 1:N){ y[i] ~ dpois(mu[i]) mu[...
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Finitely parametrizable family of univariate distributions closed under mixing

Keilson and Steutel 1972 discusses several families of characteristic functions closed under mixing, such as the even positive characteristic functions log-convex on $\Bbb R^+$. I'm interesting in a ...
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33 views

What is the PDF and CDF of a mixed exponential distribution?

Can someone derive for me the PDF and CDF of a mixed exponential distribution? Definition of a mixed exponential distribution: You have N exponential distributions with possibly different means. You ...
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21 views

How to implement Exponential Power distribution in JAGS

I would like to fit a simulated data to Exponential Power likelihood using uniform mixture with gamma mixing presented in "Scale Mixtures Distributions In Statistical Modelling" by Choy and Chan: $EP(...
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Why does not the weighted sum of gamma distribution come from weighted gamma variables?

If $Z\sim 0.3\Gamma(\alpha _1,\beta _1)+0.7\Gamma (\alpha _2,\beta_2)$, why isn't $Z=0.3X_1+0.7X_2$? $X_1\sim\Gamma(\alpha _1,\beta _1)$ and $X_2\sim\Gamma(\alpha _2,\beta _2)$?
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Does mixture of sigmoids make sense given the theories about mixture of bernoullis?

Mixture of bernoullis is the combination of bernoulli distributions, which can be illustrated by the sampling process of K bags of D coins, here is a quick tutorial about it https://cedar.buffalo.edu/~...
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52 views

Maximum likelihood estimator for a mixture of 2 distributions

Let $X_1, ..., X_n$ be iid with one of two PDFs. If $\theta = 0$, then $f(x; \theta) = 1, \ 0 < x < 1$. if $\theta = 1$, then $f(x; \theta) = \frac{1}{2\sqrt{x}}, \ 0 < x < 1$. What ...
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41 views

Derive the joint probability density function of differences of Gaussian Mixtures

Consider a 3-variate random vector $(\epsilon_0, \epsilon_1, \epsilon_2)$ which is distributed as a Gaussian mixture: (with some abuse of notation) $$ f(\epsilon_0, \epsilon_1, \epsilon_2)=\underbrace{...
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Distribution of difference of Gaussian mixtures: symmetric wrto zero?

I have the following 3-variate random vector $(X,Y,Z)$ which is distributed as a Gaussian mixture: (with some abuse of notation) $$ f(X,Y,Z)=\underbrace{w_a \mathcal{N}(\mu_a, \Sigma_a)}_{\text{...
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MLE for Mixture Model [duplicate]

$$P(W) = \sum_{k=1}^K \pi_k P(W|\mu_k)$$ $$P(W|\mu_k) = \prod_{i=1}^M(\mu_k(i))^{W(i)}$$ Here, it is a mixture model and $W$ is a one-hot encoded vector over a dictionary of size $M$ ie $\sum_{i=1}^{...
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34 views

What is the degree of freedom of semiparametric method for mixture distribution

In the semi-parametric method for density analysis, I want to compare one component semi-parametric mixture distribution and two components mixture distribution. Semi-parametric here means the shape ...
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25 views

How to forecast the individual probabilities at a later date for a mixture or cure cox model

I am following this paper https://deepblue.lib.umich.edu/bitstream/handle/2027.42/65901/j.0006-341X.2000.00227.x.pdf?sequence=1&isAllowed=y) which is titled Estimation in a Cox Proportional ...
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Why always AIC and BIC are used in mixture model than Vuong test

I am working with mixture models. I fitted more than one model to the data and then try to select the most appropriate model using different selection criteria, for example, AIC. My supervisors asked ...
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1answer
33 views

How to specify uniform distribution with same properties as normal distribution?

What I mean is, is it possible to specify a uniform random variable $U$ with random parameters $a,b$, where $a=-b$, and are generated from some other distribution, such that the marginal pdf of $U(a,b)...
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33 views

Parameterizing finite mixture distribution

Let's consider a finite mixture: $$f(x) = \sum_{i=1}^{N}w_{i}p_{i}\left(x\right)$$ where: $N$ is the number of mixed distributions $\left\{p_{1},\dots, p_{N}\right\}$ is a finite set of one-...
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29 views

How to estimate Breslow type Baseline Hazard for mixture cure model?

I have done the following bit myself: ...
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40 views

EM algorithm and AIC criteria

I am using EM algorithm to estimate the model parameters. EM-algorithm iterates until the loglikelihood is converged. After that, I need to compute AIC criteria. As known, AIC is a loglikelihood ...
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2answers
245 views

Question about the latent variable in EM algorithm

In mixture models, Expectation maximization algorithm (EM) is a commonly used method to estimate the model parameters. Suppose that I have bivariate mixture model with two mixture components, with ...
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1answer
52 views

Mixing of liquids where concentration follows normal distribution

We have several liquids where the concentration of a certain element follows a normal distribution, and we take a weighted combination of the elements. The concentrations are: $$C_i \sim \text{IID N}...
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1answer
122 views

What is a positively skewed distribution that can include zero?

I'm modelling data from a behavioural task. Participants do a few hundred trials. On each trial, they see a sequence of letters at a point on the screen and one of these letters appears surrounded by ...
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20 views

Multivariate mixture models

I am new to mixture modeling and have successfully used bernoulli mixture models to cluster datasets of binary data. My real purpose, though is to cluster datasets with mixed data types: normal, ...
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16 views

exponential waiting time model, inflated by events that never happened?

Suppose we have some data where we see the age of each sample and whether each sample received treatment. Further, for those that received treatment some (but not all) of them have a time of treatment ...
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2answers
217 views

Simulating a bimodal distribution in the range of [1;5] in R

I want to simulate a continuous data set/variable with lower/upper bounds of [1;5], while at the same time ensure that the drawn distribution can be considered as bimodal. Searching for my problem, I ...
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1answer
76 views

Convert a normal to a mixture of two normal distribution with variance equal to that of the normal

Consider a variable, say x following the normal distribution N(0,5.99). I want to translate this into a symmetric mixture of two normal distributions such that the mean of x is zero and the variance ...
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74 views

How to fit a (gaussian) mixture model to a dataset with (right) censored data in R?

I am trying to fit a mixture distribution to a dataset in R. Exploiting the R package mixtools, this goes pretty well. However, up to 20% of the data point in the dataset are right censored, therefore ...
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102 views

Sample space for mixture of two Normal distributions

I am going through the book Introduction to Probability by Joseph Blitzstein, Jessica Hwang, and I found the following problem on mixture of two Normal distributions: A certain stock has low ...
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38 views

Testing mean assumption for mixture distribution

I have claim data structured by age groups and I am trying to test the assumed claim means for each age group against the actual data. It is a mixture distribution where around 75% of the sample ...
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244 views

General conditional distributions for multivariate Gaussian mixtures

My question is similar to this one but considers a more general situation. Suppose that $ \vec{x} = (x_1, \dots, x_d) $ and let $$ p(\vec{x}) = \sum_{k=1}^{n} \pi_k \mathcal{N}(\vec{x} | \mu_k, \...
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Can “cross-validation” be used to choose a prior?

To be clear, I doubt I am using the term "cross-validation" correctly here; what I am suggesting also seems similar to "boot-strapping" and "hyperparameter tuning". Terminology is not my strength. ...
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Count the mixture weight as a free parameters for AIC

Suppose that I fit a model of three mixture components to a real data. So, I will have 3 mixture weights. Suppose that, 2 of them are turned out to be zero or very close to zero (that the data shows ...
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1answer
143 views

Assigning a scale mixture of Normal distributions for the data

it's well known that the scale mixture of normal distributions is equivalent to a Student t model, that is $$ t_{(v)}(x|\mu,\sigma^2)=\int_0^\infty N(x|\mu,\sigma^2/\lambda)\times G(\lambda|v/2,v/2)d\...
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39 views

Mixture of Poisson Regression Updating rule

The question arises from this paper https://pdfs.semanticscholar.org/5148/96bea7fa712d0c17c4bdd4aeac87f368c9c2.pdf Where the authors states an EM algorithm to model mixture of Poisson density. I'm ...
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Variance of “an observation” and problem 5.9.8 in BDA by Gelman, et. al

I would like to understand problem 5.9.8 in Bayesian Data Analysis by Gelman, et al.. In particular the problem asks: ... create a bimodal prior density for a normal mean, that is thought to be ...
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51 views

gating function for mixture of tree bayes nets

I am implementing a mixture of tree bayesian networks using bagging by generating k sets of boostrap samples and am currently using a simple uniform weight as the gating function, but am looking for ...
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29 views

Is multivariate linear regression correct in this case?

I have 20 items rated on 25 different dimensions. These items can be classified in two ways. They belong to Group A or B; also, orthogonally, they belong to Groups W, X, Y or Z. Items were rated by ~ ...
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Why does the mixture dependencies exist between variables?

Sorry if my question is clear to most of you. As a mathematics background, I really just start working with the mixture and would like to understand it in a clear way. Mixture dependencies are when ...
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Can describe a dependencies as weak dependency?

Mixture dependencies structure can be very strong. For example, 60% of the data come from the first mixture component. However, sometime we may have a weak mixture ...
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199 views

Gradient Descent vs EM

The EM algorithm is usually motivated because maximizing the log likelihood is described as being "complicated" or "difficult" due to having to take the log of the weighted sum of the likelihood; ie $...
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36 views

Mixing Distributions

Let $X|\alpha$ have a single-parameter Pareto distribution with parameters $\theta$ and $\alpha$, where theta is a known and fixed constant and $\alpha$ has an exponential distribution with parameter $...
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Sampling from mixture copula

Sampling from copula is based on inverse transformation method. However, I would like to understand the sampling algorithm from mixture copula. I really spend a long time to search, however, I could ...
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Why is the sum of two random variables a convolution?

For long time I did not understand why the "sum" of two random variables is their convolution, whereas a mixture density function sum of $f(x)$ and $g(x)$ is $p\,f(x)+(1-p)g(x)$; the arithmetic sum ...
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Two Correct Ways to Sample from a Two Component Mixture but Different Variance Results…What's happening?

Let $X_1 \sim \operatorname{Poisson}(7)$ and $X_2 \sim \operatorname{Poisson}(3)$ be two independent random variables. Let $\{0.7,0.3\}$ be our collection of mixing weights. The following R code is ...