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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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Closed form posterior for product of inverse Gamma and Normal distribution

I am currently reading a book about mixture analysis, and in the textbook a posterior for the parameters $\mu1,\mu2,\sigma_1^2,\sigma_2^2$ of a two-component gaussian mixture is derived as follows (S ...
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18 views

Can mixture model used as classification prediction

I am learning a prediction model in statistic. I read that we can used mixture model as a classification. My question is, assume we have a data which can be divided into two groups. Can we in this ...
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207 views

Joint posterior for finite mixture with normal components

I currently read the book Finite Mixture and Markov Switching Models by Sylvia Frühwirth-Schnatter and I have a question regarding Section 6.2.1 regarding finite mixtures with normal components. Maybe ...
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16 views

What does it means for “fit a less parsimonious model” in a clustering algorithm?

I'm now trying to implement the algorithm presented in https://www.stat.washington.edu/raftery/Research/PDF/fraley2005.pdf. The algorithm is the following one: First I get a mixture model for ...
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Concentration inequality for mean of Gaussian mixture

Say I have i.i.d. samples $X_1, \ldots, X_n \sim p \mathcal{N}(\mu_1, \sigma^2) + (1 - p) \mathcal{N}(\mu_2, \sigma^2)$. Then suppose I estimate the mean with the sample mean $$ \widehat{\mu} = \frac{...
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14 views

How to design Mixture of Experts where we want only one active model at a time?

I'm trying to design a Mixture of Experts where we want only one active neural network at a time. Suppose that we have 10 experts. I want to train a MoE such that only one of the experts is active ...
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4 views

Ideas for fixing manual misclassification

Imagine the following problem. Mechanics fix cars and use a set of parts which is recorded on invoices. They also write down a classification of the work that was done from a fixed set of possible ...
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89 views

Posterior Weights for Normal-Normal (known variance) model

So I have a mixture prior of: $p(\mu) = \pi p_0(\mu \mid \mu_0, \sigma^2_0) + (1-\pi) p_1(\mu \mid \mu_1, \sigma^2_1)$ and a likelihood of $$p(y\mid\mu) = (2\pi\sigma^2)^{-n/2} \exp\left(\frac{-1}{...
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1answer
38 views

Mixture of Poissons for positive and negative integers

I'm trying to design a generative model for a random variable $k\in\mathbb{Z}$. The model would work as follows. First draw $b\sim\text{Bernoulli}(p)$. If $b=1$, draw $k\sim\text{Poisson}(c)$. ...
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consistency of non-linear squares estimators in Gaussian mixture model

Does any one know or have references for the consistency of non-linear squares estimators in Gaussian mixture model? (or the consistency of qausi mles in Gaussian mixture model). Thank you. I have ...
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26 views

Binomial distribution for randomly drawn probabilities

Setting Probability theory can be a weird place sometimes. Here I was, confident in my insane math skills, trying to solve the following problem: Let $N, \alpha$ and $\beta$ be given. ...
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16 views

AIC influences by the number of the model parameters

From the different published paper about mixture models, I found that AIC is affected by the number of model components. That is due to the plenty term in ...
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8 views

Identification of mixture models with location shift

I am studying identification of mixture models of the type $$ F(x)=\sum_{j=1}^J \lambda_j G(x-\mu_j) $$ with $F,G$ denoting univariate CDFs, $G$ symmetric about $0$, $\mu_1<\mu_2<...<\mu_J$, $...
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1answer
22 views

Is it acceptable in mathematical saying that the E-step is equal to posterior probability

I am studying EM-algorithm for mixture data. I read that some authors said that, the E-step is equivalent to the calculation of the posterior probability (I think this come from Bayesian rule). So, my ...
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1answer
28 views

Point mass at zero and a chi square distribution with one degree of freedom

I am unclear about the critical value of a point mass at zero and a chi square distribution with one degree of freedom. How to find this?
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20 views

Equivalent way of rewriting a two-component mixture

I'm confused on the following equivalent way of rewriting a two-component mixture. Consider the two-component conditional mixture $$ F(z|x)=\lambda F_1(z|x)+(1-\lambda)F_2(z|x) $$ where all the $F$'...
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Simulate from a mixture of two beta distribution [closed]

I have this distribution : $X \sim 0.75\mathcal{B}e(\alpha_{X_1}=1,\beta_{X_1}=3)+0.25\mathcal{B}e(\alpha_{X_2}=5,\beta_{X_2}=1.75)$ where the density function is given from this function: ...
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A 2 component mixture is symmetric if and only if $\lambda\in \{0,1,\frac{1}{2}\}$

Consider the following mixture of two densities $$ f(x)=\lambda g(x-\mu_1)+(1-\lambda)g(x-\mu_2) $$ with $\lambda\in [0,1]$, $g(\cdot)$ symmetric around zero, $\mu_1<\mu_2$. Claim: the mixture is ...
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What is the different between the set of all model parameters and the parameter vector of the nth component

I read many articles about mixture models. I read that the author called the model parameters as "a set of all model parameters", while they said "parameter vector for the n-th component". I wonder ...
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25 views

Mixture of $K$ components

Consider a random vector $$ X\equiv \begin{pmatrix} X_1\\ X_2\\ X_3 \end{pmatrix} $$ with pdf $$f(x)=\overbrace{\sum_{k=1}^ K \frac{1}{K} f_k(x)}^{\text{finite mixture}}$$ and $\forall k=1,...,K$ $...
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40 views

Mixture of two components

Consider a univariate probability density function $p(x)$ that is a mixture of $2$ probability density functions with weights $\eta, 1-\eta$ and $\eta\in (0,1)$: $$ p(x)=(1-\eta)g(x)+\eta f(x) \...
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39 views

Calculating the probability that an observation comes from either population A or B

If I have two normal distributions A (mean = 0, variance = 4) and B (mean = 0, variance = 16), how can I calculate the probability that an observation with magnitude 2 comes from A?
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421 views

Why we do not accept the result of our simulation study as evidence of a limitation of one method

I am doing a mixture model. I have established a new method using EM-algorithm. I have simulated data from a mixture model. Then, I applied my new method to the data. The result is very satisfying. ...
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How to build a Bayesian Model to estimate the probability distribution of the parameters given the 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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20 views

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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1answer
108 views

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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2answers
51 views

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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89 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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Deriving a distribution whose pdf has the shape of a square + a triangle (a right trapezoid)

I want to the derive the PDF which looks like the sum of a triangular and uniform distribution which looks like this: To do this I have simply added the PDFs for the rectangular and triangular parts, ...
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10 views

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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76 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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1answer
88 views

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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1answer
71 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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82 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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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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36 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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51 views

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
36 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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1answer
47 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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44 views

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

I have done the following bit myself: ...
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154 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
361 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
56 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
262 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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24 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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18 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 ...