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

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Collapsed Gibbs Sampling in Mixture Models

I tried to learn how Gibbs sampling works on Mixture models by studying David Blei's notes: http://www.cs.columbia.edu/~blei/fogm/2015F/notes/mixtures-and-gibbs.pdf In the equation 28: $p(z_i = k| ...
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Composition of Normals

I.e., the data was generated from 5 normal distributions: ...
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What is the difference between a mixture model and a hierarchical model?

What is the difference between mixture and hierarchical models? Are they of the same nature with different names or they are totally different things? If there are any references, I will be happy to ...
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1answer
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EM and Kullback-Leibler divergence

Let $f$ be a density on $\mathbb{R}^{p}$. Let $f_{\theta} = \sum_{i=1}^{d} \alpha_{i}\mathcal{N}_{p}(\cdot \, ; \, \theta_{i})$ be a mixture of $d$ Gaussian distributions on $\mathbb{R}^{p}$. For each ...
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“ all of these data points come from the same distribution.” How to test?

I feel like I've seen this topic discussed here before, but I wasn't able to find anything specific. Then again, I'm also not really sure what to search for. I have a one dimensional set of ordered ...
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How to write unnormalized posterior when prior is a mixture of continuous and discrete

Suppose I want to do bayesian inference on the regression problem $\beta$ for Y = X$\beta$ + $\epsilon$ for $\epsilon_i \sim N(0,\sigma^2)$. The complication is that the prior for each component ...
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Example of computing the expectation of a discrete RV using Riemann-Stieltjes integral?

Riemann-Stieltjes integral notation is used in expectation expressions in some probability texts. Basically, dF(x) pops up in the integral rather than f(x)dx in the integral, since the CDF F(x) may ...
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Gibbs sampler gets stuck in local mode

I am very new to statistics and trying to implement a Gibbs sampler. However, according to wikipedia https://en.wikipedia.org/wiki/Gibbs_sampling and this discussion thread ...
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1answer
43 views

Efficiently sampling from mixture distribution posterior

I have the following model: $$ \begin{align} \pi_1\sim & \text{Unif}(0,1)\\ \lambda_1,\lambda_2\sim & \text{Ga}(1,1)\\ z_i\sim & \pi_1^{1(z_i=1)}\pi_2^{1(z_i=2)}\\ ...
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1answer
35 views

Mixture of Priors/Algebra?

Can someone explain how the author gets to the expression after the words "This leads to:"
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19 views

AIC of a mixture distribution

I'm trying to reproduce this paper: http://core.ac.uk/download/pdf/6394955.pdf where a latent class model/finite mixture model is used on the RAND Health insurance data. The data is freely available ...
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14 views

Metropolis-Hastings - Best sample size [duplicate]

I need to implement an algorithm to find the best sample size for the given problem: Let $X_1, ... X_n$ iid such that $X_i|\theta \sim Poisson(\theta)$. Let $\theta \sim f(\theta)$ such that ...
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64 views

Fitting Mixture of Poissons Without GLM

I am looking for a way to fit a mixture of 2 univariate Poisson distributions, but the R packages I checked, mixtools and ...
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1answer
44 views

Simulate mixture of betas

Suppose that we have $X_1, ..., X_n$ iid such that $X_i| \theta \sim Ber(\theta)$ and $\theta \sim g(\theta)$ such that $$g(\theta) = 0.6 Beta(2,1) + 0.4 Beta(1,1) = 1.2 \theta + 0.4$$ Doing the ...
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20 views

How can you model a finite mixture model where it is over a dirac point mass? [duplicate]

I am trying to model the finite mixture model case where: $$ \theta \sim \sum_{h=1}^{k}\pi_h \delta_{\theta^{*}_h} $$ where $\theta^{*}_h \sim Gamma(\alpha, \beta)$, $(\pi_1, \ldots ,\pi_k) \sim ...
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How can I fit a bayesian model with an unspecified number of mixture components to data from a normal mixture model?

Suppose today that I simulated a number of points $N$ from an equally weighted mixture of three normal distributions. We assume for convenience's sake that each of those three normal distributions ...
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60 views

Simulate a mixture distribution in R

Hi just wanna try to simulate a mixture distribution with combination of a normal distribution and a non-central t distribution, the random variable Z is defined as: $$Z = nX+(1-n)Y$$ where $$n ...
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random effect mixture model in openBUGS

what is the command for random effect mixture model in openBUGS? in the other words, How can i do random effect mixture model in openBUGS?
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151 views

Estimating a cumulative distribution function from a mixture model

I have a problem I fail to find a solution. Assume a mixture model : $F(x)=a\times G(x)+(1-a)\times H(x)$ where $F$, $G$ and $H$ are cumulative distributions of different random variables, and $a$ ...
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Sampling from compartments and proportions

Let's say we now have two stacks of powders, one is $a$ the other is $b$. The ratio of amount of $a:b$ is $p:(1-p)$. The two stacks of powders are poured together (without mixing). It forms two ...
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3answers
140 views

find peaks from response signal

My subject is to model the response of micro-organisms to the pollution of the media (one specific pollutant). We analyse the response (production of a substance) through time that is linked to the ...
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15 views

Mixing time to first event model with a count model

I want to build a model of sales productivity where productivity ramp-up involves an unknown period of zero productivity followed by productivity that follows a standard count distribution such as the ...
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Quantile function for a mixture model

Suppose a mixture model, having the following distribution function: \begin{equation} f(x) = \sum_{k = 0}^K \pi_k f_k(x) \end{equation} where $f_k(x)$ is a Poisson distribution with parameter ...
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1answer
160 views

Show that a scale mixtures of normals is a power exponential

I'm trying to show that a scale mixture of normals yields a Laplace distribution. I've gotten to the point where I have $\int N(0,\tau)\times Ga(\tau\:;\:1,\frac{\lambda^{2}}{2}) \:d\tau$ should equal ...
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27 views

Distribution of combination of events with different distributions themselves

I am asked to find the PMF of $X$, for the following definition of $X$: "There are two coins, one with probability $p_1$ of Heads and the other with probability $p_2$ of Heads. One of the coins is ...
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Building a simplex centroid mixture design around a hypothesized centroid

I'm working on a hypothetical design for a three-component mixture experiments with minimum constraints on each of the components; I have little information to guide decision-making here except some ...
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How to compute mean square error with a mixture of 2 normals?

There generated data from mixture of two normals x <- sort(c(rnorm(250,mean=-2,sd=0.5), rnorm(250,mean=3,sd=1.5))) So the true density is ...
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32 views

Understanding Metropolis-Hastings using R

I have $U\sim Uni(0,2)$ and data $(xi|U) \sim\mathcal N(U,1)$. Then $x_i : (-2,-0.5,2.3,1.6,1)$, and I have been trying to understand Metropolis-Hastings in order to code it in r. I also want to fit ...
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Estimate fraction of a known distribution in a mixture with unknown second distribution

Suppose I have a set of bulbs, which are known to be healthy. For each bulb I have a value of its brightness. The underlying distribution is not necessarily normal, and possibly have some complex ...
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26 views

Sample from a quotient of mixture distributions

I want to sample from a random variable $Z=\frac{X}{Y}$, where $X$ and $Y$ are mixtures of distributions. I figured out, how to sample from a mixture distribution (e.g. Simulating random variables ...
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72 views

Algorithm for approximating a density by a mixture density

Given a density $f(x)$ (e.g. the log-normal distribution or log-$t_{\nu=3}$ distribution), I was wondering what algorithm are known/typically used to find a mixture of distributions $g_r(x)$ from ...
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Fitting to a sum (mixture?) of components

I have a curve-fitting problem in which I'm fitting my data to a sum of components with identical functional forms but different parameters. That is $d_i = f(x_i | \theta_j) + f(x_i | \phi_j) + ...
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Gibbs sampling version for estimating Hierarchical Double Dirichlet Process Mixture of Gaussian Processes

I'm trying to implement Gibbs sampling to estimate the parameters of the following non-parametric model: $$\begin{align*} \beta|\gamma & \sim \text{GEM}(\gamma)\\ k_t|\beta & \sim \beta\\ ...
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2answers
81 views

A question on mixture model

I'm a bit confused by the conception of "mixtrue model" I'm studying hidden markov model, which is frequently referred to as a "mixture model". But I don't know what the term "mixture" implies. ...
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61 views

Calculating parameters for a mixture of beta distributions

I have two beta distributions with known parameters: ...
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33 views

Choice of base distribution for Dirichlet Process Mixture Model when data is categorical

While calculating a Dirichlet Process Mixture Model on a data set that has discrete/categorical attributes, what kind of base distribution would you assume? When the data set is completely numerical, ...
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1answer
48 views

Observed function of hidden random variables

Let's say a worker can perform 4 types of tasks in a day: A,B,C,D. Each of which tasks takes time that is distributed according to some probability distribution, say $$ T_A \sim Gamma(\alpha_A, ...
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Interpretation of Generalized Inverse Gaussian regression with GAMLSS

Background on my project: I am comparing proteins (nodes) between a network representing protein interactions in metastatic patients v/s another network representing protein interactions in ...
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Fitting a Gumbel mixture model in R

I am trying to fit some data to a Gumbel mixture model, however there doesn't seem to be any libraries that does this. I tried coding up my own MLE method, but am having problems. Any suggestions of ...
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Get quantile function of dynamic mixture model

I have a dynamic mixture distribution fitted to my risk data (i.e., I have all parameters) of Weibull and Generalized Pareto, with a Cauchy CDF mixing function, that can be written as: $mixture(x): ...
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can AIC be used to compare the fit of a single function and mixture model

I am trying to model a nonparametric probability density function with a continuous function or a mixture model, I am doing this with the R GAMLSS package. Given that a single function may have 4 ...
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Simulate from a dynamic mixture of distributions, honoring the tail

This question is a follow-up to this other question, brilliantly answered by Xi'an. I have a dynamic mixture of Weibull and GPD distributions (with a CDF Cauchy mixing function). The mixture is ...
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Is a gaussian mixture appropriate for TF-IDF?

I'm trying to fit multivariate mixtures of TF-IDF scaled variables. So these variables are weighted count proportions of words in a corpus (relative frequency in document weighted by the log of ...
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389 views

Simulate from a dynamic mixture of distributions

I need to sample from the following mixture of two distributions: $h_{\vec{\beta}}(r)=c(\vec{\beta})[(1-w_{m,\tau}(r))f_{\vec{\beta_{0}}}(r)+w_{m,\tau}(r)g_{\epsilon,\sigma}(r)]$ where ...
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85 views

Number of Gaussian mixture components needed to approximate any distribution

I remember reading an actual proven number of components, that can approximate any distribution. Somehow I think it was 18. Can someone point me to a book/article stating something of the sort? ...
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Degrees of freedom for chi-squared test for a random variable potentially from a mixture generation

I recently posted Multinomial mixture model but got no answer. Hence I take the liberty to present what I came up with on my own and a follow up question: I have two processes which generate a series ...
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Multinomial mixture model

Dear StackExchange Community I'm looking for a package or an elegant way to solve the following question with R: I have two processes which produce a number of A, C, G, T's following a multinomial ...
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Integrated Classification Likelihood computation for R package HDclassif

I'm in the process of fitting some mixture models to some data I have. As this data is high-dimensional, I used the subspace clustering package HDclassif. As the package has no option for the Akaike ...
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Can the mixture of two normal variables be normal?

The properties of gaussian mixtures are well known, see this post and Wikipedia. I am interested in the case when we have a linear model $$Y=a+bX+\epsilon$$ where $X$ and $\epsilon$ are normal with ...
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Finding the cumulative distribution of a mixture distribution of discrete and continuous variables

If I have a random variable that with probability 1/3 is a $U(1,2)$, with probability $1/3$ is $U(2,4)$ and with probability $1/3$ is a discrete rv that takes value 2 with probability $0.4$ and 3 with ...