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

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Looking for proof of conditional dependence, when the conditioning variables are linearly related

Suppose we have three random variables, $X$, $Y_1$, and $e$ (for error). Variable $e$ is independent of $X$ and $Y_1$, but $X$ and $Y_1$ are dependent. Further suppose we construct a new mixture ...
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Probability distribution on a subset of a simplex

I want to define a probability distribution on a subset of a simplex. for example, on a 3-simplex, we know that $x_1+x_2+x_3+x_4=1$ and $X \sim Dirichlet$. Would it be possible to constraint $X$ more ...
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Combining monthly distributions of data with different right-censoring levels for prediction

I have a number of pricing datasets with integer values (prices in round dollars) for the same product which are right-censored at different levels (one level per dataset). I would like to combine ...
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19 views

How to fit a mixture of quantile regressions?

I am interested in estimating a mixture of quantile regressions. I have checked several opportunities for the one-component-case as well in SAS (proc quantreg) as in R (package quantreg). But I can't ...
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Modelling of distribution

I have faced with the following distribution (it is truncated at $0.03$ for clarity): It takes values in $(0, \infty )$, unimodal (with mode around $0.0001$), heavy-tailed, and has regular "hills". ...
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How to combine two beta-binomial distributions

Say I have the following situation. I have two weighted coins: Coin 1: In the past I've seen this coin flipped 10 times, 8 of which it came up heads. So I can model the probability of $n$ heads out ...
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17 views

Models for nonnegative (incl. zero) positively skewed multivariate time series (trade volumes)

I want to build a Monte Carlo simulation that is based in part on share amounts that are traded in the market for a set of stocks. I need to be able to take into account the co-dependence of trade ...
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36 views

Tail dependence of mixture copulas

I am currently using (multivariate) mixture copulas to model a financial data set. The mixture has two components as follows: $$C_{mixture}=wC_1+(1-w)C_2$$ where $C_1$ and $C_2$ are copulas. I have ...
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18 views

Mixture Distribution of Multiples

Is there a specific name for a mixture distribution composed of a random variable and its multiples? Suppose we start with a "atomic" random variable $A$. I want to know what one calls the ...
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26 views

solution to mixture CDF / inverse CDF of finite mixture

I am currently numerically solving the follwing equation, which is a convex combination/finite mixture of two marginal CDFs $F(x)$ and $G(x)$ (actually they are joint CDFs but all arguments but x are ...
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Estimate weighted variances in mixture models

Given a generic mixture model $X$ of $k$ components, with distribution $$ f(x)=∑_i\pi_if_i(x), $$ It is easy to show that the $k-th$ moment is just the weighted mean of the $k-th$ moments of the ...
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weights in a mixture Gaussian model

If the variance of a random variable is proportional to its mean, then what is the best way of making a mixture distribution that will faithfully reconstruct a data set coming from a mixture model. ...
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15 views

mixture of Gaussians vs mixture of quadratic denominators (Cauchy)

It is known that mixture of Gaussians are dense in the set of all distribution functions. A 1-dimensional Gaussian has the following density: $$ \frac{1}{\sqrt{2\pi \sigma^2}} ...
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76 views

How can I create a topic model with a mixture of multinomials and EM?

I'm trying to create a topic model with a mixture of multinomials and the EM algorithm. I do not want to use a package. For reference, I'm implementing this in Python with numpy. Data Sets I have ...
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16 views

Is it possible to convert a Gaussian Mixture Model implementation into a Categorical Mixture Model?

I am modelling whether a customer will spend when given a voucher. I have a theory that a customer falls into one of two latent classes: call them spendthrift and miser. So I would like to fit a ...
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17 views

Mixture Density Networks — Recovering Conditionals

If we've got a network trained to generate a mixture of Gaussians and we want the conditional distributions based on a set/subset of outputs, can we easily recover these? Intuitively, yes, we should ...
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What is the difference between a mixture model and a multimodal distribution?

A distribution that is a "Mixture model" has a very similar definition as a "multimodal" distribution. Wikipedia Says: a multimodal distribution is a continuous probability distribution with two ...
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37 views

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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45 views

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
53 views

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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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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38 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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25 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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15 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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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
47 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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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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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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212 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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164 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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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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24 views

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
167 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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29 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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35 views

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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36 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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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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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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89 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. ...