Questions tagged [mixture]

A mixture distribution is one that is written as a convex combination of other distributions. Use the "compound-distributions" tag for "concatenations" of distributions (where a parameter of a distribution is itself a random variable).

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Normally its easy to solving mean and covariance of Bernoulli Distribution but how we do when its Bernoulli Distribution vector in below Questions? [duplicate]

Maybe it's so easy to solve but I confused, Could anyone explain to me how can I solve these 2 questions?
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Mixture model for a mix of normal and lognormal distributions in R

I have a distribution composed of a mixture of a normal distribution and a log-normal distribution. A simulated example that looks quite similar to what I will expect in the real data would be: ...
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What is the distribution of a mixture of exponential distributions whose rate parameters follow a gamma distribution?

I want to know the theoretical distribution of a mixture of exponential distributions whose rate parameters are distributed according to a gamma distribution: $$ y\sim\text{Exp}(\theta), \quad\text{...
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What pitfalls should we avoid with Heidelberger-Welch convergence

I'm working through validating a Bayesian mixture model for multi-species occupancy with a collaborator. Initially, we relied on coda::heidel.diag to alert us to ...
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Mixture of two chi-squared distributions

I have a question about something I found in the book Frailty models in survival analysis (Wieneke, 2011). He talks about testing signficance of the frailty term. According to him, Claeskens et al. (...
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Mixture models and K-means clustering similarities: What is the covariance matrix?

Simple question: Is the covariance matrix used in the algorithm the covariance between the groups (Ie. Covariance between group 1 and group 2, group 1 and group 3 and so forth)? Or is it between the ...
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Monte Carlo integration and mixture distribution

How could I estimate an integral using Monte Carlo method when I have a mixture distribution? For example I want to estimate the below integral: $I=\int_{0}^{1}f(x)dx$ And my distribution is: $p(x)=...
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Variance of a Mixture [Solved] [duplicate]

Problem: Let $f_1(y)$ and $f_2(y)$ be density functions, and let $a$ be a constant such that $0\le a\le 1.$ Consider the function $f(y)=af_1(y)+(1-a)f_2(y).$ Show that $f(y)$ is a density function. ...
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What is the variance of the mixture distribution that is made up of 2 of the same distribution? [duplicate]

Let's say that I have two exponential distributions with a mean of 10. Now consider the mixture distribution where there is 50-50 chances of getting either one of the two exponential. What would be ...
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35 views

Can Gaussian mixture models with different amount of populations have the same pdf

Concretely, denote $f_{\mu, \sigma}$ for the pdf of a normally distributed random variable with mean $\mu$ and standard deviation $\sigma$. Is it possible that $$\displaystyle \sum_{i = 0}^n p_i f_{\...
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How to sample from a mixture of densities of transformed random variables?

Suppose we are given a set of $m$ random variables, $X_1$, $X_2$, ..., $X_m$, defined over the same set $\mathcal{X}$, with known densities $p_{X_i}$, for $i=1$, $2$, ..., $m$. Assume that getting ...
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182 views

Compute mean and variance of mixture of Gaussians given mean/variance of component gaussians [duplicate]

Given $N$ means and variances $\{\mu_1,\mu_2,....\mu_N\}$ , $\{\sigma_1^2,\sigma_2^2,....\sigma_N^2 \}$ ,and the fact that combined they make a gaussian mixture, how do I compute for that mixture $M$, ...
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Number of events in a segment if waiting times are drawn from a mixture of two exponential distributions

What is the probability for $n$ events to occur over a period of time $t$, if the duration of each individual event is $\tau_1$ with the probability $p$ and $\tau_2$ with the probability of $(1-p)$? ...
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Extendability of finite sequences of exchangeable random variables?

The old paper Binomial Mixtures and Finite Exchangeability by G. R. Wood makes reference (see Table 2) to a 1971 conjecture of Crisma that gives a formula for the probability that an exchangeable ...
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Convolutional roots of mixture of exponential distributions

In the reference book on infinite divisibility and generalised gamma convolution, BONDESSON, Lennart. Generalized gamma convolutions and related classes of distributions and densities. Springer ...
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Can the likelihood ratio estimate multivariate confidence levels?

Wilks' theorem describes the log-ratio between the highest likelihood of a distribution $\mathcal{L}$ (aka the dominant mode, given at $\vec{x}_{m}$) and the likelihood of a distribution at a given ...
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160 views

Calculation of AIC in finite mixture modeling

I have a question about calculation the AIC to find my optimal amount of clusters. I am applying mixture modeling with the EM algorithm. I know the formula AIC = -2ln(log-lik) + 2k. These are my log-...
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mixture models in 1 d [duplicate]

I have been reading the following slides available at: http://www.inf.ed.ac.uk/teaching/courses/iaml/2011/slides/em.pdf and I found the following formula: in this part the author is trying to ...
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Impute Missing Data Values with Mixture Models

Suppose I have a dataset with $o$ representing a collection of data dimensions with observed values and $d$ representing missing dimensions. The mixture model consists of discrete variables $Z = {1, .....
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Confidence interval on mean of mixture distribution with normal components

I have 50 sacks, each containing material from one of 10 varieties of hemp. Each sack has a different weight. I don't know what variety is in each sack, or how much material I have of each variety. ...
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Is the Mixture model just one type of Latent variable model?

Is the Mixture model just one type of Latent variable model? If not, then what is the relationship between Mixture models and Latent variable models?
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186 views

Why is weighing random observations according to their probability from all distributions wrong?

Is sampling all distributions n times and then talking out i numbers from each sample, where i is probability of that distribution * n, wrong? Suppose $$ 0.3\!\times\mathcal{N}(0,1)\; + \;0.5\!\...
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Random Samples from Spliced Distribution

I am studying Clauset, Shalizi, and Newman, Power Law Distributions in Empirical Data (preprint available here) in R. Packages used: ...
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Modelling a random variable that is mostly zero, but otherwise exponential (PyMC3)

I'm new to probabilistic programming, and have run into problems of this kind a few times now. Simply put: I often find myself wanting to model a random variable that mostly has some nice, continuous ...
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Convergence of empirical quantiles to theoretical quantiles - mixed type distribution

It is well known that under certain (not too restrictive) conditions empirical quantiles of a distribution converge to the corresponding theoretical quantiles in probability as the sample size $n \to \...
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Variance analysis for state dependent mixture importance sampling estimators

Let $(E,\mathcal E,\lambda)$ be a measure space $k\in\mathbb N$ $q_i:E\to[0,\infty)$ be $\mathcal E$-measurable with $$\int q_i\:{\rm d}\lambda=1$$ and $\nu_i:=q_i\lambda$ for $i\in\{1,\ldots,k\}$ $...
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Clusters keep switching in Gibbs sampling of Dirichlet Process Mixture Model

All the code and data for this question is on GitHub (stackexchange.R script). I've got multivariate Bernoulli data that I'd like to analyse using Bayesian Mixture ...
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How to latently cluster regressors based on regressors' relationship with the outcome?

What is the best way/method to model patterns across coefficients and reduce number of coefficients based on these patterns? We have hundreds of regressors on the same scale and try to reduce the ...
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Significance of modes in a distribution

I have several datasets with angular measurements, i.e. circular values from 0 to $2\pi$. These datasets tend to have peaks at 0 and/or $\pi$, and I need to tell if the peaks are detected/significant. ...
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119 views

Estimating two-component mixture of Weibull distributions?

Is there any existing package (preferably in Python or Matlab) to estimate the parameters of a two-component Weibull mixture model? And failing that, I am hoping to get some pointers towards rolling ...
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Predication of MDN

How to get the prediction of Mixture Density Networks? In MDN one models the conditional density: $P(y^i |x^i) = \sum_{j=1}^{m} \alpha_j(x^i)\phi({y^i|x^j})$, so I guess one just sample from the ...
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latent variables in EM algorithm are assumed to be i.i.d from multinomial distribution, from what they are idependent

In EM algorithm we introduce a latent variables, say $z_i$, $i=1,...n$, $n$ is the number of the mixture component. These variables ($z_i$) are assumed to be independent and identically distributed ...
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How to find quantiles and likelihoods of mixture distributions?

My PDF: M was estimated and found to be 5. I need to work out the quartiles for the PDF above. In addition, I need to use different methods of estimation to estimate the parameters. So far I've ...
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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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688 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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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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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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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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167 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
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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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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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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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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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326 views

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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43 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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60 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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426 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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4answers
99 views

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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377 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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