Questions tagged [mixture]

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

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24 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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54 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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56 views

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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38 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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61 views

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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Given distribution of $X$ and $X|Y=y$, is it possible to find distribution of $Y$?

What the title says! My intuition is NO since in Bayesian statistics we typically specify the prior and likelihood, and from those two we can compute the posterior and so on. We can interpret $Y$ = ...
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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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Mixture Densities weights

I'm supposed to find the mixture weights and densities of all the mixture components. Should i find the normalizing constant in this case then work from there? Any hints or solutions will be much ...
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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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133 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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31 views

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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54 views

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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1answer
62 views

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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1answer
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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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32 views

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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1answer
72 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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35 views

How is jaccard similarity used to find the similarity between Bootstrap samples when measuring stability of EM?

Im reading the answer on "how to determine number of clusters in EM algorithm". How to tell if data is "clustered" enough for clustering algorithms to produce meaningful results? One of the ...
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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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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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685 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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1answer
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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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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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1answer
24 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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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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1answer
147 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
43 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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1answer
29 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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1answer
30 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
78 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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1answer
24 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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1answer
275 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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1answer
41 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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1answer
56 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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425 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
90 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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1answer
329 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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0answers
41 views

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
60 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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1answer
206 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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