Questions tagged [mixture-distribution]

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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Recommended/Commonly Used Likelihoods for TF-IDF Observations in Mixture Models?

What are recommended or commonly used likelihoods for TF-IDF observations in mixture models? The below related questions ask about whether a Multinomial likelihood can be used (and if I understand ...
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How to evaluate the loss on a Gaussian Mixture Model?

I successfully modeled my data using a Gaussian Mixture Model in scikit-learn but I can't figure out how I should say "how good" the model is by calculating the loss. My first thought was to ...
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Does the sum of two variables second order stochastic dominate the mixture of two variables

hello I have two independent variable P and Q. They are both non-negative. Let $\alpha \in (0,1)$. Now I define two new variables on them: The first variable is the sum of the two variables $$R_1:=\...
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Modeling ideas on biologics manufacturing data with complex genealogy

This question is about finding options to model the data for a biologics drug manufacturing. The manufacturing process is divided into upstream and downstream, where the output material from an ...
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123 views

Estimating individual components of a mixture distribution

I am trying to jointly estimate the components of a mixture distribution. I have a sampling from a mixture, XY, composed of X ...
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Choosing the Dirichlet prior in a mixture model

Consider the following mixture model with $K < \infty$ components, $$ f\left(x \mid \theta_{1}, \ldots, \theta_{K}, \pi_{1}, \ldots, \pi_{K}\right)=\sum_{k=1}^K \pi_{k} \varphi\left(x \mid \theta_{...
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Compute likelihood of mixture distribution while avoiding floating point problems

I have a mixture distribution with likelihood function $$ L(\theta) = \prod_{i=1}^N \sum_{k=1}^K f(X_i|\theta_k) \lambda_k $$ where $N$ is the sample size, $K$ is the number of component, $\theta_k$ ...
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32 views

Mixture model when K=1

Assume that I want to estimate the parameters of a distribution like for example the Gaussian distribution, but I have the code only for the estimation of the parameters of a mixture of Gaussian ...
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Testing a sample is drawn from a mixture

Suppose you have data $(Y_1, \dots, Y_N)$ drawn from a finite mixture population with $K$ components and you estimate the model parameters $(\theta_1,\dots, \theta_K)$ and the mixture weights $(\phi_1,...
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Chain rule for KL divergence, conditional measures

The chain rule for KL divergence is widely seen in the theoretical machine learning literature and generally referenced to [1, Theorem 2.5.3]: $$ \text{KL}[p(x, y) \mid q(x, y)]= \text{KL}[p(x) \mid q(...
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Bayespy model construction

I am trying to solve a classification model using a bayesian approach. In particular I am using as reference the following work: Modeling Analysts’ Recommendations via Bayesian Machine Learning The ...
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Estimate parameters of a concrete categorical mixture model (information retrieval)

Let $f_{i,d}$ be the frequency of the word $i$ in the document $d$ and $l_d$ be the length of the document $d$. Then $P(X = i \mid D = d) = \frac{f_{i,d}}{l_{d}}$ is the probability of drawing the ...
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Transform Copula Marginals to come from Mixture Distribution

I am trying to create a joint distribution that has a specific copula (e.g. Clayton) and whose marginals come from a mixture distribution (e.g. the mixture of two Gaussian distributions). My idea was ...
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Expectation of ratios of probability density functions

I'm trying to solve/simply the expression below:- $\large \mathbb{E_{x \sim b(x)}} B\ [log\left(1 - \frac{A\ a(x)}{2\ c(x)}\right)]$, or $B \large \int_{x}b(x)log\left(1 - \frac{A\ a(x)}{2\ c(x)}\...
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simulation of mixture GARCH models

I want to simulate data that follow a mixture - GARCH specification. The conditional density of the return series $r_t $, based on information up to time t is given by $ f_{t-1}(r_t;\theta) = \sum_{i=...
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How to design an algorithm for k means-like problem?

I have some observation about a random variable x and I want to estimate the parameters of its probability distribution function. For example: data = {1, 2, 3, 4, 5} and choose normal distribution as ...
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Estimating weights of known component distributions in a mixture distribution

Given $n$ probability density functions ($p_1$, ..., $p_n$) with known distributions, what are the ways of estimating the weights ($w_1$, ..., $w_n$) of these component distributions given a sample ...
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How to sample from the distribution of the difference between two dependent mixture models

How to sample from the distribution of the difference between two dependent mixture models I have two distributions that represent a value before and after an event. Each of these distributions is ...
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Fitting mixture model of Gaussians and uniform distributions to real data

I have times series of wind direction and velocity. For now, I leave aside the velocity and focus on the distribution of wind directions. Over there, there is usually three main wind directions, and ...
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What is the nature of $r_i^t$?

Using Expectation Maximization (EM) algorithm, I want to vary the number of clusters used according to $K = [2,4, ... 50]$ for a normal distribution initialized randomly (...
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How to find the Percentile of this distribution?

Let say, we have 2 IIDs $X, Y \sim N \left( 0, 1 \right) + \eta $ Now $\eta$ has a discrete distribution with values 0(-10) with probabilities 0.991 & 0.009 respectively. Also assume that $Z = X+Y$...
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Nonparametric mixture estimation

Let's assume that we have two samples $\{X_i\}_{i=1..N}$ and $\{Y_i\}_{i=1..M}$ corresponding to random variables $X$ and $Y$. Let there also be a sample $\{Z_i\}_{i=1..K}$ corresponding to random ...
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Method of Moments for Mixture distribution

The restaurant owner also wants to reconfigure her seating layout, and has asked you for help in modeling her clients. She gives you a dataset of past reservations, and tells you that she gets a mix ...
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Evaluating (Uniform) Expectations over Non-simple Region

Background. Let $V = (X,Y)$ be a random vector in 2-dimensions uniformly distributed over two disjoint regions $R_X \cup R_Y$ defined as follows: $$ \begin{align} R_X &= ([0,1] \times [0,1]) \...
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Bivariate random effect problems in selection models (Mixture Cure model)

I am currently working on a mixed effects selection model. The selection model is a logistic model with a Gaussian random effect. The principal model is a survival model with a Gaussian random effect (...
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How do I interpretably identify mixture distributions?

I am trying to measure the impact of a binary variable $B$ on continuous $y$, with some covariates $X$. I fit an OLS regression and a mixture model of $2$ OLS regressions. The mixture model fits much ...
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60 views

How to compare two set of PMFs?

I'm facing some challenge and I don't know the correct approach for this. I'm having two sets of PMFs $S_1, S_2$ and I need to compare (distance like Jensen–Shannon) $S_1$ with $S_2$. What's the best ...
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68 views

A Gaussian scale mixture representation of the logistic distribution

Can the logistic distribution with density function $$f(x) = \frac{e^{-x}}{\left(1 + e^{-x}\right)^2}$$ be represented as a Gaussian scale mixture? In other words, if \begin{align*} X|V &\sim N(0, ...
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164 views

Categorical mixture model when mixture components are not PDFs (don't sum to 1)

I constructed a model that behaves the way I want, it successfully recovers parameters from simulated data, etc. However, I get the feeling that I re-invented the wheel, so to speak - surely someone ...
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9 views

Likelihood of a truncated (in)homogenous Poisson Process

I am studying a Poisson process of events that have occured until a given time. Theses occurences $T_n $a re observed by the observer only if the delay $U_n$ between occurence and time of report ...
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52 views

Taking Expectation Over Inverse Sum of Indicator Functions?

I'm working with a zero inflated Poisson distribution that has the following pmf: $$f(y|w,\lambda)=wI[y=0]+(1-w)\frac{e^{-\lambda}\lambda^{y}}{y!}$$ I would like to find the expectation of the ...
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Combination of distributions [reference request]

Consider a random variable $X:\Omega \rightarrow E$ that combines multiple distributions: $$X(\omega)\sim\begin{cases} N(0,\sigma^2),\hspace{0.2cm} \text{if $\omega =0$}\\ \text{beta}(0,1), \hspace{0....
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How to make inference on cluster-specific parameters in a Bayesian mixture model

Suppose I have a mixture model, for example of the kind $$ y_i \mid w, \{\theta_h\}, H \sim \sum_{h=1}^H w_h f(y_i \mid \theta_h) \\ P(H=h) = q_h \\ w \mid H \sim Dirichlet(\alpha) \\ \theta_1, \ldots,...
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27 views

Finding the mode of multiple sample modes

I'm running an experiment where I'm collecting samples of different size (numeric data only) and computing the mean, median and mode of each sample. I'm interested in finding out the mode across all ...
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31 views

Interpretation of a mixture model

I want to have a mixture model like $\lambda \cdot P(s'|s, a) + (1- \lambda) \cdot P'(s'|s)$ where $P$ and $P'$ are conditional distributions and $\lambda \in [0,1]$ is a weight. I have two questions ...
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Detect if there is actually two populations in a sample

I have been counting stomata on fossil leaf material to apply a known relationship between stomatal index and CO2. I thought that the material was all from one population (one species at a given site)....
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Estimating tail deviation from Q-Q plots

I am running experiments and for certain cases am able to find a suitable distribution for the data. However, in most cases, depending on a certain parameter, the observed vs fitted distributions have ...
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77 views

Show that the sum of two random variables is a mixture

Take any $({\lambda},{\mu},F,G)$ such that 1) $\lambda\equiv (\lambda_1,..., \lambda_J)$, $\lambda_j\in (0,1)$ for each $j=1,...,J$ and $\sum_{j=1}^J \lambda_j=1$ 2) $\mu\equiv (\mu_1,..., \mu_J)$ ...
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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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115 views

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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238 views

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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46 views

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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60 views

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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2answers
143 views

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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131 views

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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1answer
46 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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20 views

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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1k 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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