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Questions tagged [dirichlet-distribution]

The Dirichlet distribution refers to a family of multivariate distributions, which are the generalization of the univariate beta distribution.

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Bayesian posterior pmf for weighted dice with uniform prior

We want to find posterior probability mass function for dice tossing with uniform prior. We are interested in rolling of weighted dice. The outcome is 1,2,...,6. We assume that prior probability ...
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Dirichlet expected sufficient statistics

If $\theta$ is a ($k$-dimensional) Dirichlet distribution, the sufficient statistics are $\log\theta_i, i = 1,\ldots, k$. It can be shown that if the Dirichlet has parameter $\alpha = (\alpha_1, \...
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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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Relation between Chinese restaurant process (CRP) vs Dirichlet Process (DP)

There are multiple questions regarding the topic and but I have not found an answer to my doubts. Particularly: On Wikipedia Dirichlet Process page, regarding the connection between the Chinese ...
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Ranking/Sorting Star Ratings - Bayesian Credible Interval

I recently started analyzing episode polling data from a website that uses a 1-10 rating system. I've been reading about ranking star rating systems using Bayesian Credible Interviews as explained by ...
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Excellent fit, zero convergence hierarchical dirichlet model in JAGS

I am fitting a hierarchical dirichlet model to some data in JAGS. My samples (referred to as cores in the code) are observations of the relative abundance of 3 ...
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weighted sum of posterior Dirichlet distributions

I have the following distribution: $q(\vec\theta) = \frac{\sum_k \alpha_k}{\sum_k \beta_k \alpha_k} (\sum_k\theta_k\beta_k) \frac{\Gamma(\sum_k \alpha_k)}{\prod_k\Gamma(\alpha_k)} \prod_k \theta_{k}^{...
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convergence issues moving from dirichlet to multinomial-dirichlet in JAGS (implemented in R)

I am modeling microbiome count data, using a multinomial dirichlet. The number of times I observe each microbial "species" depends on its fractional abundance within a microbial community, and the ...
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Definition of distribution conditioned on both a categorical and Dirichlet prior

If we have a conditional categorical distribution, with unknown parameters, we can represent with a table, as in the example below: \begin{align*} &z \quad P(z|\theta)\\ &0 \quad \theta_0\\ &...
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Distribution of *conditional* frequencies when frequencies follow a Dirichlet distribution

Context: we have a large number of individuals characterized by two binary traits; call these $T$ with values $\{0,1\}$, and $T'$ with values $\{0',1'\}$. So there are four types of individuals: $00'$,...
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Recovering $\theta$ in Dirichlet-Multinomial (Polya) distribution

I'm working on Latent Dirichlet Allocation with Collapsed Gibbs Sampling. LDA has two Dirichlet-Multinomial distribution and one of them is a document-topic distribution that determines the ...
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Marginal Likelihood of Multinomial Dirichlet model

To find the marginal likelihood of the multinomial Dirichlet model, I tried the following: $$\int_\theta p(N|\theta)p(\theta)d\theta=\frac{n!}{n_1!...n_K!}\frac{\Gamma(\sum_{k=1}^K\alpha_k)}{\Pi_{k=1}^...
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How to specify a zero-inflated Dirichlet model in JAGS/BUGS

There was a recent publication discussing the advantages of the zero-inflated dirichlet for microbiome count data which is compositional (you are modeling a matrix of species relative abundance data ...
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Trouble specifying a hierarchical dirichlet model in JAGS

I have a sampling design where samples (cores) are taken within plots. Those plots are then nested within sites. There are multiple sites. I would like to get a hierarchical site-level estimate of ...
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dirichlet likelihood simplified

I am looking for derivation of eqn 5 in C.Moody's paper https://arxiv.org/pdf/1605.02019.pdf where it says the loss function coming from dirichlet enforcement of sparsity is $L^d=\lambda\sum_{jk}(\...
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Dirichlet-Multinomial Distribution with many zero counts

Short version: Is there someway to make the dirchlet-multinomial distribution sensitive to the presence of zero counts? Long version: I am attempting model metric positions in musical data. You can ...
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Marginal medians of the Dirichlet distribution

I am working with a 3 dimensional Dirichlet distribution with parameters $\alpha_1,\alpha_2,\alpha_3>0$. I have been trying to figure out a useful 'median' concept for this distribution. The vector ...
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dirichlet model returning “Support of observed nodes is not fixed” when monitoring DIC in JAGS

I am fitting a multivariate dirichlet model to species relative abundances using the runjags package in R, which is a wrapper for the ...
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Maximum a posteriori on Multinomial distribution with a Dirichlet prior can result in negative probabilities?

I am doing a maximum a posteriori (MAP) estimation of a Multinomial distribution $M(c_1,\dots,c_n|p_1,\dots,p_n)$ with a Dirichlet prior $D(p_1,\dots,p_n|\alpha_1,\dots,\alpha_n)$. The experimental ...
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Bayesian inference of non-homogeneous Markov transition matrix

The data consists of several discrete-time Markov chains, indexed by a global time. I assume all the chains are governed by the same transition matrix, but that this can change in time. I want to ...
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Using a Dirichlet distribution to estimate an expected response and uncertainity of response

I have an $N$ dimentional Dirichlet distribution $\text{Dir}\left(\alpha\right)$ that describes the number of times different responses, $r\in\left(1, 2, 3, \ldots N\right)=R$, have been observed. I ...
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Understanding Dirichlet Process Mixtures

I have been reading a ton of stuff online and have a question about my understanding of Dirichlet Process Mixtures. First some basics on what I understand: Dirichlet Distribution: multivariate ...
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multinomial model with some certain parameters

I might be asking a naive questions here, sorry. Imaging I have 4 categories, each one has a probability of $\theta_i$ being selelcted, $i=1..4$ and sum of $\theta_i$ is 1. For this simple ...
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Convergence issue dirichlet model JAGS, implemented in R

I have data on the relative abundances of 3 species, stored in the matrix r.spp.y. Species 1 has a negative relationship with the variable ...
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Expected value of the columns in a matrix generated from an Indian Buffet Process

Assume that a binary matrix $\mathbf{Z}$ of size $N \times K$ is drawn from an Indian Buffet process: $\mathbf{Z} \sim \text{IBP}(\alpha)$. The probability of $\mathbf{Z}$ given $\boldsymbol \pi \...
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Explanation of Formal Definition of Dirichlet Process

I am reading about the Dirichlet process and I can understand the construction from Chinese restaurant process or stick-breaking process or Polya urn scheme. Now I am trying to understand why ...
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parameter estimation in Bayesian Network with many zeros (in R)

I have a very small BN with only 4 nodes but have a large simulated dataset with around 36 000 observations. I want to use this dataset for parameter learning in bayesian networks. However, my ...
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Including features in a Dirichlet model with Markov dynamics

I have a fantasizing about a model here, so please keep in mind that this is not even half-baked: I have categorical time series data $y_t\sim\text{Cat}(y\ |\ \lambda_t)$ with a hidden variable $\...
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Summation of an infinite series involving a gamma function, in the context of estimating a Dirichlet prior

I have an unknown multinomial distribution $P^*$ over potentially unbounded set $\Sigma=\{1,2,\ldots,L\}$ from which a training set $\{x^1,\ldots,x^N\}$ has been observed. The observations form the ...
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Dirichlet disatribution: Connection between marginal and stick breaking distribution

Let suppose to have a probability vector $\boldsymbol{\pi} = (\pi_1,\pi_1,\dots , \pi_K)$, where by definition $\pi_K = 1-\sum_{j=1}^{K-1} \pi_j$, Dirichlet distributed with parameters $(\alpha_1,\...
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Measuring covariance among multiple y-variables using a dirlichet model, implemented in JAGS in R

I have data on 3 species abundances as fractional compositions. Within each sample, the sum of all species abundances is 1. Essentially, each observation is a proportional abundance. Here is an ...
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Normalization constant for uniform distribution over categorical distributions

Suppose we have a uniform distribution over all categorical distributions p for m categories, where the pdf has the form $$ f(x) = \left\{\begin{aligned} &c, && 0 \le p_i \le 1, i = 1, ......
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Is this a correct usage of Dirichlet-Multinomial model?

I have an interesting problem here. Say a person views a number of different websites and each website has a known demographic distribution. For example, the website xyz.com is 20% age 15-30, 60% ...
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Dirichlet process and clustering

How does clustering with a Dirichlet process as prior work? I am confused as to if the parameter $\alpha_i$ $\sim$ DP is found via clustering or is used to cluster. I undestrand how it can be used to ...
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Calculate Variance from Dirichlet-like Distribution Empirically

I'm interested in the proportion of time that a sensor is in a particular state. The sensor tells me the amount of time that it's in each state, which I will denote by $X = \{ X_1, X_2, X_3\}$. I ...
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Covariance between X and ln(Y) with X and Y beta random variables

I have a Dirichlet distribution $D(\alpha_1,\alpha_2,\alpha_3)$, with $\alpha=\sum_{i=1}^3 \alpha_i$. I know that the marginal distributions are beta distributions. Consider for instance the first 2 ...
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How to qualify this distribution (it was derived from a multinomial-dirichlet)? Does it exist? Is there a name for?

What is the name of the distribution below? Is it a Dirichlet-compound multinomial distribution? If not, why? \begin{equation} \begin{split} P[\theta_1|\theta_2,\theta_3,\boldsymbol{z},\...
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Purpose of Dirichlet noise in the AlphaZero paper

In DeepMind's AlphaGo Zero and AlphaZero papers, they describe adding Dirichlet noise to the prior probabilities of actions from the root node (board state) in Monte Carlo Tree Search: Additional ...
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Proof of Normalization of Dirichlet Distribution

Bishop defines the Dirichlet distribution (for M states) as: $p_M(\mu | \alpha) = \frac{\Gamma(\alpha_1 + ... + \alpha_M)}{\Gamma(\alpha_1)\Gamma(\alpha_2) \cdots \Gamma(\alpha_M)} \prod_{k=1} ^ M \...
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Mode of dirichlet distribution with parameters between 0 and 1

I have a dirichlet distribution with three dependent variables.What is the range in which the dirichlet parameters (alphas) should lie? I read in a ...
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141 views

Parameters of a dirichlet distribution and posterior mode

What is the range in which the dirichlet parameters (alphas) should lie? I saw the condition that alphas must be greater than 0. Then can I have alpha values between 0 and 1?
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The Bayes credible interval / Bayes credible region of the posterior distribution of a multinomial Dirichlet conjugate pairs

I have a posterior distribution of Dirichlet form with new parameters (alpha1, x1), (alpha 2, x2) and (alpha 3, x3) and the posterior mode of each dependent variable as the Bayes estimator. I wish to ...
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298 views

Find marginal distribution of $K$-variate Dirichlet

I've already seen https://math.stackexchange.com/questions/1064995/marginal-of-dirichlet-distribution-is-beta-integral, but need to extend this to the $K$-variate case. We have $\mathbf{x} = \begin{...
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1answer
530 views

Posterior mode, posterior mean and posterior variance of a posterior distribution of dirichlet form

What is the significance of finding the posterior mean, posterior mode and posterior variance in dirichlet - multinomial conjugate pair bayesian estimation? Are all of them equally important while ...
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1answer
400 views

Maximum likelihood estimation of a Dirichlet distribution multivariate parameters

Is it necessary to find the 'maximum likelihood estimates' of prior dirichlet parameters after finding their initial values through the 'method of moments ' to find posterior probabilities through ...
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1answer
256 views

Predictive Density for Dirichlet Multinomial

I am wondering what the predictive distribution of a Dirichlet-Multinomial distribution is. In this tutorial (page 24), the predictive density is simple and something like "pseudo samples." However, ...
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2answers
148 views

How is the generalised Beta function defined

I've been reading about the Dirichlet distribution and have become somewhat confused as to how to evaluate its normalisation term. Specifically, I'm quite happy with the relationship between the beta ...
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Entropy of Dirichlet distributed vector

Suppose I have two Dirichlet distributed vectors $X$ and $Y$ such that $ X \sim \text{Dirichlet}(\alpha) $, $ Y \sim \text{Dirichlet}(\beta) $ with fixed vectors of hyperparameters $\alpha$ and $\beta$...
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Bayesian smoothing using Dirichlet prior : why not MAP?

I am reading about smoothing methods for language model ( I am working on unigram model). If you are not familiar with unigram model, it is closely related to multinomial distribution (with the ...
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Is 1 - Dirichlet variable also a Dirichlet?

Just a simple question regarding the properties of Dirichlet distribution: Suppose $(X_1, \ldots, X_K) \sim Dir(\alpha_1, \ldots, \alpha_K)$, can we express the distribution for $(1-X_1, \ldots, 1-...