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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Treating missing data in making Bayesian inference

Suppose we have two biased coins $X_1,X_2$ that are possibly correlated to each other. In each round, when both the coins are tossed, there can be four possible outcomes: $(HH,HT,TH,TT).$ Let's ...
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Gibbs sampling for mixture with Dirichlet prior?

I want to sample from the distribution of a mixture distribution. The hierarchical model is $x_i\sim f$, where: $$f(x\mid \theta_1,\dots,\theta_p, w_1,\dots,\omega_p) = \sum_{j=1}w_p\varphi(x\mid\...
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Modeling Bayesian inference using Dirichlet conjugate

I'm trying to formalize my research question and want to know whether the following set up makes any sense or not. Suppose there are two coins $a$ and $b$. Probability of tossing heads are given by $...
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38 views

Deriving conditional probability of bivariate bernoulli by using Dirichlet

While I was working on my research project, I found it difficult to derive a conditional probability from Dirichlet dist. Consider two Bernoulli trials that are possibly correlated with each other. ...
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Moments of the Dirichlet distribution [duplicate]

I was reading the Wikipedia article of the Dirichlet distribution which gives a general equation for the moments of a Dirichlet distributed random variable $X=(X_1,\cdots,X_K) \sim Dir(\boldsymbol{\...
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Bayesian inference using Dirichlet: muddled outcome case

In relation to my previous question (Bayesian inference for Beta distribution after an uncertain outcome), Suppose that $$(x_1,x_2,x_3)\sim Dirichlet(a_1,a_2,a_3)$$ and an associated Mutinoulli ...
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LDA: weight distributions of inferred documents

I have trained a two-topic Latent Dirichlet Allocation (LDA) model on a corpus and I am now inferring on a test corpus (the nature of the corpus is irrelevant). During inference, for each new document ...
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22 views

Difference between two dimensions sampled from Dirichlet distribution

Say I'm doing Bayesian inference on a Dirichlet-Multinomial model: $$ x \in [1,2,3]; \\ x \sim Multinomial(p_1, p_2, p_3); \\ p_1, p_2, p_3 \sim Dirichlet(\alpha_1, \alpha_2, \alpha_3); \\ \alpha_n =...
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From beta distribution to Dirichlet: Estimation of the concentrantion parameters

Searching at least 3 hours about the connection between beta distribution and dirichlet. My problem is: I have a collection of random variables $X_i \sim Beta(a_i, b_i)$. The parameters $a_i$ and $...
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37 views

Applying de Finetti's representation theorem to Dirichlet distribution

Let's begin from the de Finetti–Hewitt–Savage theorem: for an exchangeable sequence of random variables we can always write $$ p(x_1, x_2,\cdots) = \int \prod p(x_i | L) P(dL) $$ where $L$ is a latent ...
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dirichlet distribution and excessively large numerator

what I am trying to do is calculating posterior probability using dirichlet distribution as my prior. the situation is like this. a web log have three variables A, B, C, and each variable's value is ...
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Clarifying Dirichlet Process Mixture Probability Terms

Suppose I have a Dirichlet Process Mixture model defined as follows: $\alpha \sim G(a,b)\\ \pi|\alpha \sim \text{Dir}(\alpha)\\ z|\pi \sim \text{Cat}(\pi)\\ $ where $G$ is just a standard Gamma ...
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Multinomial-dirichlet with fractional counts

Suppose a lepidopterologist wants to estimate the relative proportions of three different species of butterfly. They go out into the field and count $N$ butterflies and record the number of each ...
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39 views

Is fair to compare Dirichlet to a Multivariate Beta regression?

I am conducting some analysis on my data I found a strange behavior and would greatly appreciate some guidance or suggestions. I am trying to investigate the effect of a categorical variable (cl) to ...
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Dirichlet Multinomial Posterior Predictive Distribution for Language Model

I have been trying to teach myself about Bayesian analysis, and whilst I have been through the theory several times, I am struggling to actually apply it. I have found some questions online to ...
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162 views

Coding a simple Stick-Breaking Process in Python

I've just red the great 2012 blog post of Edwin Chen about Dirichlet Process with companion code in R and Ruby. Then I'm trying to translate the Stick-Breaking Process from R to Python. I've got this ...
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How is the mode in Dirichlet-Multinomial calculated?

The mode in Dirichlet-Multinomial is $$ \mathrm{Mode}(\pi_i) = \frac{\alpha_i + x_i - 1}{\sum_{j=1}^k (\alpha_j + x_j -1)} $$ Could you point out how is it calculated please? What is the importance ...
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What are the possible estimates of the parameters of the multinomial distribution?

The expected value of the parameters of the multinomial distribution (taking into account the Dirichlet prior $D(\alpha)$ and the posterior Dirichlet-Multinomial) is: $\pi_i = α_i+ x_i / \sum_{j} α_j+...
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239 views

How to use the Dirichlet prior for estimating the multinomial parameters? [closed]

I know that the multinomial distribution gives the likelihood of some vector D of occurrences to happen given a probability vector (parameters) P' i.e. P(D|P'). Now with a Dirichlet prior we are ...
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Understanding Multivariate Logistic Normal pdf

I'm trying to understand the Multivariate Logistic Normal distribution, in order to plot its pdf and compare it with a Dirichlet distribution. I believe I can follow the pdf derivation for the ...
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102 views

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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34 views

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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1answer
295 views

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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61 views

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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296 views

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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84 views

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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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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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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147 views

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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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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110 views

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 ...