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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Distribution of the mean of a Dirichlet-distributed distribution

Suppose that $(f_0,\dotsc,f_N)$, with $f_n\ge0, \sum_n f_n=1$, is a distribution (set of normalized weights or frequencies) having a Dirichlet distribution with parameters $\alpha_n$: $$\mathrm{p}(f_0,...
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How to parameterize variational Dirichlet distribution

I am learning about variational inference and am implementing a couple of things from scratch. I am trying to build a Gaussian mixture model where the prior on the mixture component selection is a ...
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Deriving the marginal multivariate Dirichlet distribution

I am trying to understand how my professor (see derivation below) has derived the multivariate marginal distribution of a subvector of $\theta_j$´s from a Dirichlet distribution. I understand ...
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Parametrization of Dirichlet distribution

Hej! Consider I have a Dirichlet distribution with 4 variables, where the mean (u) values of these are known. $(u1+u2+u3+u4=1)$ Now, I want to obtain the parameters of the Dirichlet distribution ($\...
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Using categorical data to build a Dirichlet distribution

I am building a graphical model. I have some categorical data $\boldsymbol{\mu}$ where they are generated by $p(\boldsymbol{\mu}|\boldsymbol{s},\mathbf{A})=\prod_k\prod_j\mathbf{A}_{ij}^{\mu_is_j}$. ...
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Cumulative sum of a Dirichlet distribution?

Assume $X_1, X_2, ..., X_n \sim Dir(\alpha_1, \alpha_2, ..., \alpha_n)$ What is the joint distribution of $X_1, X_1+X_2, ..., \sum_{i=1}^n X_i$?
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Calculate log likelihood of Dirichlet distribution using Gamma distribution

Let $X_1,\dots,X_{k+1}$ be mutually independent random variables, each having a gamma distribution with parameters $\alpha_i,i=1,2,\dots,k+1$ show that $Y_i=\frac{X_i}{X_1+\cdots+X_{k+1}},i=1,\dots,k$,...
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What Distribution Do I need?

Suppose I am drawing coloured balls from a bag. The ball can be red, green or blue. The probabilities of drawing a red, green or blue bag are uncertain, but I have confidence bounds for the ...
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Dirichlet process Posterior notation

While reading through Yee Whye Teh's tutorial on Dirichlet Process, I came across the Posterior distribution for DP. After observing some draws $\theta_1,\theta_2, ... ,$, the posterior is defined as $...
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Measure of base distribution $G_0(A)$ in the Dirichlet process

I am studying the Dirichlet Process and I came across this rule that states that given a partition $A_i's$ over the space $\Theta$, which represents the space in which $G_0$ is drawn from, $$(G(A_1),G(...
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Is there a statistical distribution whose values are bounded $[-1,1]$ and sum to 1?

The Dirichlet distribution contains values that are bounded $[0,1]\in \mathbb{R}$ and sum to $1$. Is there a parametric distribution or similar method whose values do the same but reach as low as $-1$?...
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Probability that a Linear Combination of Dirichlet Random Variables is a Distribution

I've been putting a lot of thought on this problem, but it seems I ran out of ideas. Any help would be appreciated! Suppose we generate two probability vectors $\boldsymbol{\theta}_1, \boldsymbol{\...
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Restriction and dependence in the Dirichlet distributon

The Dirichlet distribution is sometimes said that it is "too restrictive and imposes strong conditions on the dependence between components". What is the reason?
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Overall variance of a Dirichlet distribution

I have observations on proportion of individuals at different age groups. I'm doing some simulation experiments, and I need to introduce error in the observed data. For that, I'm simulating data from ...
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Marginal Cluster assignments for Dirichlet Process mixture model

I am watching Tamara' Broderick video on Dirichlet Process mixture models where she talks about computing $p(z_n = k | z_1,z_2,..z_{n-1})$ at ardoun 16:06. The z's are drawn from $$\rho_1 \sim beta(...
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How to derive the expectation of $\ln \mu_j$ in Dirichlet distribution

I have derived the mean and variance of $\mu_j$ in Dirichlet distribution $\text{Dir}(\mu_1, \cdots, \mu_K|\alpha_1, \cdots, \alpha_k)$. On https://en.wikipedia.org/wiki/Dirichlet_distribution, it ...
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What is a non-informative choice of parameters for a Dirichlet distribution?

Dirichlet distribution is a conjugate prior for multinomial distribution. I want to impose a non-informative prior over sampling weights $\pi$ for a draw $x=(x_1,…,x_N)$ from a multinomial ...
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On the choice of prior in Bayesian Bootstrap

Let $d=(d_1,…,d_K)$ be a vector of all the possible values that the data $x=(x_1,…,x_N)$ could possibly take. Then, each $x_i$ is modeled as being drawn from the $K$ possible values where the ...
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Uniform posterior on bounded space [duplicate]

In a particular Bayesian problem, I have encountered a choice of parameters that leads to a uniform posterior distribution. Given prior \begin{equation} p(\boldsymbol{\pi}) =Dirichlet(\boldsymbol{\...
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How to model proportions with a hierarchical structure?

I have thinking about how to model proportions for a problem with hierarchical structure. In the problem, I have observations of users over multiple days, where each observation is a proportion of ...
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Softmax vs the Dirichlet distribution

As far as I understand one can in principle model the distribution over a set of $k$ categories using e.g.: the Dirichlet distribution A softmax model. As far as I can tell, both use $k$ parameters ...
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How does the uniform Dirichlet PDF integrate to 1?

For the uninformative 3-dimensional Dirichlet prior ${\rm Dir}(1, 1, 1)$, I understand that the probability density function (PDF) evaluates uniformly to 2, and the support are all three-dimensional ...
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On posterior in the Bayesian bootstrap

The Bayesian bootstrap was introduced by Rubin (1981) as a Bayesian analog of the original bootstrap. Given dataset $X=\{x_1, \dots, x_N\}$, instead of drawing weights $\pi_{n}$ from the discrete set $...
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Finding the log Jacobian of a transformation from lower to high dimensions, specifically in the case of normal cdf to Dirichlet?

I have random variables $\mu$ and $\sigma$ that I have transformed and I am interested in finding their joint distribution given the following information I have. Particularly, I need help finding the ...
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Fitting/Inference for Dirichlet Process/CRP for clustering

Please excuse my ignorance on this topic, I don't have much experience in nonparametric Bayes. I read about Dirichlet process clustering and the Chinese Restaurant Process analogies. I think I ...
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Understanding Dirichlet clustering

I am trying to understand Dirichlet processes and how they are used for clustering. I am following this overview. I understand Dirichlet distributions (ie. generalizations of the Beta, used for ...
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Dirichlet distribution vs Multinomial distribution?

Both Dirichlet and multinomial distributions are distributions over vectors, and both Dirichlet and multinomial distributions are constrained so that all of the elements of these vectors sum to a ...
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log likelihood function causes huge differences with similar inputs

I am not sure whether there is a solution to this problem, but here goes. My problem is that in my function, I am taking log of a matrix and then taking its mean, and doing this for two very similar ...
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Posterior probability distribution from multinomial sample

I want to get the posterior from a multinomial sample and want to know if the following derivation is correct. Suppose when drawing (with replacement) a sample of $N$ balls from a urn with $K$ ...
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How to design a joint distribution of random variables when their summation is known

Suppose we have $D$ random variables $X_1, \ldots, X_D$, and $X_i\in[0,1]$ for $i=1,\cdots,D$. Can we somehow design a joint distribution $P(X_1, \ldots, X_D)$, such that $\sum_{i=1}^{D}X_i=s$, where $...
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Why use MCMC sampling when using conjugate priors?

I've been getting to grips with some Bayesian modelling, but one thing is confusing the heck out of me when I look at tutorials and worked-through problems online. I'm looking at a problem with a ...
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Question about possible typo in a tutorial about the stick-breaking model of the Dirichlet distribution

I am reading a tutorial on the Dirichlet distribution: http://mayagupta.org/publications/FrigyikKapilaGuptaIntroToDirichlet.pdf and I think there is a typo in Step 2 of the stick-breaking model of ...
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Proving independence relationship

Let $X_1,X_2,X_3$ be continuous positive random variables satisfying $X_1+X_2+X_3<1$ and the following independence relations $$\frac{X_1}{X_1+X_2}\perp \!\!\!\perp \frac{X_3}{1-X_1-X_2}~ and$$ $$\...
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Proposal Function For Variables That Sum To 1 (Dirichlet Prior)

Recently I've been trying to use MCMC to infer a set of 50 random variables (species frequencies) that sum to 1 with the Metropolis-Hastings algorithm. However, the algorithm is not working well ...
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Normalized subvectors of Dirichlet, mutually independent?

Let$$X=(X_1,\cdots,X_k)\sim Dir(\alpha_1,\cdots,\alpha_k)$$ According to this reference, the independence of the two vectors, $$\bigg(\frac{X_1}{X_1+\cdots+X_j},\cdots,\frac{X_j}{X_1+\cdots+X_j}\...
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Inference on Dirichlet hyper-parameter

I'm working on a Gibbs sampler for a (somewhat custom version of) Latent Dirichlet Allocation model. In short, I have data that comes from a $K$-dimensional Dirichlet-Multinomial distribution, i.e. $$...
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LDA alpha equivalent in structural topic model

I'm using an implementation of the structural topic model (stm), written in R using the stm package. I want to reduce the number of topics that are prevalent in ...
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If (x1,x2,…xn) follow a dirichlet distribution with all alphas=1, then I think xi~U(0,1)/summation[U(0,1)]. If yes, how to prove it?

If (x1,x2,...xn) follow a dirichlet distribution with all alphas=1, then I think xi~U(0,1)/summation[U(0,1)]. If yes, how to prove it? Note that xi are all positive and summation(xi)=1
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I want to represent x1, x2, …, xn (where their sum =1) by Dirichlet distribution. What alpha's should I select if x1, x2,… have the same pdf

I want to represent x1, x2, ..., xn (where their sum =1) by Dirichlet distribution. What alpha's should I select if x1, x2,...,xn have the same probability density function? all 0 < xi < 1. In ...
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could someone please give a concrete example to illustrate the Dirichlet distribution prior for bag-of-words?

I am aware of the notion of the Dirichlet distribution, a multivariate generalization of the beta distribution. To get parameters of the Dirichlet distribution prior for bag-of-words, this CMU ...
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Aggregation with an overlap: Dirichlet distribution

Suppose that we have $$(p_1,p_2,p_3,p_4)\sim Dirichlet(a_1,a_2,a_3,a_4),$$ where $p_4=1-p_1-p_2-p_3.$ When we add random variables for example, $p_1+p_2$ and $p_3+p_4$, the resulting distributions ...
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What is the name of this distribution?

I came across this: a categorical distribution with $K=10,000$ parameters (categories), and we take only few samples from this distribution, say $N=400$ (the point is $N < K$). Now, obviously, not ...
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Is it possible to fit a Dirichlet Regression to changing response variables?

A toy problem to illustrate my issue: We put a 100 people in a room with 10 candy bars. Each bar is different and has a different brand, flavor, size, color, etc. We ask each person in the group to ...
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Bayesian inference out of partial information - Dirichlet example

Suppose we have two coins $X_1$ and $X_2$. They are possibly biased and correlated coins. The heads probability of each coins is denoted by $p_1$ and $p_2$ which we don't know at the beginning. The ...
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Probability that a random variable is smaller than another in a random vector

Suppose that a random vector $X=(X_1,X_2,X_3)$ follows a Dirichlet distribution with a shape parameter $(a_1,a_2,a_3).$ What I want to calculate is the probability of $X_1>X_2$ and I want to ...
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Bayesian update for Beta distribution

I'm wondering how to find a posterior of a beta distribution when the "new information" is not an outcome of a binomial trial. Let $p$ be the probability of Head of a (biased) coin toss. As usual in ...
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Bayesian Inference: Prior in Chinese Restaurant Process

For the Chinese restaurant process, as used in Dirichlet Process mixture models, we have a prior that data point i belongs to cluster j, where c is an indicator. n represents the total number of data ...
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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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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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