Questions tagged [dirichlet-distribution]

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

Filter by
Sorted by
Tagged with
0
votes
0answers
9 views

Single-value alpha parameter for Dirichlet distribution

I'm trying to implement an event schema induction method from a paper from 2015. The authors use a generative approach to learn a language model. For this, they use a lot of probability distributions ...
0
votes
0answers
19 views

Using the methods of moments in R for the dirichlet distribution

I'm trying to build a distribution of transition probabilities to randomly sample from in a Markov model where individuals can transition from one health state to another (assume that in the image ...
4
votes
1answer
130 views

CDF of Dirichlet Distribution

We know that a random variable $p=(p_{1}, p_{2},..., p_{K})$ which follows a $\textit{Dirichlet}$ distribution with parameters $\textbf{a} = (a_{1}, a_{2},..., a_{K})$ has as pdf $$f(p) = \frac{1}{B(\...
0
votes
0answers
35 views

How is a convex combination of Dirichlet-distributed variables distributed?

Let $X = (X_1, \dots, X_K) \sim \operatorname{Dir}(\alpha_1, \dots, \alpha_K)$ and define the convex combination $Y = \sum_{i=1}^{K} c_i X_i$. In the case of $K=2$, the constraint $\sum_{i=1}^{K} X_i =...
2
votes
1answer
19 views

Bayesian (continuous) logistic regression model with Beta likelihood?

I have a problem where my target variable are continuous/float values in the range [0,1]. If my data were integers in {0,1} this would be a simple logistic regression / Bernoulli likelihood problem. ...
0
votes
0answers
28 views

Dirichlet-distribution and its correlation?

I have the following variables that follow a beta distribution: ...
3
votes
1answer
133 views

Mean of Generalization of the Dirichlet Distribution

I know that if $X_{1},X_{2},...X_{n}$ are independent $\mathrm{Gamma}(\alpha_{i},\theta)$ - distributed variables (notice they all have the same scale parameter $\theta$) and $Y_{i}=\frac{X_{i}}{\sum_{...
2
votes
1answer
36 views

How to generate data from a generalized Dirichlet distribution?

I need to generate data from a generalized Dirichlet distribution in Python to test my model, but I have no idea how can I proceed with that, can anyone guide me?
1
vote
1answer
18 views

Concentration Bounds for categorial distribution with good Dirichlet prior

I would like to know if there are any standard methods for analyzing the concentration bounds (for example Hoeffding's bound) for a multinomial distribution modelled with a Dirichlet prior, with the ...
0
votes
0answers
33 views

Sampling with a Dirichlet likelihood in PyMC3

I am trying to implement the independent fusion classifier model from Trick & Rothkopf A Normative Model of Classifier Fusion (arXiv:2106.01770) using PyMC3, but am running into a problem with ...
3
votes
1answer
56 views

How to visualize Dirichlet distribution (with more than 3 targets)?

I want to plot a Dirichlet distribution $\operatorname{Dir}(\alpha), \alpha=[\alpha_1, \alpha_2, \ldots,\alpha_n]$. However, when I google it, almost all of the results consider 3 targets ($n=3$), and ...
2
votes
1answer
62 views

Stationary distribution of a Markov chain with a random transition matrix

Consider a Markov chain $\{X_t\}$ on a finite state $\mathcal{S} = \{1,\dots, S\}$ space whose transition matrix $P$ is populated by elements of the form $$ p_{ij} = P(X_{t+1} = j | X_t = i)$$ and we ...
1
vote
0answers
25 views

Log-likelihood of a finite mixture distribution (PDF overflowing)

I'm trying to use a finite mixture of Dirichlet distributions in a project, but am encountering problems with the PDF becoming so large for input values close to 0 that it overflows to infinity (as ...
0
votes
0answers
15 views

Bayesian Multinomial in rating

I have a dataset that has two columns:2)The rating of a product from 0:5 and 2) the numbers of votes.Can i make a Bayesian analysis with rating following a multinomial distribution (categories 1-5) ...
2
votes
1answer
39 views

Upgrading weight parameters to random variable in Gaussian mixtures

In a Gaussian mixture model we model a density like: $p(\mathbf{x}|\pi,\mu,\sigma)=\sum \pi_i N(\mathbf{x}|\mu_i,\sigma_i)$ [1] where $\pi,\mu$ and $\sigma$ are parameters. I would like to know if the ...
3
votes
1answer
121 views

Difficulties in computing the derivatives of the Dirichlet distribution

I need to compute the first derivatives of the Dirichlet distribution, defined in the following way: $$r(P; \pi, \rho) = \frac{\Gamma(c)}{\prod_{i=1}^{k} \Gamma(c \pi_i)} \cdot \prod_{i=1}^{k} P_i^{c\...
0
votes
0answers
30 views

Calculate the Wasserstein distance between the dirichlet and uniform distributions

I am reading a couple of papers that each have their own way of constructing skewed data partitions from a uniform (global) data set. The data set consists of entries with a label $y$, with in total $...
2
votes
0answers
13 views

Question about the distribution of the average of Dirichlet-distributed random variables

Suppose that each in a set of $n$ random variables $\boldsymbol{X}_1, .., \boldsymbol{X}_n$ are Dirichlet-distributed with parameters $\boldsymbol{\alpha}_i$, where $i$ is an index for the random ...
2
votes
0answers
39 views

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_{...
1
vote
0answers
13 views

On the distribution of a scaled sum of a Dirchlet random variable

Consider $(X_{1},\dots,X_{K})=X\sim \text{Dir}(\alpha)$ and a vector $v=(v_{1},\dots ,v_{K})\in\mathbb{R}^{K}$. Is there a parametric density function for the distribution of: $Xv^{T}=vX^T=\sum^{K}_{i=...
0
votes
0answers
17 views

I'm trying to identify the posterior distributions in LDiA. Are theta and phi in this PGM the posterior distributions of LDiA?

My understanding is that alpha and beta are the dirichlet priors, and the posteriors would be theta and phi. And since theta and phi are from the same family of distributions as alpha and beta, the ...
0
votes
0answers
23 views

In the dirichlet distribution, does each x represent a particular distribution on the simplex?

The dirichlet distribution takes this form: I'm trying to understand how we end up with a vector from this formula rather than a scalar. The only way I see how is that each x itself is a vector ...
0
votes
0answers
67 views

Understanding Dirichlet Distribution Variance

I need some help in understanding the variance/standard deviation in the Dirichlet distribution. I apologize in advance for the lack of latex. In the Beta distribution, as the shape parameters ...
1
vote
1answer
25 views

In Latent Dirichlet allocation, is the following formula the probability of observing a single document, or an entire corpus?

This is the formula in question: Source: https://en.wikipedia.org/wiki/Latent_Dirichlet_allocation
3
votes
0answers
20 views

Bounding values of a Dirichlet distribution

Consider $k$ random variables $X_1, X_2, \ldots, X_k$ such that $(X_1, X_2, \ldots, X_k)$ follow a $\text{Dirichlet}(1, 1, \ldots, 1)$ distribution. For a large enough $k$, I am trying to bound/find ...
0
votes
0answers
12 views

Bayesian network: graph synthesis & data sampling

Input (What I have): some Bayesian networks (both graph structure and conditional probability distribution (cpd)) and corresponding categorical datasets (e.g. bnlearn repo). Output (What I want): ...
3
votes
0answers
98 views

A clarification in the original Dirichlet Process paper by Ferguson

I am reading the paper "Bayesian Analysis of Some Nonparametric Problems" by Ferguson where the Dirichlet process is introduced. There is a proposition 5 where the joint distribution of ...
0
votes
0answers
24 views

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 ...
0
votes
0answers
15 views

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,...
0
votes
0answers
17 views

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 ...
4
votes
1answer
193 views

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 ...
1
vote
1answer
41 views

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 ($\...
4
votes
0answers
160 views

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 $p(\boldsymbol{\mu}|\boldsymbol{s},\mathbf{A})=\prod_k\prod_j\mathbf{A}_{ij}^{\mu_is_j}$. ...
0
votes
0answers
36 views

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 $X_1, X_1+X_2, ..., \sum_{i=1}^n X_i$?
0
votes
1answer
93 views

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 $Y_i=\frac{X_i}{X_1+\cdots+X_{k+1}},i=1,\dots,k$,...
0
votes
2answers
37 views

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 ...
5
votes
2answers
412 views

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 $-1$?...
2
votes
1answer
121 views

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{\...
2
votes
0answers
15 views

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?
0
votes
0answers
61 views

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 ...
1
vote
0answers
19 views

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(...
1
vote
2answers
279 views

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 ...
3
votes
2answers
451 views

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 ...
0
votes
1answer
155 views

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{\...
1
vote
0answers
26 views

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 ...
0
votes
2answers
393 views

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 ...
0
votes
1answer
71 views

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 ...
2
votes
2answers
300 views

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 $...
0
votes
2answers
87 views

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 ...
4
votes
2answers
1k views

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 ...

1
2 3 4 5 6