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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Choosing a probability distribution for 4D data: dirichlet challenges and alternatives
I'm seeking the right distribution for my 4D data, where the sum of values in each sample equals one. Currently, I've chosen to employ the Dirichlet distribution. However, upon applying this ...
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Do you know if this re-scaled Dirichlet kernel is known in the literature? How to sample from it?
In a Bayesian analysis, I came across the following distribution that results ends up looking like a re-scaled Dirichlet distribution. The motivation comes from looking at probabilities $x_1, \ldots, ...
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Dirichlet distribution with correlated components?
I am working with models that use Dirichlet distributions. However, I want to account for correlations between components. If this question is a duplicate, I'd also appreciate any pointers to the ...
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A confusion about computing transformation of random variables
Let $(X,Y)$ be a pair of random variables with joint pdf $f_{XY}$. Let $(U,V)$ be two random variables obtained from $(X,Y)$ by $U = u(X,Y)$ and $V = v(X,Y)$ where $u$ and $v$ are, say, nice ...
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number of parameters in Dirichlet Mixture Model clustering (non-bayesian)
I made a function that implements the clustering algorithm in the research article "Clustering compositional data using Dirichlet mixture model" (2022). I am now trying to figure out which ...
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Dirichlet distribution centered at a gaussian random field?
I created three (100x100) synthetic images using values randomly drawn from a Dirichlet distribution as shown below. Each image represent the abundance values of some surface components (soil, water ...
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Power of Uniform Order Statistics
I know that if $U$ is a uniform r.v. in $(0,1)$, then $U^a\sim Beta(1/a,1)$ with $a>0$.
On the other hand, if $U_{(1)}\leq \cdots\leq U_{(n)}$ are the uniform order statistics, then, with $U_{(0)}=...
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Stick-breaking construction of Dirichlet distribution vs Dirichlet process
Let $F_0$ be some probability measure and $\alpha > 0$ be the concentration parameter. I can draw a random distribution from $F\sim \mathrm{DP}(\alpha, F_0)$ using the stick-breaking construction:
\...
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Dirichlet Regression output and using the calculated coefficients in regression model
I am very new to Dirichlet Regression and trying to make sense of the output and the regression coefficients. I am doing a biomass study and have tested the following variables (DBHH, DBH + H, DBH and ...
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Bayesian inference based on a 3$\times$3 contingency table
How do I make inferences about population parameters based on a 3$\times$3 table of observations? In "Bernoulli's Fallacy", Aubrey Clayton provides this (Table 5.8).
Democrat
Republican
...
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Inferring relative frequencies in partially observed Dirichlet-Multinomial
I am working with this example. I have a set of multinomial data with $N$ rows and $D+1$ categories drawn from buckets of different size, $M_n$: $$ \mathbf{p}_{D+1}~\sim~\text{Dirichlet}(\mathbf{a}_{D+...
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Bayesian reparametrization are they equivalent?
Suppose that we are in a Bayesian context, we we have the following matrix $n,$ $K\times K,$ as parameter, and we assume that
$$n_{ij}\sim Pois(w*w_{ij})$$
where $w\sim Gamma(N+1,1)$ and $w_{ij}$ is ...
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Classifying changes in Dirichlet distribution over time?
I'm interested in studying user preferences regarding streaming content. Given a discrete number of categories (ex: adventure, horror, comedy, family, drama) and the amount of time a given user ...
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Distribution of the ratio of Dirichlet/Gamma variates
It can be seen that the following random variates have the same distribution:
$\frac{X_1 + X_3}{X_2 + X_3}$, where $(X_1, X_2, X_3) \sim \text{Dirichlet} (\alpha_1, \alpha_2, \alpha_3)$
$\frac{Y_1 + ...
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Computation of ratio with Dirichlet distribution
I would like to compute ratio of proportions coming from a Dirichlet distribution. My understanding is that each proportion should be treated as a random variable and therefore I should use Taylor ...
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Advice on how to solve a constrained KL Divergence problem between a Dirichlet and a Logistic Normal
I would like some advice or path to follow to solve the following problem.
Consider a random variable $Y$ that follows a Dirichlet distribution $Y \sim Dir(\alpha)$. Let $X$ be a member of the ...
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How can we measure the "fit" between the softmax outputs and Dirichlet distribution?
For simplicity, I'll consider classification with 3 classes.
Then, softmax outputs can be considered as the set of points in 2-simplex.
I want to measure the 'fit' of this softmax output with target ...
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What is the expected value of the logarithm of a variable of the Generalized Dirichlet?
For a Dirichlet $\operatorname{Dir}\left(\mu_1, \cdots, \mu_K \mid \alpha_1, \cdots, \alpha_k\right)$,
\begin{equation}
\mathbb{E}\left[\ln \left[\mu_j\right]\right]=\psi\left(\alpha_j\right)-\psi\...
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Ordinal regression - 'induced Dirichlet' conditional posterior distribution
I am trying to implement the 'induced Dirichlet' prior model proposed by Michael Betancourt (from section 2.2 of his ordinal regression case study here: https://betanalpha.github.io/assets/...
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Dirichlet Mixture Model manually
I am implementing the Dirichlet Mixture Model using the EM algorithm in R, but am experiencing issues with the results. I generated two binomial distributions with fractions of (70%, 30%) and means of ...
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Cross validation on bootstrap data
I am performing a dirichlet model for different species using a small sample size (between 8 to 20 samples per each).
Since my dataset is small, I bootstrap my data with 1000 iterations, averaging 3 ...
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Combining Dirichlet and Gamma-Normal distributions
I have a model that describes 2 dimensional data where each data points is define as d = [category, x].
The category dimension can take 3 different values with respective probability $p_1$, $p_2$ and $...
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Dirichlet Process posterior with partially observed data
Suppose I dipose of a set of independant observed couples $(x_1,y_1),...,(x_N, y_N)$ from a joint distribution $P(x,y)$. Furthermore, I suppose that the random distribution $P$ as a Dirichlet prior
$P\...
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Marginal density of dirichlet distribution
I'm studying BRML.
In this book, a Dirichlet distribution is defined as
$$
p(\alpha | u) = \frac{\Gamma(\sum_{q=1}^Q u_q)}{\prod_{q=1}^Q \Gamma(u_q)} \delta_0 \left( \sum_{q=1}^{Q} \alpha_q - 1 \right)...
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Comparing two randomly loaded dice
Say I have two six-sided dice, A and B, which are loaded in different ways, and I'd like to compare their probability distributions.
So far I've constructed the priors for the probabilities $\vec\pi = ...
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compute Dirichlet distribution parameter from known mean distribution
For a particular Bayesian study I am going to apply Dirichlet distribution as my proposal random number generator. I am going to update the distribution parameter every trial based on a given ...
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Interpreting the quantities sampled from a Dirichlet distribution
Suppose you sample $M$ vectors from $Dirichlet_K(\alpha)$. You then show a histogram summarizing the distribution of the $M$ values that were sampled for dimension $k = 1$ (i.e. the first dimension, ...
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Why does latent dirichlet allocation (LDA) fail when dealing with large and heavy-tailed vocabularies?
I'm reading the 2019 paper Topic Modeling in Embedding Spaces which claims that the embedded topic model improves on these limitations of LDA. But why does LDA have these limitations—why does it fail ...
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Let $H$ be the base distribution of a Dirichlet process. How is this process well-defined in case $H(B_1) = 0$?
I have read that the parameters of Dirichlet distribution must be strictly positive.
The Dirichlet distribution of order $K \geq 2$ with parameters $\alpha_{1}, \ldots, \alpha_{K} \color{blue}{> 0}...
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Bayesian updates for Dirichlet-multinomial with Gamma prior
Let
$$
\begin{aligned}
X_i &\sim \text{Dir-multinom}(X\mid\lambda)\\
\lambda_{j} &\sim \text{Gamma}(\lambda_j\mid\alpha,\beta)\\
\end{aligned}
$$
where $i$ iterates over observations, $j$ ...
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Computing a prior from two components in Naive Bayes
Given a model parameter $\theta$ that is composed of two distributions in a Naive Bayes classifier, how is $P(\theta)$ typically computed in practice?
More specifically, from the article of Nigam et ...
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Mixtures of Dirichlet multivariates or Dirichlet processes
I am exploring the properties of Dirichlet distributions and their parameters. When mixing two Dirichlet distributed random bivariates
$$\mathbf{X}\equiv(X_1,X_2)\sim\text{Dir}(\alpha_1,\alpha_2)$$
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Dirichlet distribution parameters from known variances
Let's assume, I know the variances of Dirichlet distribution parameters. Let these variances be:
$Var[X_1], ..., Var[X_n]$.
Is there a analytical solution to derive the parameter value alpha_i given ...
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Sum of squares for a Dirichlet distribution
I have some data that takes the form of vectors $(a_0,...,a_n)$ lying on the simplex $\Sigma a_i = 1$ (all $a_i$'s non-negative). I have noticed that the maximum $\max_i a_i$ is very highly correlated ...
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Reparameterization trick for the Dirichlet distribution
Summary:
My aim is to create a (probabilistic) neural network for classification that learns the distribution of its class probabilities. The Dirichlet distribution seems to be choice. I am familiar ...
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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 ...
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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(\...
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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 =...
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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. ...
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Dirichlet-distribution and its correlation?
I have the following variables that follow a beta distribution:
...
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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_{...
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Approximating the Logit-Normal by Dirichlet
There is a known approximation of the Dirichlet Distribution by a Logit-Normal, as presented in wikipedia.
However, I am interested in the reverse, can I approximate a logit-normal by a Dirichlet?
I.e....
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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?
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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 ...
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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 ...
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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)$$
and we ...
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Log-likelihood of a finite mixture distribution (PDF overflowing)
I'm trying to use a finite mixture of Dirichlet distributions in a project, but am encountering problems with the PDF becoming so large for input values close to 0 that it overflows to infinity (as ...
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Upgrading weight parameters to random variable in Gaussian mixtures
In a Gaussian mixture model we model a density like:
$p(\mathbf{x}|\pi,\mu,\sigma)=\sum \pi_i N(\mathbf{x}|\mu_i,\sigma_i)$ [1]
where $\pi,\mu$ and $\sigma$ are parameters.
I would like to know if the ...
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Difficulties in computing the derivatives of the Dirichlet distribution
I need to compute the first derivatives of the Dirichlet distribution, defined in the following way:
$$r(P; \pi, \rho) = \frac{\Gamma(c)}{\prod_{i=1}^{k} \Gamma(c \pi_i)} \cdot \prod_{i=1}^{k} P_i^{c\...
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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 ...
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