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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Mapping two Dirichlet Distributions into a comparative Dirichlet
Assume I observe some draws from 2 choice options, and want to infer the probabilities of various outcomes, e.g. non-negative integers up to a limit L. I could simply use 2 Dirichlet distributions to ...
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What is a representation of positive numbers summing to one that can be sampled via HMC?
I have a probability density $f(x): \mathbb{R}^n \rightarrow \mathbb{R}$ whose argument vector $x$ satisfies the constraints that all elements are positive and sum to unity. I need to generate samples ...
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Zero-Inflated Dirichlet
I want to set up a model that will rely on something similar to a zero-inflated Dirichlet distribution. As such, I'm trying to figure out how a zero-inflated Dirichlet distribution is set up from the ...
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How to calculate the expectation of the following Dirichlet distribution and Beta distribution?
This is a question from my research, related to the derivation of the variational EM algorithm with mean-field assumption about LDA-based model.
We all know, given that $\boldsymbol{\theta} \sim \...
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How to derive the expectation of $\log[a \theta_k + b]$ in Dirichlet distribution?
Given that $\boldsymbol{\theta} \sim \mathrm{Dir}(\boldsymbol{\alpha})$, then $E_{p(\boldsymbol{\theta} \mid \boldsymbol{\alpha})}[\log{\theta_k}] = \Psi(\alpha_k) - \Psi(\sum_{k'=1}^K \alpha_{k'})$, ...
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Sampling quantities with a fixed sum ("string cutting"), but those quantities have to be discrete
I would like to sample 6 quantities that are guaranteed to add up to 600, each with a mean of 100. I want to be control the amount of variance around 100 (same variance for all 6 quantities, but need ...
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Dirichlet/multinomial dirichlet model with autocorrelation
I need to estimate an inferential statistical model of a variable that is a set of 8 proportions that sum to 1. The data repeat for 25 years and the series is an AR1 process. Is there a statistical ...
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Results dirichlet regression - brms vs DirichelReg comparison
I am new to Dirichlet regression, but I am trying to understand why model outputs are potentially different when I use two different R packages, and how I could interpret the slope and intercept ...
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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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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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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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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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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 ...