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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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:
$$r(P; \pi, \rho) = \frac{\Gamma(c)}{\prod_{i=1}^{k} \Gamma(c \pi_i)} \cdot \prod_{i=1}^{k} P_i^{c\...
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Question about the distribution of the average of Dirichlet-distributed random variables
Suppose that each in a set of $n$ random variables $\boldsymbol{X}_1, .., \boldsymbol{X}_n$ are Dirichlet-distributed with parameters $\boldsymbol{\alpha}_i$, where $i$ is an index for the random ...
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Choosing the Dirichlet prior in a mixture model
Consider the following mixture model with $K < \infty$ components,
$$
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_{...
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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=...
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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
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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 ...
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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 ...
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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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