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

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Generate a random set of numbers with fixed sum and desired means and variances?

The Dirichlet distribution allows you to generate a sample of numbers $x_i$ with a prescribed sum, say $\sum_i x_i = 1$. Moreover, the parameters $\alpha$ allow some degree of control on the means of ...
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Estimator for E[A]/E[B]^2

If I have 2 random variables $A$ and $B$, with unknown means (denoted $\mu_A$ and $\mu_B$), and I want to estimate $\mu_A/\mu_B^2$ from $n$ samples of each, $\frac{\sum A_i/n}{(\sum B_i/n)^2}$ is ...
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19 views

What is the mode of Dirichlet-Multinomial (Polya) distribution?

What is the ML estimate of the parameter $e_i$ for the Dirichlet-Multinomial (Polya) distribution defined below? $p(\mathbf{x}|\mathbf{e}) = \frac{N!}{\prod_i^d x_i!}\frac{\Gamma(A)}{\Gamma(N+A)}\...
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Bayesian A/B/C testing

I built a Bayesian A/B testing tool - i.e. one which models A and B having posteriors $Beta(\alpha_i, \beta_i)$ where $\alpha_i, \beta_i$ are updated every iteration. After T iterations, I compare ...
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Multivariate marginal Dirichlet distribution

For a vector $X = (x_1, \dots, x_m)$, let $\mathcal{C}(X) = \frac{1}{x_1+\dots+x_m}(x_1, \dots, x_m)$. If $(X_a, X_b)$ follows a Dirichlet distribution with parameters $(\alpha_{a_1}, \dots, \alpha_{...
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Interpreting the entropy of a Dirichlet distribution

I was looking for a measure to interpret the "spikiness" of categorical histograms. So, if it becomes unnaturally skewed towards a certain value at a given time, I want a metric that will show some ...
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what is this property? $\int p(x,\pi)d\pi=p(x|E[\pi])$?

Sorry if the title does not make sense, from the answer of this question Mistake in derivation about categorical distribution and Dirichlet distribution? it can be shown that say $p(x|\pi)$ follows ...
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Mistake in derivation about categorical distribution and Dirichlet distribution?

$p(x|\pi)$ follows the categorical distribution (the multinomial with one observation), where $\sum\pi_i=1$ and $x$ is a one-hot vector, and $p(\pi|\alpha)$ follows the Dirichlet distribution. $p(x|\...
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Characterize linear transformation of Dirichlet distributed variables

Let $X=(X_1,....,X_K)\sim{}\text{Dir}(\alpha_1,...,\alpha_K)$ be a Dirichlet distribution with parameters $\alpha_1,...,\alpha_K$. Let $A$ be a non-singular linear map and $(Y_1,....,Y_K)=A(X_1,....,...
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How to sample weights for weighted kernels?

I'm using a SVM classifier with a weighted RBF kernel. My dataset has 17 features. In the RBF kernel I will use a weight for each feature. Of course the weights must sum to one. For choosing the best ...
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how does the beta function make the total probability of Dirichlet distribution integrate to 1? [duplicate]

the Dirichlet distribution $$Dir(\boldsymbol\mu|\boldsymbol\alpha)=\frac{1}{B(\boldsymbol\alpha)}\prod_K\mu_k^{\alpha_k-1}$$ How is the normalization term $B(\boldsymbol\alpha)$ calculated? or in ...
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Figuring out distribution parameters for specified modes

I'm trying to find out, which parameters $\alpha_i$ should Dirichlet distribution have in order to have as modes some specified $x_i$. The formula for mode for Dirichlet distribution is this (from ...
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42 views

How to deduct the coefficient of the Dirichlet distribution?

I am studying the textbook Introduction to Probability Models by Sheldon Ross. In Section 3.6 page 151 (10th edition), the author uses the Dirichlet distribution to deduct the coefficient of a "...
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23 views

Statistical test to compare Dirichlet distributions

I want assess if two groups are statistically significantly different. I assume the data points in both groups are generated from a Dirichlet distribution - is there an appropriate test for this? ...
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32 views

What does it mean to integrate over a random measure?

I'm currently looking at a paper of Dirichlet process random effects model and the model specification is as follows: $$ \begin{align*}y_{i} &= X_{i}\beta + \psi_{i} + \epsilon_{i}\\ \psi_{i} &...
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Prior estimation for Dirichlet Process Clustering

I wrote this code for Dirichlet Process Clustering using Chinese Restaurant Process in which a parameter ...
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35 views

Dirichlet Process Clustering Prior

I'm doing dirichlet process clustering where dirichlet priors are used as: with CRP representation as: First customer will always choose first table. Second will choose already occupied table with ...
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Converting Dirichlet distribution to distribution on the log-linear parameters

Dirichlet prior/posterior provides a probability density on distributions over a multinomial variable. It has the form : $P(P) \varpropto \prod_i{P_i^{\alpha_i-1}}$ I can also describe the ...
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Dirichlet group assignment

As in the Dirichlet clustering, the dirichlet process can be represented by the following: Chinese Restaurant Process Stick Breaking Process Poly Urn Model For instance, if we consider ...
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Latent Dirichlet allocation: how to derive $\theta$ given $\alpha$?

I have been studying Latent Dirichlet allocation (LDA) since quite long. I have a confusion in Dirichlet priors. For example, if I consider 3 topics and take $\alpha_1 = 0.1$, $\alpha_2 = 0.1$ and $\...
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1answer
41 views

which classifier to choose for probability histogram-like features

I have a populations of 500 elements. Each element is represented by a 10 dimension feature vector which sum of element is equal to 1 (you can think about it as a histogram of probabilities). In ...
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Dirichlet multivariable regression with temporal autocorrelation?

BACKGROUND: I have some animal behaviour data. The time allocated by a group of animals to different behaviours per minute was recorded repeatedly until the end of the experiment. Therefore, I have ...
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How can we plot realizations form a posterior distribution of a Dirichlet Process? What bin size to use?

In a Dirichlet Process, where $P \sim DP(\alpha P_0)$ and $y_i \sim P$ are iid, the posterior distribution of a Dirichlet Process $P$ is usually given as: $$ P \mid y_1, \ldots y_n \sim DP\left(\...
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Is there any kernel defined for simplex domain (i.e. probability vector)?

I am wondering if there is any kernel function that is specifically designed for simplex domain. By "simplex domain," I mean a set whose elements are probability vectors. For example, 3-D simplex may ...
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Truncated Dirichlet process vs Dirichlet distribution

As the topic suggests I was wondering what the main differences are in using one over the other. Suppose for sake of simplicity the Dirichlet distribution has all parameters set to $\alpha$. All I ...
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Dirichlet conjugate update derivation

I am attempting to derive the update equations for the conjugate to the Dirichlet distribution, as outlined here: http://mathoverflow.net/questions/20399/conjugate-prior-of-the-dirichlet-distribution ...
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Dirichlet Prior for Multinomial

The Dirichlet function is the conjugate prior of the multinomial. So the posterior is also Dirichlet given some observations. If e.g. I observe the counts $X=(10,3,4)$ from 17 trials (10 for class 1, ...
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Jeffrey's Prior for Dirichlet

As I understand, Jeffreys' Prior for any distribution is an objective one, and in the wiki, the Jeffreys' prior is written to be 0.5. On the other hand, I've found these notes which state that the ...
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33 views

Naive Bayes where Feature Space is LDA Output

I understand that the output for Latent Dirichlet Allocation is a distribution over K topics. Suppose I have a Dx(K+1) matrix, where rows are documents and columns are the topic distribution + one ...
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63 views

Probability of account win/loss using Bayesian Statistics

I am trying to estimate the probability of winning or losing an account, and I'd like to do this using Bayesian Methods. I'm not really that familiar with these methods, but I think I understand the ...
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73 views

Dirichlet Distribution with constraints

This is a follow up to an earlier question that I asked here: How should I measure my confidence in this estimate of this finite distribution In that question I asked how I could estimate a finite ...
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How to fit a Dirichlet Process prior?

I currently am working with the galaxies dataset in the R package "DPpackage". I am trying to assume that each of my datapoints $y_i \sim P$ with $P \sim DP(\alpha P_0)$, which denotes a Dirichlet ...
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Modelling distributions evolving with time

Say we have N probability vectors of dimensionality D describing how a process evolves in N time steps, something like multidimensional time series. What's the best way to model this sequence in ...
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Why doesn't PDF of Dirichlet Distribution seem to integrate to 1?

I have been trying to find the expected value of a function of a random variable with a Dirichlet distribution by integrating its product with the Dirichlet density function over a simplex in R. To ...
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Marginal probability function of the Dirichlet-Multinomial distribution

I can't seem to find a written out derivation for the marginal probability function of the compound Dirichlet-Multinomial distribution, though the mean and variance/covariance of the margins seem to ...
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What is the mode of the Dirichlet distribution when some $\alpha_i < 1$

Suppose $X \sim \mathcal D(\alpha_1, \ldots, \alpha_p)$, and suppose $\alpha_i < 1$ for some $\alpha_i$. In this case, the density is unbounded, and so no proper "mode" can exist. On the other ...
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Generate data from Dirichlet distribution? [closed]

I want to generate Dirichlet distributed data, but I don't know how to do it. Could you please help me?
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Polya Urn - formulas

I am trying to understand Dirichlet processes and Polya's Urn, following this excellent article. One thing that I am struggling with, is to understand in the example below what why the number of ...
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dirichlet process - formula explanatation

i am self-studying the Dirichlet process, from among others http://www.stat.ubc.ca/~bouchard/courses/stat547-sp2011/notes-part2.pdf, and struggling with the following basic question: I understand ...
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About LDA model, I need a true expert to tell me that what is the real benefits of the Dirichlet prior? [closed]

Well,you know ,the only difference between pLSI and LDA is that the latter has a Dirichlet prior,thus the number of model parameters do not increase with the size of corpus,and this avoid the ...
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Where to find Wikipedia's wiki.dat for LDA, an example used in Vowpal Wabbit? [closed]

I am following Vowpal Wabbit's tutorial on LDA (Latent Dirichlet Allocation) found here. wiki.dat is used in it as a Wikipedia corpus or supplementary Wikipedia dictionaries, maybe. It must be ...
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Distribution of the largest fragment of a broken stick (spacings)

Let a stick of length 1 be broken in $k+1$ fragments uniformly at random. What is the distribution of the length of the longest fragment? More formally, let $(U_1, \ldots U_k)$ be IID $U(0,1)$, and ...
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Gaussian mixture models and Dirichlet Processes - choosing alpha

Can someone point me to a good empirical method or rule of thumb for choosing alpha in a Dirichlet Processe mixture? (Alpha as outlined here: http://scikit-learn.org/stable/modules/mixture.html#...
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Simplification of the marginalized multinomial distribution

The likelihood expression for multinomial count variables $\{n_1, \dots, n_K\}, \sum_{k=1}^K {n_k} = N$, conditionned by parameters $\{\pi_1, \dots, \pi_K\}, \sum_{k=1}^K {\pi_k} = 1$ is given as: \...
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Uniform Sampling from Intersections of Faces of Simplices

I'm trying to sample uniformly on the intersections of faces of several simplicies, with all coordinates being non-negative. That is, given constraints $$A\vec{w}=\vec{b} \ \ and \ \ \vec{w} \geq \vec{...
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Dirichlet predictive distribution

I need to derive a posterior predictive density for observation T+1 given that the posterior distribution of $(x_{1}, x_{2}, 1-x_{1}-x_{2})$ is a Dirichlet distribution with parameter $(\alpha_{1}, \...
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What is the median of the Dirichlet Distribution?

The Dirichlet is a multivariate generalisation of the Beta distribution, and Wikipedia states that the median of the Beta distribution is the inverse of the incomplete Beta function evaluated at $\...
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How to sample from a distribution while the sample point simultaneously needs to satisfy an inequality constraint?

I have a Dirichlet distribution $P(x)$ , where $x$ is a point on a simplex. It is very easy to sample from this Dirichlet distribution. But I have a set of non-linear, non-convex constraints $f_i(x)&...
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Multinomial distribution - where is the normalising constant?

I've been reading up on Multinomial/Dirichlet priors and came across this note. I'm wondering why the normalising constant for the multinomial distribution drops out in the derivation of the joint ...