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Questions tagged [dirichlet-process]

A family of stochastic processes whose realizations are probability distributions

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1answer
75 views

Clarifying Dirichlet Process Mixture Probability Terms

Suppose I have a Dirichlet Process Mixture model defined as follows: $\alpha \sim G(a,b)\\ \pi|\alpha \sim \text{Dir}(\alpha)\\ z|\pi \sim \text{Cat}(\pi)\\ $ where $G$ is just a standard Gamma ...
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29 views

Dirichilet Process Mixture with dependent likelihood

For a Dirichlet Process Mixture model, is there a version where the conditional distribution of data is not independent? For example, with an autoregressive likelihood. $\theta_n \sim DP(\alpha, G_0)...
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88 views

Coding a simple Stick-Breaking Process in Python

I've just red the great 2012 blog post of Edwin Chen about Dirichlet Process with companion code in R and Ruby. Then I'm trying to translate the Stick-Breaking Process from R to Python. I've got this ...
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30 views

Relation between Chinese restaurant process (CRP) vs Dirichlet Process (DP)

There are multiple questions regarding the topic and but I have not found an answer to my doubts. Particularly: On Wikipedia Dirichlet Process page, regarding the connection between the Chinese ...
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40 views

Gibbs sampler for Dirichlet Process concentration parameter

I am trying to implement a Gibbs sampler for Hierarchical Dirichlet process, but I cannot seem to correctly estimate the concentration parameters. I therefore started testing just this part of a ...
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38 views

HDP: Gibbs sampler implementation

I am trying to recreate the model proposed by Gao et al. (2011), based on the Hierarchical Dirichlet Process proposed by Teh and al. (2005). To estimate the model (let's call it iHDP) I need to ...
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12 views

Layman's explanation for Beta Stacey Dirichlet Process

I would really appreciate it if someone would explain the Beta Stacey Dirichlet process and its application scenarios in layman's terms.
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62 views

Dirichlet-Multinomial Distribution with many zero counts

Short version: Is there someway to make the dirchlet-multinomial distribution sensitive to the presence of zero counts? Long version: I am attempting model metric positions in musical data. You can ...
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43 views

How to interpret graphical model for Dirichlet process mixture for variational inference?

I am working through this paper by Blei and Jordan, which introduces variational inference for Dirichlet process mixtures. They derive an evidence lower bound (ELBO) function based on a stick breaking ...
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71 views

How to select the best Baysian GMM with Dirichlet process

I am very new to GMM models and currently trying to make a unsupervised clustering on data. The data has 34 features (dimension). I am thinking of using GMM with Dirichlet process and trying to code ...
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49 views

How to actually draw a distribution from a given Dirichlet process?

I am wondering if there's an algorithm to actually draw a distribution from a given Dirichlet process. The closest thing I've came across is the stick-breaking construction of a Dirichlet process, but ...
3
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1answer
101 views

Understanding Dirichlet Process Mixtures

I have been reading a ton of stuff online and have a question about my understanding of Dirichlet Process Mixtures. First some basics on what I understand: Dirichlet Distribution: multivariate ...
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1answer
75 views

Explanation of Formal Definition of Dirichlet Process

I am reading about the Dirichlet process and I can understand the construction from Chinese restaurant process or stick-breaking process or Polya urn scheme. Now I am trying to understand why ...
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1answer
204 views

Stick-breaking construction of Dirichlet process

In the stick-breaking construction of Dirichlet (let me base things on Sethuraman's construction - slide 6 of this) do we sample one $\phi$ vector from the base distribution $H$ and use it for ...
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1answer
69 views

Why is the Dirichlet Process not a completely random measure?

A completely random measure assigns independent mass to nonintersecting subsets. I cannot figure, however, how the Dirichlet Process does not qualify as a CRM? Aren't the atoms all independent from ...
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64 views

Dirichlet process and clustering

How does clustering with a Dirichlet process as prior work? I am confused as to if the parameter $\alpha_i$ $\sim$ DP is found via clustering or is used to cluster. I undestrand how it can be used to ...
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39 views

Clustering and Dirichlet process' parameter

I am reading a paper in which they describe a bayesian model in which the prior $a_i$ is defined as a Dirichlet Process (DP). They say: "We use a DP to find the optimal $a_i$ via clustering". Later on ...
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24 views

Q about closed form of distribution

Suppose you have a K boxes, each with associated probability $p_i$. You start putting balls in the boxes with probability of putting a ball in a box being proportional to $(n_i+1)\cdot p_i$ where $n_i$...
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66 views

Applications of Hierarchical Dirichlet Process to Continuous Data

I read Yee Whye Teh et al.'s paper on Hierarchical Dirichlet Process. In section 5, they show sampling algorithm using base distribution H and data distribution F. One of their applications is HDP-...
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1answer
320 views

Predictive Density for Dirichlet Multinomial

I am wondering what the predictive distribution of a Dirichlet-Multinomial distribution is. In this tutorial (page 24), the predictive density is simple and something like "pseudo samples." However, ...
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219 views

Collapsed Gibbs sampler on Hierarchical Dirichlet Process Mixture Model

I am trying to design a collapsed Gibbs sampler on a mixture model based on Hierarchical Dirichlet Process ($g\sim DP(\gamma, b)$, $\pi\sim DP(\alpha, g)$ ). Should I resample from the posterior of ...
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1answer
141 views

In a Dirichlet process, can the base distribution be discrete?

Must it be continuous? Note we are talking about the base distribution. The sampled distribution is discrete. 1) If the base distribution is continuous, drawing from it will get a new value (a new ...
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1answer
34 views

functionals of Dirichlet process mixture

I want to solve (or simulate) /int f(x)G(dx) where f is a function(or phrase of x) and G is DPM(Dirichlet Process mixture prior). Muliere (1998) approximate this ...
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500 views

Dirichlet Process Posterior

I'm reading through a tutorial on the Dirichlet process (http://www.stats.ox.ac.uk/~teh/research/npbayes/Teh2010a.pdf) and have a small question. Given a draw from a Dirichlet process $G\sim\textsf{DP}...
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132 views

Dirichlet Process vs. Mixture Models with Many Mixtures

The Dirichlet Process prior is a Bayesian non-parametric prior to model your data as coming from an infinite mixture of distributions. Since your data is finite, only a finite number of these mixture ...
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30 views

DPMM asymptotics for finite mixtures?

My understanding is that using a Dirichlet process mixture model for a mixture with finitely many components will result in a misspecified model. Are there any asymptotic results or bounds on the ...
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255 views

Need a good book on Dirichlet Process

Was curious if there is any good book on Dirichlet Process (DP). I'm reading the DBA 3 by Andrew Gelman and I didn't understand it well enough. I was curious if there is a book that go into it a bit ...
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91 views

A Problem with Dirichlet process mixture model (DPMM)

I know that DPMM is a non-parametric Bayesian clustering model which can automatically determine the number of clusters based on data. However, which kind of clusters does the model find? For example,...
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284 views

Gibbs sampling in the Hierarchical Dirichlet Process

For an inference problem using a Dirichlet Process prior, one can derive a "basic" Gibbs sampling scheme, where we have a conditional for any parameter $\theta_i$ given the samples $x_i$ and all the ...
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281 views

Understanding Bayesian Histogram

I'm reading about Dirichlet Process Models in "Bayesian Data Analysis" by Gelman et. al. To motivate the idea, they start with a section on Bayesian histograms. I am a little confused about their ...
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1answer
622 views

What does the base distribution of the Dirichlet Process mean?

So far I only really understand the Dirichlet Process through its various metaphors. For the Polya Urn scheme, my understanding is that the "base distribution" is the original distribution of colors ...
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1answer
128 views

Nonparametric topic modeling: hierarchical dirichlet vs. Indian buffet?

The hierarchical dirichlet process (Teh 2005) allows you to discover unlimited topics to describe a document. An alternative process, the Indian Buffet process (Griffiths 2011) is another ...
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2answers
427 views

Posterior of parameter for Chinese Restaurant Process

I have a Chinese Restaurant Process with (unknown) concentration parameter $\theta$. After $n$ customers have been seated, I observe the number of non-empty tables and the number of people at each ...
2
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1answer
209 views

Check on intuition behind infinite mixture models for clustering

I'm trying to better understand the intuition and practical application of infinite mixture models (Dirichlet Process) and finite mixture models. For example, say I have a data set on which I run a ...
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1answer
54 views

Dirichlet Process with known mean

A Dirichlet process $\operatorname{DP}(\alpha,G_0)$ can be thought of as a distribution of distributions. I would like an object similar to a Dirichlet process, but which has support only on ...
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1answer
306 views

How can we convert values proportional to probabilities to Bernoulli probabilities?

According to Wikipedia, the parameter in a Bernoulli distribution should be $0<p<1$. I am reading this famous paper proposing Hierarchical Dirichlet Process, and on page 1580, A.6 and the ...
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63 views

Given a pdf which is a mixture of Gaussians, how do I infer the position (mean), variance, and number of Gaussians?

I have the following data, which when plotted as a histogram, are a mixture of Gaussians: I would like to write an algorithm that would infer: (1) the number of "peaks" or normal distributions in ...
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1answer
127 views

How does this Sampler work for the Concentration parameter of Dirichlet Process?

I am puzzled by how this Gibbs sampler on section 6 of Escobar & West (1995) works. To put it in simple words, the aim is to sample $\alpha$. The defined terms are: $$\eta\sim \texttt{Beta}(a,b)$$ ...
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86 views

Bayesian nonparametrics vs model selection using Minimum Message Length

As we know mixture models are important tools in density estimation and in general in statistical machine learning. I have always used nonparametric Bayesian mixture models to avoid the problem of ...
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1answer
140 views

Notation for base distribution in Dirichlet process

I have a (hopefully simple) question about notation when defining a DP. I have read a lot of papers on DPs, but this is not clear to me, or at least I have not noticed a convention. Say that I am ...
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1answer
135 views

Sampling Concentration Parameter of DP according to Escobar and West

I am reading Escobar&West paper and in particular am interested in their Gibbs sampler for the concentration parameter of Dirichlet Process (eq 13, eq 14). Given this, $$p(\alpha,\eta|k)\propto p(...
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1answer
97 views

Escobar and West Sampler for Dirichlet Process Parameters

I am reading Escobar&West paper and in particular am interested in their Gibbs sampler for the concentration parameter of Dirichlet Process. The issue I have is at the end of their section 6, ...
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60 views

Sampling concentration parameter of DP via Slice sampling?

Is there a published work which shows how sampling the Dirichlet Process's concentration parameter can be done via Slice sampling?
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1answer
104 views

How to Gibbs sample proportional to a probability

I am reading this tutorial on Hierarchical Chinese Restaurant Process. On pdf page 141 (slide title: MCMC Problem Specification for N-grams) it says: $$F(s_{1,k})=\frac{\alpha^{S'_1+s_{1,k}}}{(\alpha)...
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1answer
329 views

Gibbs sampling with constraints

I am reading tutorials on Gibbs sampling for partition sampling in Dirichlet Process (Chinese Restaurant Process), and have been struggling to understand the terminology used in the tutorials. To ...
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1answer
264 views

Probability mass function and Random measure

I have seen this in a few tutorial and papers on Dirichlet Process, where people refer to the probability mass function of stick-breaking-process as a ...
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53 views

Finding the posterior probability of a maximum value from a dirichlet distribution

I have a modeling problem that can be reconceptualized in a simpler way like so: Imagine I have an infinite urn with red, green, and blue balls in it with the proportions pr, pg, and pb (these ...
7
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1answer
176 views

Dirichlet process mixture MCMC

I'm reading Markov Chain Sampling Methods for Dirichlet Process Mixture Models by Radford M. Neal. Equation (3.6) states that $$ \text{If } c=c_{j} \text{ for some } j\neq i: P\left(c_{i}=c\;|\;c_{-i}...
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1answer
153 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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1answer
250 views

Chinese Restaurant Process

I want to implement Chinese Restaurant Process representation of Dirichlet Process for random partitions. The problem setup is as follows: I have some data (customers) which I have to randomly ...