Questions tagged [nonparametric-bayes]

Bayesian methods for infinite dimensional parameter spaces.

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48 views

How to interpret this plot?

Please give me an explanation based on this Non-linear estimation results from the aggregate model (HOSP). This figure shows the non-linear effect of age (AGE), education (EDU), family size (FAMSZ), ...
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Are the Beta and Bernoulli Processes Normalized Random Measures with Independent Increments?

James et al. 2009 introduced the notion of "Normalized Random Measures with Independent Increments." Do the Beta process and Bernoulli process belong to this family?
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what does z|G mean in nonparametric bayes?

when saying z is distributed according to G, where G comes from a dirichlet process, i saw this expression: z|G ~ G is this same meaning with z ~ G ?
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Online Stochastic Variational Inference for Dirichlet Process Mixture Models

There's a 2013 NeurIPS paper I'm trying to understand, Online Learning of Nonparametric Mixture Models via Sequential Variational Approximation. I have a few questions: Equation 2, which defines a ...
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34 views

Mean of draws of Dirichlet Process

Lets assume a Dirichlet process random measure in stick-breaking notation $G=\sum^\infty_{i=1} p_i \delta_{\lambda_i}$, such that $\lambda_i\sim H$ from some base distribution H, with point mass $\...
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55 views

Why nonparametric maximum likelihood of mixture is convex

Consider $x_i \sim N(\mu_i, 1)$ where $i = 1, \ldots, n$ and assume $\mu_i$ is generated i.i.d. from an unknown distribution $F$. We are interested in estimating the unknown $\mu_i$. One way to solve ...
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Reason for Poisson Distribution in Indian Buffet Process?

I have a two part question regarding the Indian Buffet Process. Suppose we consider $IBP(\alpha)$. In the IBP, the $n$th customer samples $\lambda_n \sim Poisson(\alpha/n)$ new dishes. Why does the ...
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32 views

Existence of Indian Buffet Dish Distribution?

The Chinese Restaurant Process has an associated distribution, the Chinese Restaurant Table Distribution, which describes the probability of the number of non-empty tables after N customers have been ...
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1answer
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Expected Number of Dishes in Indian Buffet Process?

I'm sure this question has an answer somewhere online, but I can't find it. Suppose I have an Indian Buffet Process with $T$ customers and concentration parameter $\alpha$. For those unfamiliar with ...
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114 views

Why does Chinese Restaurant Table Distribution look like a Gaussian Distribution?

The Chinese Restaurant Table Distribution describes the probability distribution for the number of non-empty tables in the Chinese Restaurant Process after $T$ customers have been seated. Specifically,...
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106 views

Chinese Restaurant Process: Expected cardinality (number of customers) of each block (table)?

Short version of the question: The Chinese Restaurant Process defines a distribution over partitions of $[T] := \{1, ...., T\}$. What is the expected cardinality of the $t$th block, where $t \in \{1, ....
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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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R fitted() function applied to rbart (Bayesian Additive Regression Trees w/ random effects) object in the dbarts package

I'm having a hard time figuring out what the output of fitted() applied to a rbart object means. Specifically, I fit my data ...
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Dirichlet Process vs Hierarchical Dirichlet Process: coupling among transitions on infinite HMM

I'm new to nonparametric Bayesian, and I am reading a paper about beam sampling for the infinite hidden Markov model. In the paper, it is mentioned that since there is no coupling among the ...
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Dirichlet process Posterior notation

While reading through Yee Whye Teh's tutorial on Dirichlet Process, I came across the Posterior distribution for DP. After observing some draws $\theta_1,\theta_2, ... ,$, the posterior is defined as $...
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Measure of base distribution $G_0(A)$ in the Dirichlet process

I am studying the Dirichlet Process and I came across this rule that states that given a partition $A_i's$ over the space $\Theta$, which represents the space in which $G_0$ is drawn from, $$(G(A_1),G(...
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1answer
38 views

What is the Dirichlet Proces Mixture Models posterior

I am trying to understand Dirichlet Process Mixture models. One of the videos I have been watching is by Tamara Broderick. I think it is a very good introductory video to Dirichlet Process mixture ...
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Why are discrete random measures not dominated?

A statement I’ve seen without proof in many books and papers is that random probability measures obtained by normalizing a completely random measure, such as the Dirichlet process, are not dominated ...
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362 views

Bayesian Nonparametric Latent feature model

For quite a long time I've been trying to understand the paper "Bayesian Nonparametric Latent feature model" (by Zoubin Ghahramani et al.) [http://mlg.eng.cam.ac.uk/zoubin/papers/GhaGriSol06.pdf]. In ...
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Exchangabilty in CRF generative model for HDP

In the HDP Setting, the groups (or documents) are assumed exchangeable between them, and the samples (or words) within each group/topic are also exchangeable. Note the Setting chapter here However, ...
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167 views

Kernel density estimate vs Dirichlet process mixture

Nowadays the Dirichlet process mixture (DPM) seems to be the default Bayesian approach for density estimation. My question is why not simply use the kernel density estimate (KDE) to model the density? ...
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118 views

Dirichlet process mixture modelling for a Gaussian likelihood

Let $\mathcal{Y} = (\mathbf{y}_1, \dots, \mathbf{y}_N)$ be data observed, such that each $\mathbf{y}_i \in \mathbb{R}^2$. Now conditional on unobserved cluster centres (means) $\mathcal{X} = (\mathbf{...
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852 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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Do Stochastic Processes such as the Gaussian Process/Dirichlet Process have densities? If not, how can Bayes rule be applied to them?

The Dirichlet Pocess and Gaussian Process are often referred to as "distributions over functions" or "distributions over distributions". In that case, can I meaningfully talk about the density of a ...
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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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Bayesian nonparametric estimate of median [closed]

I've been working on estimating the population median of a variable with a complex distribution that is not easily characterized as a parameterized probability distribution. So far, the best I have ...
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Bayesian Model Fit for Binary Data

I am looking for methods (and preferably references) for assessing model fit for Bayesian analyses of binary data. Specifically, I am fitting Bayesian parametric and nonparametric item response models ...
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Gaussian Process: vector valued response

Gaussian Processes (GPs) define a prior over functions that can be updated to a posterior once we have observed data. I've been working with scalar-valued GPs, i.e. functions $f: \mathbb{R}^{d} \...
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259 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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Help me understand the Bayesian kernel density estimation (Sibisi and Skilling, 1996)

Sibisi and Skilling (1996, also mentioned in the 1997 paper) define Bayesian kernel density as $$ f(x) = \int dx' \,\phi(x')\, K(x, x') \tag{2} $$ Here the kernel $K$ is an assigned smooth ...
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Normality test for likert scale [duplicate]

I have a questionnaire data that consist of composite Likert scale and discrete Likert items. Likert scales or variables are the sum of few Likert items while the one-item variables are represented ...
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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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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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432 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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414 views

About Hierarchical Latent Dirichlet allocation (hLDA)

I am reading Blei et al.'s paper (Hierarchical Topic Models and the Nested Chinese Restaurant Process) about hLDA. I am confused about the details of deriving the posterior of $p(\boldsymbol c_m| \...
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2answers
864 views

Chinese Restaurant process (CRP)

I am trying to understand the Chinese Restaurant process (CRP) and Weighted Chinese Restaurant process (WCRP) described in a research paper "Automatic Discovery of Cognitive Skills"- Robert V. Lindsey,...
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1answer
337 views

Question about implementing nested Chinese Restaurant Process (nCRP)

I am trying to follow the original paper on nCRP by Blei et al., 2010 and am confused with it's implementation. The authors describe the analogy for an nCRP as follows: A tourist arrives at the ...
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1answer
169 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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1answer
290 views

Proof of neutrality for dirichlet distribution

I am trying to learn the fields of bayesian non-parametric approaches. I am going thru this manuscript: http://mayagupta.org/publications/FrigyikKapilaGuptaIntroToDirichlet.pdf I am bit stuck with: ...
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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
134 views

resampling hyperparameters in a Hierarchical Dirichlet Process

The sampling scheme for the hyper-parameters of hierarchical dirichlet process (HDP) is explained in the appendix of the original paper by Teh et al. I agree that the auxiliary variable $s_j$ is a ...
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1answer
271 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
498 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 ...
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1answer
189 views

What does $\in$ mean vs $=$ in probability? What does $d\phi$ mean?

In the following lecture notes on Bayesian nonparametrics http://stat.columbia.edu/~porbanz/papers/porbanz_BNP_draft.pdf, I often see something like \begin{align} P[\Phi_{i}\in d\phi|...]\\ P[\Phi_{i}=...
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1answer
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Gaussian Process Regression for piecewise linear response functions

I am performing Gaussian Process Regression (without noise) for response functions which are piecewise linear. My question: Does there exist a covariance function, such that sample paths from a ...
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1answer
195 views

Marginalizing over a Chinese Restaurant Process prior

I am reading a paper by Kemp et al. and there is a part about marginalising over a Chinese Restaurant Process and I am quite clueless about how could one marginalise over such a prior! The details of ...
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Clustering methods for unknown number of clusters

Matrix $X=[x_1,...,x_i,...,x_N]$ is a data-set containing $N$ data-points that each data-point $x_i$ is a vector of $D$ dimensions. Each dimension is a feature. The number of clusters ($K$) is unknown....
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predictive distribution of linear bayesian regression with unknow $\Sigma$ and $\Omega$

The posterior distribution for weights in linear regression setup is \begin{equation} \begin{aligned} B &| Y,X \sim \mathcal{N}(\mu, \Lambda) \\ \mu &= \Lambda X^{\mathsf{T}}\Sigma^{-1}Y \\ \...
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1answer
106 views

Is it possible to define the mean of a varying distribution?

Suppose $(p_1,\ldots,p_k)$ be the vector of multinomial parameters and $$(p_1,\ldots,p_k)\sim \mbox{Dirichlet}(\alpha_1,\ldots,\alpha_k).$$ Let's define a function $f(p_1,\ldots,p_k) \in \mathbb{R}$. ...
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108 views

Proof that the Chinese restaurant process corresponds to Dirichlet process?

Let $(S, \mathcal{S})$ be a Polish space. Is there a nice proof of the fact that if the people are seated in a restaurant according to Chinese restaurant process, and then for each table, we sample a ...