Questions tagged [nonparametric-bayes]

Bayesian methods for infinite dimensional parameter spaces.

Filter by
Sorted by
Tagged with
25
votes
3answers
3k views

Is there a Bayesian approach to density estimation

I am interested to estimate the density of a continuous random variable $X$. One way of doing this that I learnt is the use of Kernel Density Estimation. But now I am interested in a Bayesian ...
15
votes
4answers
10k views

Gaussian Processes: How to use GPML for multi-dimensional output

Is there a way to perform Gaussian Process Regression on multidimensional output (possibly correlated) using GPML? In the demo script I could only find a 1D example. A similar question on CV that ...
11
votes
1answer
3k views

Covariance matrix for Gaussian Process and Wishart distribution

I'm reading through this paper on Generalised Wishart Processes (GWP). The paper calculates the covariances between different random variables (following Gaussian Process) using squared exponential ...
11
votes
0answers
490 views

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 ...
10
votes
2answers
2k views

Introductory textbook on nonparametric Bayesian models?

I'd like to wrap my head around this topic but learning from white-papers and tutorials is hard because there are many gaps which are usually filled in textbooks. If it is important I have ...
10
votes
2answers
6k views

PyMC for nonparametric clustering: Dirichlet process to estimate Gaussian mixture's parameters fails to cluster

Problem setup One of the first toy problems I wanted to apply PyMC to is nonparametric clustering: given some data, model it as a Gaussian mixture, and learn the number of clusters and each cluster's ...
10
votes
1answer
308 views

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 ...
10
votes
1answer
2k views

Putting a prior on the concentration parameter in a Dirichlet process

Most of this is background, skip to the end if you already know enough about Dirichlet process mixtures. Suppose I am modeling some data as coming from a mixture of Dirichlet processes, i.e. let $F \...
9
votes
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} &...
7
votes
3answers
1k views

Books for learning non parametric Bayesian model

Having studied parametric Bayesian statistics during the two last years, I plan to begin to self-study non parametric Bayesian model during this summer and look for recommendations. I would like the ...
7
votes
2answers
2k views

How adding covariance noise in Gaussian processes to prevent overfitting?

I am told, in Gaussian Processes, adding covariance function noise to others, say SEiso or Materns, cause a better result, since it prevents from over fitting. I appreciate if someone could put more ...
7
votes
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,...
6
votes
2answers
1k views

What are some applications of Chinese restaurant processes?

What are some applications of Chinese restaurant processes? I'm trying to learn a bit about non-parametric Bayesian methods, starting with Dirichlet processes and CRPs, but all the tutorials I've ...
6
votes
2answers
638 views

Understanding the construction of Dirichlet process

I'm trying to understand the construction process of DP, however, with little background in measure theory, the original papers are hard to read, but I believe the ideas behind these papers can be ...
5
votes
5answers
5k views

Simple introduction to MCMC with Dirichlet process prior?

I'm looking for a simple and easy to read introduction to using MCMC with a dirichlet process prior. Or perhaps using MCMC in any machine learning scenario, eg Gaussian Process. I've been circling ...
5
votes
1answer
5k views

How do I use the GPML package for multi dimensional input?

I have downloaded the Gaussian Processes for Machine Learning (GPML) package (gpml-matlab-v3.1-2010-09-27.zip) from the website, and I can run the regression example (demoRegression) in Octave. It ...
5
votes
2answers
4k views

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....
5
votes
1answer
2k views

understanding of effect of $\alpha$ in Dirichlet distribution

When reading the topic modeling tutorial written by Blei, KDD 2011 tutorial I was confused about a set of diagrams which aim to show the effect of $\alpha$ in Dirichlet distribution. For example, for ...
4
votes
1answer
4k views

stick breaking model of Dirichlet process

I have a question regarding sticking-breaking model of Dirichlet process, which is defined as follows: There are further statements that I am not clear that how to derive equation 1 from that ...
4
votes
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}=...
4
votes
1answer
1k views

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 ...
4
votes
1answer
113 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,...
4
votes
1answer
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{...
4
votes
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 ...
4
votes
1answer
898 views

Bayesian Updating for a Discrete Rating Value

I have an item for which I slowly collect rating values on a website. It is a movie item on a website and at the beginning it has no rating but I assign it a Gaussian prior $N(\mu_0, \sigma_0^2)$. A ...
4
votes
1answer
1k views

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} \...
4
votes
1answer
125 views

Learning parameters of non-parametric Bayesian models

I have a sample of Chinese restaurant process which I want to model as Pitman–Yor process. How do I determine parameters of Pitman-Yor model from given sample? For Dirichlet process I would just use ...
4
votes
1answer
165 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? ...
4
votes
1answer
1k views

Pitman-Yor processes in R or Python

I am looking for a good tool in R or Python or any other implementation that can help to me generate sampling from hierarchical Pitman-Yor processes (HPY) (one of the recent and popular nonparametric ...
3
votes
1answer
46 views

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 ...
3
votes
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 $f(p_1,\ldots,p_k) \in \mathbb{R}$. ...
3
votes
0answers
96 views

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 ...
3
votes
0answers
194 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 ...
3
votes
0answers
90 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 ...
3
votes
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 ...
3
votes
0answers
233 views

Estimating parameters of categorical distribution from sum-of-outcome data

Let $X$ be a categorical random variable with possible outcomes $o_1,...,o_n \subset [l, u]$ (real numbers with a known lower bound $l>0$ and known upper bound $u$) that occur with probability $p(X ...
3
votes
0answers
1k views

Non-parametric estimate of conditional expectation

I have a (fairly smooth) function $f$ and a sample $\{(x_i,y_i)\}_{i=1,\ldots,N}$ from the joint distribution of the random variables $X$ and $Y$. I would like to estimate the conditional expectation ...
3
votes
0answers
235 views

Default priors for mixtures

Suppose we have $p$ dimensional vectors $Y_i$ which we model with $f_Y (y |\theta) = \sum \pi_k N(y | \mu_k, \Sigma_k)$ with $\theta$ being a catch all for the model parameters (the number of ...
2
votes
1answer
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 ...
2
votes
1answer
24 views

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 ...
2
votes
1answer
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, ....
2
votes
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 ...
2
votes
1answer
288 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: ...
2
votes
0answers
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), ...
2
votes
0answers
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 ...
2
votes
0answers
53 views

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 ...
2
votes
0answers
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 ...
2
votes
0answers
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 ...
2
votes
0answers
32 views

Is there Bayesian search theory about search through the space of Bayesian models?

Consider we start with a specific Bayesian model, say an infinite mixture model with a Dirichlet Process as a prior. I know there are wildly many variants on this theme, from the Hierarchical ...
2
votes
0answers
158 views

Points to keep in mind while implementing a nonparametric bayesian inference procedure from scratch

I have been trying to implement a Bayesian inference procedure from scratch for a specific problem, but I have implemented the procedure, and it doesn't seem to work. Since, I can't just post the ...