October 11th NP Bayes Meetup

In today’s NP Bayes discussion group we returned to the 2006 Hierarchical Dirichlet Process (HDP) paper by Teh et al to discuss sampling-based inference. We spent most of our time sorting through the notational soup needed to specify the HDP variables and their relationships between one another. This led to a brief discussion of implementation issues, and finally a description of the three Gibbs sampling techniques presented in the paper.

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A Hierarchical Pitman-Yor Model of Natural Language

In the lab meeting on 9/17, we discussed the hierarchical, non-parametric Bayesian model for discrete sequence data presented in:

Wood, Archambeau, Gasthaus, James, & Teh,  A Stochastic Memoizer for Sequence Data.  ICML, 2009.

The authors extend previous work that used hierarchically linked Pitman-Yor processes to model the predictive distribution of a word given a context of finite length (an n-gram model), and here consider the distribution of words conditioned on a context of unbounded length (an \infty-gram model). The hierarchical structuring allows for the combination of information from contexts of different lengths, and the Pitman-Yor process allows for power-law distributions of words similar to those seen in natural language.  The authors develop the sequence memoizer and use coagulation and fragmentation operators to marginalize and reduce the computational complexity and create a collapsed graphical model on which inference is more efficient. The model is shown to perform well (i.e. have low perplexity) compared to existing models when applied to New York Times and Associated Press data.

NP Bayes reading group (9/27): hierarchical DPs

Our second NPB reading group meeting took aim at the seminal 2006 paper (with >1000 citations!) by Teh, Jordan, Beal & Blei on Hierarchical Dirichlet Processes. We were joined by newcomers Piyush Rai (newly arrived SSC postdoc), and Ph.D. students Dan Garrette (CS) and Liang Sun (mathematics), both of whom have experience with natural language models.

We established a few basic properties of the hierarchical DP, such as the the fact that it involves creating dependencies between DPs by endowing them with a common base measure, which is itself sampled from a DP. That is:

  • G_0 \sim DP(\gamma, H)     (“global measure” sampled from DP with base measure H and concentration \gamma).
  • G_j|\alpha_0,G_0 \sim DP(\alpha_0,G_0)  (sequence of conditionally independent random measures with common base measure G_0, e.g., G_j are distributions over clusters from data collected on different days)

Beyond this, we got bogged down in confusion over metaphors and interpretations, unclear whether G_j‘s were topics or documents or tables or restaurants or ethnicities, and were hampered by having two different version of the manuscript floating around with different page numbers and figures.
This week: we’ll take up where we left off, focusing on Section 4 (“Hierarchical Dicirhlet Processes”) with discussion led by Piyush.  We’ll agree to show up with the same (“official journal”) version of the manuscript, available: here.

Time: 4:00 PM, Thursday, Oct 4.
Location: SEA 5.106
Please email pillow AT mail.utexas.edu if you’d like to be added to the announcement list.

Revivifying the NP Bayes Reading Group

After a nearly 1-year hiatus, we’ve restarted our reading group on non-parametric (NP) Bayesian methods, focused on models for discrete data based on generalizations of the Dirichlet and other stick-breaking processes.

Thursday (9/20) was our first meeting, and Karin led a discussion of:

Teh, Y. W. (2006). A hierarchical Bayesian language model based on Pitman-Yor
processes. Proceedings of the 21st International Conference on
Computational Linguistics and the 44th annual meeting of the
Association for Computational Linguistics. 985-992

In the first meeting, we made it only as far as describing the Pitman-Yor (PY) process, a stochastic process whose samples are random probability distributions, and two methods for sampling from it:

  1. Chinese Restaurant sampling (aka “Blackwell-MacQueen urn scheme”), which directly provides samples \{X_i\} from distribution G \sim PY with G marginalized out.
  2. Stick-breaking, which samples the distribution G = \sum \pi_i \delta_{\phi_i} explicitly, using iid draws of Beta random variables to obtain stick weights \pi_i.

We briefly discussed the intuition for the hierarchical PY process, which uses PY process as base measure for PY process priors at deeper levels of the hierarchy (applied here to develop an n-gram model for natural language).

 
Next week: We’ve decided to go a bit further back in time to read:

Teh, Y. W.; Jordan, M. I.; Beal, M. J. & Blei, D. M. (2006). Hierarchical dirichlet processes. Journal of the American Statistical Association 101:1566-1581.

Time: Thursday (9/27), 4:00pm.
Location: Pillow lab
Presenter: Karin

note: if you’d like to be added to the email announcement list for this group, please send email to pillow AT mail.utexas.edu.

Using size-biased sampling for certain expectations

Let \{\pi_i\}_i be a well defined infinite discrete probability distribution (e.g., a draw from Dirichlet process (DP)). We are interested in evaluating the following form of expectations: E\left[ \sum_i f(\pi_i) \right] for some function f (we are especially interested when f = -\log, which gives us Shannon’s entropy). Following [1], we can re-write it as

E\left[ \sum_i \frac{f(\pi_i)}{\pi_i} \pi_i \right] = E\left[ E[ \frac{f(X)}{X} | \{\pi_i\}]\right]

where X is a random variable that takes the value \pi_i with probability \pi_i. This random variable X is better known as the first size-biased sample \tilde{\pi_1}. It is defined by \Pr[ \tilde \pi_1 = \pi_i | \{\pi_i\}_i] = \pi_i. In other words, it takes one of the probabilities \pi_i among \{\pi_i\}_i with probability \pi_i.

For Pitman-Yor process (PY) with discount parameter d and concentration parameter \alpha (Dirichlet process is a special case where d = 0), the size biased samples are naturally obtained by the stick breaking construction. Given a sequence of independent random variables V_n distributed as Beta(1-d, \alpha+n d), if we define \pi_i = \prod_{k=1}^{i-1} (1 - V_k) V_i, then the set of \{\pi_i\}_i is invariant to size biased permutation [2], and they form a sequence of size-biased samples. In our case, we only need the first size biased sample which is simply distributed as V_1.

Using this trick, we can compute the entropy of PY without the complicated simplex integrals. We used this and its extension for computing the PY based entropy estimator.

  1. Jim Pitman, Marc Yor. The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. The Annals of Probability, Vol. 25, No. 2. (April 1997), pp. 855-900, doi:10.1214/aop/1024404422
  2. Mihael Perman, Jim Pitman, Marc Yor. Size-biased sampling of Poisson point processes and excursions. Probability Theory and Related Fields, Vol. 92, No. 1. (21 March 1992), pp. 21-39, doi:10.1007/BF01205234

NP Bayes Reading Group: 6th meeting

This week, our discussion focused on estimating the hyperparameter for Dirichlet process models. We began by working through a couple of theorems in Antoniak (1974) for mixtures of Dirichlet processes. Importantly, it can be shown, in the language of Chinese restaurant processes, that the number of occupied tables and number of samples are sufficient to find a distribution over the Dirichlet hyperparameter.
Given a gamma prior for the hyperparameter, we worked through the derivation of a posterior distribution for the hyperparamter given the number of occupied tables and number of observations given by Escobar & West (1995). This results in an easily samplable mixture of two gamma distributions which can be added to the Gibbs sampling scheme we reviewed last week.

NP Bayes Reading Group: 5th meeting

This week we discussed how to apply a Dirichlet process-based method to real problems. We focused on the Infinite Gaussian Mixture Model and its uses in spike sorting (Wood & Black, 2008). In this model, the observed data come from an unknown (and potentially infinite) number of multivariate Gaussians. Our goal is to cluster the observations that come from the same Gaussian. This requires an MCMC approach. We chose to examine a collapsed Gibbs sampler which takes advantage of conjugate priors (the inverse Wishart for the multivariate normal). Combined with last week’s results on exchangeability, a Gibbs sweep merely needed to examine each observation given the current labeling of all other observations. The prior for the clustering was the given by the Chinese Restaurant Process and the likelihood became a multivariate Student-t. Next week, we will see how sample over the hyperparameter for the CRP.