The study of point cloud data sampled from a stratification, a collection of manifolds with possible different dimensions, is pursued in this paper. We present a technique for simultaneously soft clustering and estimating the mixed dimensionality and density of such structures. The framework is based on a maximum likelihood estimation of a Poisson mixture model. The presentation of the approach is completed with artificial and real examples demonstrating the importance of extending manifold learning to stratification learning.

Scoring structures of undirected graphical models by means of evaluating the marginal likelihood is very hard. The main reason is the presence of the parti- tion function which is intractable to evaluate, let alone integrate over. We propose to approximate the marginal likelihood by employing two levels of approximation: we assume normality of the posterior (the Laplace approximation) and approxi- mate all remaining intractable quantities using belief propagation and the linear response approximation. Em- pirically, we find that our procedure has about two orders of magnitude better accuracy than standard BIC methods for small datasets, but deteriorates when the size of the dataset grows.

The additive clustering model is widely used to infer the features of a set of stimuli from their similarities, on the assumption that similarity is a weighted linear function of common features. This paper develops a fully Bayesian formulation of the additive clustering model, using methods from nonparametric Bayesian statistics to allow the number of features to vary. We use this to explore several approaches to parameter estimation, showing that the nonparametric Bayesian approach provides a straightforward way to obtain estimates of both the number of features used in producing similarity judgments and their importance.

We present a new statistical framework called hidden Markov Dirichlet process (HMDP) to jointly model the genetic recombinations among possibly infinite number of founders and the coalescence-with-mutation events in the resulting genealogies. The HMDP posits that a haplotype of genetic markers is generated by a sequence of recombination events that select an ancestor for each locus from an unbounded set of founders according to a 1st-order Markov transition process. Conjoining this process with a mutation model, our method accommodates both between-lineage recombination and within-lineage sequence variations, and leads to a compact and natural interpretation of the population structure and inheritance process underlying haplotype data. We have developed an efficient sampling algo rithm for HMDP based on a two-level nested Polya urn scheme. On both simulated and real SNP haplotype data, our method performs competitively or significantly better than extant methods in uncovering the recombination hotspots along chromosomal loci; and in addition it also infers the ancestral genetic patterns and offers a highly accurate map of ancestral compositions of modern populations.

Computation of a satisfactory control policy for a Markov decision process when the parameters of the model are not exactly known is a problem encountered in many practical applications. The traditional robust approach is based on a worstcase analysis and may lead to an overly conservative policy. In this paper we consider the tradeoff between nominal performance and the worst case performance over all possible models. Based on parametric linear programming, we propose a method that computes the whole set of Pareto efficient policies in the performancerobustness plane when only the reward parameters are subject to uncertainty. In the more general case when the transition probabilities are also subject to error, we show that the strategy with the "optimal" tradeoff might be non-Markovian and hence is in general not tractable.

Although theorists have interpreted classical conditioning as a laboratory model of Bayesian belief updating, a recent reanalysis showed that the key features that theoretical models capture about learning are artifacts of averaging over subjects. Rather than learning smoothly to asymptote (reflecting, according to Bayesian models, the gradual tradeoff from prior to posterior as data accumulate), subjects learn suddenly and their predictions fluctuate perpetually. We suggest that abrupt and unstable learning can be modeled by assuming subjects are conducting in- ference using sequential Monte Carlo sampling with a small number of samples -- one, in our simulations. Ensemble behavior resembles exact Bayesian models since, as in particle filters, it averages over many samples. Further, the model is capable of exhibiting sophisticated behaviors like retrospective revaluation at the ensemble level, even given minimally sophisticated individuals that do not track uncertainty in their beliefs over trials.

A general modeling framework is proposed that unifies nonparametric-Bayesian models, topic-models and Bayesian networks. This class of infinite state Bayes nets (ISBN) can be viewed as directed networks of'hierarchical Dirichlet processes' (HDPs) where the domain of the variables can be structured (e.g. Existing models, such as nested-DP, Pachinko allocation, mixed membership sto- chastic block models as well as a number of new models are described as ISBNs. Two experiments have been performed to illustrate these ideas.

For infants, early word learning is a chicken-and-egg problem. One way to learn a word is to observe that it co-occurs with a particular referent across different situations. Another way is to use the social context of an utterance to infer the in- tended referent of a word. Here we present a Bayesian model of cross-situational word learning, and an extension of this model that also learns which social cues are relevant to determining reference. We test our model on a small corpus of mother-infant interaction and find it performs better than competing models. Fi- nally, we show that our model accounts for experimental phenomena including mutual exclusivity, fast-mapping, and generalization from social cues.

A method is proposed for semiparametric estimation where parametric and non- parametric criteria are exploited in density estimation and unsupervised learning. This is accomplished by making sampling assumptions on a dataset that smoothly interpolate between the extreme of independently distributed (or id) sample data (as in nonparametric kernel density estimators) to the extreme of independent identically distributed (or iid) sample data. This article makes independent simi- larly distributed (or isd) sampling assumptions and interpolates between these two using a scalar parameter. The parameter controls a Bhattacharyya affinity penalty between pairs of distributions on samples. Surprisingly, the isd method maintains certain consistency and unimodality properties akin to maximum likelihood esti- mation.

Point-based algorithms have been surprisingly successful in computing approx- imately optimal solutions for partially observable Markov decision processes (POMDPs) in high dimensional belief spaces. In this work, we seek to understand the belief-space properties that allow some POMDP problems to be approximated efficiently and thus help to explain the point-based algorithms' success often ob- served in the experiments. We show that an approximately optimal POMDP so- lution can be computed in time polynomial in the covering number of a reachable belief space, which is the subset of the belief space reachable from a given belief point. We also show that under the weaker condition of having a small covering number for an optimal reachable space, which is the subset of the belief space reachable under an optimal policy, computing an approximately optimal solution is NP-hard. However, given a suitable set of points that "cover" an optimal reach- able space well, an approximate solution can be computed in polynomial time.