The popular Lasso approach for sparse estimation can be derived via marginalization of a joint density associated with a particular stochastic model. A different marginalization of the same probabilistic model leads to a different non-convex estimator where hyperparameters are optimized. Extending these arguments to problems where groups of variables have to be estimated, we study a computational scheme for sparse estimation that differs from the Group Lasso. Although the underlying optimization problem defining this estimator is non-convex, an initialization strategy based on a univariate Bayesian forward selection scheme is presented. This also allows us to define an effective non-convex estimator where only one scalar variable is involved in the optimization process. Theoretical arguments, independent of the correctness of the priors entering the sparse model, are included to clarify the advantages of this non-convex technique in comparison with other convex estimators. Numerical experiments are also used to compare the performance of these approaches.

When modeling a probability distribution with a Bayesian network, we are faced with the problem of how to handle continuous variables. Most previous work has either solved the problem by discretizing, or assumed that the data are generated by a single Gaussian. In this paper we abandon the normality assumption and instead use statistical methods for nonparametric density estimation. For a naive Bayesian classifier, we present experimental results on a variety of natural and artificial domains, comparing two methods of density estimation: assuming normality and modeling each conditional distribution with a single Gaussian; and using nonparametric kernel density estimation. We observe large reductions in error on several natural and artificial data sets, which suggests that kernel estimation is a useful tool for learning Bayesian models.

Dawid, Kjaerulff and Lauritzen (1994) provided a preliminary description of a hybrid between Monte-Carlo sampling methods and exact local computations in junction trees. Utilizing the strengths of both methods, such hybrid inference methods has the potential of expanding the class of problems which can be solved under bounded resources as well as solving problems which otherwise resist exact solutions. The paper provides a detailed description of a particular instance of such a hybrid scheme; namely, combination of exact inference and Gibbs sampling in discrete Bayesian networks. We argue that this combination calls for an extension of the usual message passing scheme of ordinary junction trees.

Stochastic simulation algorithms such as likelihood weighting often give fast, accurate approximations to posterior probabilities in probabilistic networks, and are the methods of choice for very large networks. Unfortunately, the special characteristics of dynamic probabilistic networks (DPNs), which are used to represent stochastic temporal processes, mean that standard simulation algorithms perform very poorly. In essence, the simulation trials diverge further and further from reality as the process is observed over time. In this paper, we present simulation algorithms that use the evidence observed at each time step to push the set of trials back towards reality. The first algorithm, "evidence reversal" (ER) restructures each time slice of the DPN so that the evidence nodes for the slice become ancestors of the state variables. The second algorithm, called "survival of the fittest" sampling (SOF), "repopulates" the set of trials at each time step using a stochastic reproduction rate weighted by the likelihood of the evidence according to each trial. We compare the performance of each algorithm with likelihood weighting on the original network, and also investigate the benefits of combining the ER and SOF methods. The ER/SOF combination appears to maintain bounded error independent of the number of time steps in the simulation.

In this letter, we derive the optimal discriminant functions for modulation classification based on the sampled distribution distance. The proposed method classifies various candidate constellations using a low complexity approach based on the distribution distance at specific testpoints along the cumulative distribution function. This method, based on the Bayesian decision criteria, asymptotically provides the minimum classification error possible given a set of testpoints. Testpoint locations are also optimized to improve classification performance. The method provides significant gains over existing approaches that also use the distribution of the signal features.

Copulas allow to learn marginal distributions separately from the multivariate dependence structure (copula) that links them together into a density function. Vine factorizations ease the learning of high-dimensional copulas by constructing a hierarchy of conditional bivariate copulas. However, to simplify inference, it is common to assume that each of these conditional bivariate copulas is independent from its conditioning variables. In this paper, we relax this assumption by discovering the latent functions that specify the shape of a conditional copula given its conditioning variables. We learn these functions by following a Bayesian approach based on sparse Gaussian processes with expectation propagation for scalable, approximate inference. Experiments on real-world datasets show that, when modeling all conditional dependencies, we obtain better estimates of the underlying copula of the data.

The classical mixture of Gaussians model is related to K-means via small-variance asymptotics: as the covariances of the Gaussians tend to zero, the negative log-likelihood of the mixture of Gaussians model approaches the K-means objective, and the EM algorithm approaches the K-means algorithm. Kulis & Jordan (2012) used this observation to obtain a novel K-means-like algorithm from a Gibbs sampler for the Dirichlet process (DP) mixture. We instead consider applying small-variance asymptotics directly to the posterior in Bayesian nonparametric models. This framework is independent of any specific Bayesian inference algorithm, and it has the major advantage that it generalizes immediately to a range of models beyond the DP mixture. To illustrate, we apply our framework to the feature learning setting, where the beta process and Indian buffet process provide an appropriate Bayesian nonparametric prior. We obtain a novel objective function that goes beyond clustering to learn (and penalize new) groupings for which we relax the mutual exclusivity and exhaustivity assumptions of clustering. We demonstrate several other algorithms, all of which are scalable and simple to implement. Empirical results demonstrate the benefits of the new framework.

By developing data augmentation methods unique to the negative binomial (NB) distribution, we unite seemingly disjoint count and mixture models under the NB process framework. We develop fundamental properties of the models and derive efficient Gibbs sampling inference. We show that the gamma-NB process can be reduced to the hierarchical Dirichlet process with normalization, highlighting its unique theoretical, structural and computational advantages. A variety of NB processes with distinct sharing mechanisms are constructed and applied to topic modeling, with connections to existing algorithms, showing the importance of inferring both the NB dispersion and probability parameters.

This paper works through the optimization of a real world planning problem, with a combination of a generative planning tool and an influence diagram solver. The problem is taken from an existing application in the domain of oil spill emergency response. The planning agent manages constraints that order sets of feasible equipment employment actions. This is mapped at an intermediate level of abstraction onto an influence diagram. In addition, the planner can apply a surveillance operator that determines observability of the state---the unknown trajectory of the oil. The uncertain world state and the objective function properties are part of the influence diagram structure, but not represented in the planning agent domain. By exploiting this structure under the constraints generated by the planning agent, the influence diagram solution complexity simplifies considerably, and an optimum solution to the employment problem based on the objective function is found. Finding this optimum is equivalent to the simultaneous evaluation of a range of plans. This result is an example of bounded optimality, within the limitations of this hybrid generative planner and influence diagram architecture.

Directed acyclic graphs have been used fruitfully to represent causal strucures (Pearl 1988). However, in the social sciences and elsewhere models are often used which correspond both causally and statistically to directed graphs with directed cycles (Spirtes 1995). Pearl (1993) discussed predicting the effects of intervention in models of this kind, so-called linear non-recursive structural equation models. This raises the question of whether it is possible to make inferences about causal structure with cycles, form sample data. In particular do there exist general, informative, feasible and reliable precedures for inferring causal structure from conditional independence relations among variables in a sample generated by an unknown causal structure? In this paper I present a discovery algorithm that is correct in the large sample limit, given commonly (but often implicitly) made plausible assumptions, and which provides information about the existence or non-existence of causal pathways from one variable to another. The algorithm is polynomial on sparse graphs.