Regression models with MoPs Bayesian networks

We present a model of Bayesian network for continuous variables, where densities and conditional densities are estimated with B-spline MoPs. We use a novel approach to directly obtain conditional densities estimation using B-spline properties. In particular we implement naive Bayes and wrapper variables selection. Finally we apply our techniques to the problem of predicting neurons morphological variables from electrophysiological ones

Bayesian Analysis of Switching ARCH Models

We consider a time series model with autoregressive conditional heteroskedasticity that is subject to changes in regime. The regimes evolve according to a multistate latent Markov switching process with unknown transition probabilities, and it is the constant in the variance process of the innovations that is subject to regime shifts. The joint estimation of the latent process and all model parameters is performed within a Bayesian framework using the method of Markov Chain Monte Carlo simulation. One iteration of the sampler involves first a multi-move step to simulate the latent process out of its conditional distribution. The Gibbs sampler can then be used to simulate the parameters, in particular the transition probabilities, for which the full conditional posterior distribution is known. For most parameters, however, the full conditionals do not belong to any well-known family of distributions. The simulations are then based on the Metropolis-Hastings algorithm with carefully chosen proposal densities. We perform model selection with respect to the number of states and the number of autoregressive parameters in the variance process using Bayes factors and model likelihoods. To this aim, the model likelihood is estimated by combining the candidate's formula with importance sampling. The usefulness of the sampler is demonstrated by applying it to the dataset previously used by Hamilton and Susmel who investigated models with switching autoregressive conditional heteroskedasticity using maximum likelihood methods. The paper concludes with some issues related to maximum likelihood methods, to classical model selection, and to potential straightforward extensions of the model presented here.

Selection and Estimation for Mixed Graphical Models

We consider the problem of estimating the parameters in a pairwise graphical model in which the distribution of each node, conditioned on the others, may have a different parametric form. In particular, we assume that each node's conditional distribution is in the exponential family. We identify restrictions on the parameter space required for the existence of a well-defined joint density, and establish the consistency of the neighbourhood selection approach for graph reconstruction in high dimensions when the true underlying graph is sparse. Motivated by our theoretical results, we investigate the selection of edges between nodes whose conditional distributions take different parametric forms, and show that efficiency can be gained if edge estimates obtained from the regressions of particular nodes are used to reconstruct the graph. These results are illustrated with examples of Gaussian, Bernoulli, Poisson and exponential distributions. Our theoretical findings are corroborated by evidence from simulation studies

The distribution of a linear predictor after model selection: Unconditional finite-sample distributions and asymptotic approximations

We analyze the (unconditional) distribution of a linear predictor that is constructed after a data-driven model selection step in a linear regression model. First, we derive the exact finite-sample cumulative distribution function (cdf) of the linear predictor, and a simple approximation to this (complicated) cdf. We then analyze the large-sample limit behavior of these cdfs, in the fixed-parameter case and under local alternatives.Comment: Published at http://dx.doi.org/10.1214/074921706000000518 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Multiscale Dictionary Learning for Estimating Conditional Distributions

Nonparametric estimation of the conditional distribution of a response given high-dimensional features is a challenging problem. It is important to allow not only the mean but also the variance and shape of the response density to change flexibly with features, which are massive-dimensional. We propose a multiscale dictionary learning model, which expresses the conditional response density as a convex combination of dictionary densities, with the densities used and their weights dependent on the path through a tree decomposition of the feature space. A fast graph partitioning algorithm is applied to obtain the tree decomposition, with Bayesian methods then used to adaptively prune and average over different sub-trees in a soft probabilistic manner. The algorithm scales efficiently to approximately one million features. State of the art predictive performance is demonstrated for toy examples and two neuroscience applications including up to a million features

New Estimation Approaches for the Hierarchical Linear Ballistic Accumulator Model

The Linear Ballistic Accumulator (Brown & Heathcote, 2008) model is used as a measurement tool to answer questions about applied psychology. The analyses based on this model depend upon the model selected and its estimated parameters. Modern approaches use hierarchical Bayesian models and Markov chain Monte-Carlo (MCMC) methods to estimate the posterior distribution of the parameters. Although there are several approaches available for model selection, they are all based on the posterior samples produced via MCMC, which means that the model selection inference inherits the properties of the MCMC sampler. To improve on current approaches to LBA inference we propose two methods that are based on recent advances in particle MCMC methodology; they are qualitatively different from existing approaches as well as from each other. The first approach is particle Metropolis-within-Gibbs; the second approach is density tempered sequential Monte Carlo. Both new approaches provide very efficient sampling and can be applied to estimate the marginal likelihood, which provides Bayes factors for model selection. The first approach is usually faster. The second approach provides a direct estimate of the marginal likelihood, uses the first approach in its Markov move step and is very efficient to parallelize on high performance computers. The new methods are illustrated by applying them to simulated and real data, and through pseudo code. The code implementing the methods is freely available.Comment: 35 pages, 6 figures, 7 table

Conditional Density Estimation by Penalized Likelihood Model Selection and Applications

In this technical report, we consider conditional density estimation with a maximum likelihood approach. Under weak assumptions, we obtain a theoretical bound for a Kullback-Leibler type loss for a single model maximum likelihood estimate. We use a penalized model selection technique to select a best model within a collection. We give a general condition on penalty choice that leads to oracle type inequality for the resulting estimate. This construction is applied to two examples of partition-based conditional density models, models in which the conditional density depends only in a piecewise manner from the covariate. The first example relies on classical piecewise polynomial densities while the second uses Gaussian mixtures with varying mixing proportion but same mixture components. We show how this last case is related to an unsupervised segmentation application that has been the source of our motivation to this study.Comment: No. RR-7596 (2011