Learning Mixtures of Truncated Basis Functions from Data

In this paper we investigate methods for learning hybrid Bayesian networks from data. First we utilize a kernel density estimate of the data in order to translate the data into a mixture of truncated basis functions (MoTBF) representation using a convex optimization technique. When utilizing a kernel density representation of the data, the estimation method relies on the specification of a kernel bandwidth. We show that in most cases the method is robust wrt. the choice of bandwidth, but for certain data sets the bandwidth has a strong impact on the result. Based on this observation, we propose an alternative learning method that relies on the cumulative distribution function of the data. Empirical results demonstrate the usefulness of the approaches: Even though the methods produce estimators that are slightly poorer than the state of the art (in terms of log-likelihood), they are significantly faster, and therefore indicate that the MoTBF framework can be used for inference and learning in reasonably sized domains. Furthermore, we show how a particular subclass of MoTBF potentials (learnable by the proposed methods) can be exploited to significantly reduce complexity during inference

Learning Conditional Distributions using Mixtures of Truncated Basis Functions

Mixtures of Truncated Basis Functions (MoTBFs) have recently been proposed for modelling univariate and joint distributions in hybrid Bayesian networks. In this paper we analyse the problem of learning conditional MoTBF distributions from data. Our approach utilizes a new technique for learning joint MoTBF densities, then propose a method for using these to generate the conditional distributions. The main contribution of this work is conveyed through an empirical investigation into the properties of the new learning procedure, where we also compare the merits of our approach to those obtained by other proposals

Hybrid Bayesian Networks Using Mixtures of Truncated Basis Functions

This paper introduces MoTBFs, an R package for manipulating mixtures of truncated basis functions. This class of functions allows the representation of joint probability distributions involving discrete and continuous variables simultaneously, and includes mixtures of truncated exponentials and mixtures of polynomials as special cases. The package implements functions for learning the parameters of univariate, multivariate, and conditional distributions, and provides support for parameter learning in Bayesian networks with both discrete and continuous variables. Probabilistic inference using forward sampling is also implemented. Part of the functionality of the MoTBFs package relies on the bnlearn package, which includes functions for learning the structure of a Bayesian network from a data set. Leveraging this functionality, the MoTBFs package supports learning of MoTBF-based Bayesian networks over hybrid domains. We give a brief introduction to the methodological context and algorithms implemented in the package. An extensive illustrative example is used to describe the package, its functionality, and its usage

Learning Mixtures of Polynomials of Conditional Densities from Data

Mixtures of polynomials (MoPs) are a non-parametric density estimation technique for hybrid Bayesian networks with continuous and discrete variables. We propose two methods for learning MoP approximations of conditional densities from data. Both approaches are based on learning MoP approximations of the joint density and the marginal density of the conditioning variables, but they differ as to how the MoP approximation of the quotient of the two densities is found. We illustrate the methods using data sampled from a simple Gaussian Bayesian network. We study and compare the performance of these methods with the approach for learning mixtures of truncated basis functions from data

Bose-Fermi mixtures in 1D optical superlattices

The zero temperature phase diagram of binary boson-fermion mixtures in two-colour superlattices is investigated. The eigenvalue problem associated with the Bose-Fermi-Hubbard Hamiltonian is solved using an exact numerical diagonalization technique, supplemented by an adaptive basis truncation scheme. The physically motivated basis truncation allows to access larger systems in a fully controlled and very flexible framework. Several experimentally relevant observables, such as the matter-wave interference pattern and the condensatefraction, are investigated in order to explore the rich phase diagram. At symmetric half filling a phase similar to the Mott-insulating phase in a commensurate purely bosonic system is identified and an analogy to recent experiments is pointed out. Furthermore a phase of complete localization of the bosonic species generated by the repulsive boson-fermion interaction is identified. These localized condensates are of a different nature than the genuine Bose-Einstein condensates in optical lattices.Comment: 18 pages, 9 figure

Conditional density approximations with mixtures of polynomials

Mixtures of polynomials (MoPs) are a non-parametric density estimation technique especially designed for hybrid Bayesian networks with continuous and discrete variables. Algorithms to learn one- and multi-dimensional (marginal) MoPs from data have recently been proposed. In this paper we introduce two methods for learning MoP approximations of conditional densities from data. Both approaches are based on learning MoP approximations of the joint density and the marginal density of the conditioning variables, but they differ as to how the MoP approximation of the quotient of the two densities is found. We illustrate and study the methods using data sampled from known parametric distributions, and we demonstrate their applicability by learning models based on real neuroscience data. Finally, we compare the performance of the proposed methods with an approach for learning mixtures of truncated basis functions (MoTBFs). The empirical results show that the proposed methods generally yield models that are comparable to or significantly better than those found using the MoTBF-based method