Copula-type Estimators for Flexible Multivariate Density Modeling using Mixtures

Copulas are popular as models for multivariate dependence because they allow the marginal densities and the joint dependence to be modeled separately. However, they usually require that the transformation from uniform marginals to the marginals of the joint dependence structure is known. This can only be done for a restricted set of copulas, e.g. a normal copula. Our article introduces copula-type estimators for flexible multivariate density estimation which also allow the marginal densities to be modeled separately from the joint dependence, as in copula modeling, but overcomes the lack of flexibility of most popular copula estimators. An iterative scheme is proposed for estimating copula-type estimators and its usefulness is demonstrated through simulation and real examples. The joint dependence is is modeled by mixture of normals and mixture of normals factor analyzers models, and mixture of t and mixture of t factor analyzers models. We develop efficient Variational Bayes algorithms for fitting these in which model selection is performed automatically. Based on these mixture models, we construct four classes of copula-type densities which are far more flexible than current popular copula densities, and outperform them in simulation and several real data sets.Comment: 27 pages, 3 figure

Cumulative Distribution Functions As The Foundation For Probabilistic Models

This thesis discusses applications of probabilistic and connectionist models for constructing and training cumulative distribution functions (CDFs). First, it is shown how existing tools from the copula literature can be combined to build probabilistic models. It is found that this simple construction leads to numerical and scalability issues that make training and inference challenging. Next, several innovative ideas, combining neural networks, automatic differentiation and copula functions, introduce how to assemble black-box probabilistic models. The basic building block is a cumulative distribution function that is straightforward to construct, composed of arithmetic operations and nonlinear functions. There is no need to assume any specific parametric probability density function (PDF), making the model flexible and normalisation unnecessary. The only requirement is to design a computational graph that parameterises monotonically non-decreasing functions with a constrained range. Training can be then performed using standard tools from any neural network software library. Finally, factorial hidden Markov models (FHMMs) for sequential data are presented. It is shown how to leverage cumulative distribution functions in the form of the Gaussian copula and amortised stochastic variational method to encode hidden Markov chains coherently. This approach enables efficient learning and inference to model long sequences of high-dimensional data with long-range dependencies. Tackling such complex problems was impossible with the established FHMM approximate inference algorithm. It is empirically verified on several problems that some of the estimators introduced in this work can perform comparably or better than the currently popular models. Especially for tasks requiring tail-area or marginal probabilities that can be read directly from a cumulative distribution function