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Feng Li
-- doctoral thesis --
Bayesian Modeling of Conditional Densities
This thesis develops models and associated Bayesian inference methods for flexible univariate and multivariate conditional density estimation. The models are flexible in the sense that they can capture widely differing shapes of the data. The estimation methods are specifically designed to achieve flexibility while still avoiding overfitting. The models are flexible both for a given covariate value, but also across covariate space. A key contribution of this thesis is that it provides general approaches of density estimation with highly efficient Markov chain Monte Carlo methods. The methods are illustrated on several challenging non-linear and non-normal datasets.
In the first paper, a general model is proposed for flexibly estimating the density of a continuous
response variable conditional on a possibly high-dimensional set of covariates. The model
is a finite mixture of asymmetric student-t densities with covariate-dependent mixture weights.
The four parameters of the components, the mean, degrees of freedom, scale and skewness, are
all modeled as functions of the covariates. The second paper explores how well a smooth mixture
of symmetric components can capture skewed data. Simulations and applications on real data
show that including covariate-dependent skewness in the components can lead to substantially improved
performance on skewed data, often using a much smaller number of components. We also
introduce smooth mixtures of gamma and log-normal components to model positively-valued response
variables. In the third paper we propose a multivariate Gaussian surface regression model
that combines both additive splines and interactive splines, and a highly efficient MCMC algorithm
that updates all the multi-dimensional knot locations jointly. We use shrinkage priors to
avoid overfitting with different estimated shrinkage factors for the additive and surface part of the
model, and also different shrinkage parameters for the different response variables. In the last paper
we present a general Bayesian approach for directly modeling dependencies between variables
as function of explanatory variables in a flexible copula context. In particular, the Joe-Clayton copula
is extended to have covariate-dependent tail dependence and correlations. Posterior inference
is carried out using a novel and efficient simulation method. The appendix of the thesis documents
the computational implementation details.
Key words: Bayesian inference; density estimation; smooth mixtures; surface regression; copulas;
Markov chain Monte Carlo.
ISBN 978-91-7447-665-1