Conditional Copula Inference and Efficient Approximate MCMC

This thesis consists of two main parts. The first part focuses on parametric conditional copula models that allow the copula parameters to vary with a set of covariates according to an unknown calibration function. Flexible Bayesian inference for the calibration function of a bivariate conditional copula is introduced. The prior distribution over the set of smooth calibration functions is built using a sparse Gaussian Process prior for the Single Index Model. The estimation of parameters from the marginal distributions and the calibration function is done jointly via Markov Chain Monte Carlo sampling from the full posterior distribution. A new Conditional Cross Validated Pseudo-Marginal criterion is used to perform copula selection and is modified using a permutation-based procedure to assess data support for the simplifying assumption. The first part concludes with methods for establishing data support for the simplifying assumption in a bivariate conditional copula model. After splitting the observed data into training and test sets, the method proposed will use a flexible Bayesian model fit to the training data to define tests based on randomization and standard asymptotic theory. I discuss theoretical justification for the method and implementations in alternative models of interest: Gaussian, Logistic and Quantile regressions. The performance is studied via simulated data. The second part of the thesis focuses on approximate Bayesian methods. Approximate Bayesian Computation (ABC) and Bayesian Synthetic Likelihood (BSL) are popular simulation based methods for sampling from the posterior distribution when the likelihood is not tractable but simulations for each parameter are easily available. However these methods can be computationally inefficient since a large number of pseudo-data simulations is required. I propose to use perturbed MCMC versions of ABC and BSL algorithms and attempt to significantly accelerate these samplers. The main idea of the proposed strategy is to utilize past samples with k-Nearest-Neighbor approach for likelihood approximation. This general method works for ABC and BSL and greatly reduces computational cost and number of required simulations for these samplers. Performance and computational advantage are examined via series of simulation examples. The second part concludes with theoretical justifications and convergence properties of the proposed strategies.Ph.D