Stephen G. Walker

Nhat Ho, Stephen G. Walker

Taking the Fourier integral theorem as our starting point, in this paper we focus on natural Monte Carlo and fully nonparametric estimators of multivariate distributions and conditional distribution functions. We do this without the need for any estimated covariance matrix or dependence structure between variables. These aspects arise immediately from the integral theorem. Being able to model multivariate data sets using conditional distribution functions we can study a number of problems, such as prediction for Markov processes, estimation of mixing distribution functions which depend on covariates, and general multivariate data. Estimators are explicit Monte Carlo based and require no recursive or iterative algorithms.

Nhat Ho, Stephen G. Walker

Starting with the Fourier integral theorem, we present natural Monte Carlo estimators of multivariate functions including densities, mixing densities, transition densities, regression functions, and the search for modes of multivariate density functions (modal regression). Rates of convergence are established and, in many cases, provide superior rates to current standard estimators such as those based on kernels, including kernel density estimators and kernel regression functions. Numerical illustrations are presented.

Simon Lyddon, Chris Holmes, Stephen Walker

In this paper we revisit the weighted likelihood bootstrap, a method that generates samples from an approximate Bayesian posterior of a parametric model. We show that the same method can be derived, without approximation, under a Bayesian nonparametric model with the parameter of interest defined as minimising an expected negative log-likelihood under an unknown sampling distribution. This interpretation enables us to extend the weighted likelihood bootstrap to posterior sampling for parameters minimizing an expected loss. We call this method the loss-likelihood bootstrap. We make a connection between this and general Bayesian updating, which is a way of updating prior belief distributions without needing to construct a global probability model, yet requires the calibration of two forms of loss function. The loss-likelihood bootstrap is used to calibrate the general Bayesian posterior by matching asymptotic Fisher information. We demonstrate the methodology on a number of examples.