Stephen G. Walker

We show that a special case of method of moment estimator derived from the Stein class coincides with the class of generalized score matching estimator. Choosing a suitable weight function for generalized score matching is not straightforward. However, by placing it within the method of moment framework we can alleviate this problem by extending the Stein class to generalized method of moments. As a consequence we can work with a number of functions and hence derive generalized score matching estimators with optimal properties.

Uncertainty associated with statistical problems arises due to what has not been seen as opposed to what has been seen. Using probability to quantify the uncertainty the task is to construct a probability model for what has not been seen conditional on what has been seen. The traditional Bayesian approach is to use prior distributions for constructing the predictive distributions, though recently a novel approach has used density estimators and the use of martingales to establish convergence of parameter values. In this paper we reply on martingales constructed using score functions. Hence, the method only requires the computing of gradients arising from parametric families of density functions. A key point is that we do not rely on Markov Chain Monte Carlo (MCMC) algorithms, and that the method can be implemented in parallel. We present the theoretical properties of the score driven martingale posterior. Further, we present illustrations under different models and settings.

The paper undertakes Bayesian style inference using posterior distributions. The key difference is that we use an assumption of a conditionally identically distributed (c.i.d.) sequence rather than the more common exchangeable sequence. We show that there remains the existence of a prior and posterior while the updating mechanism is achieved through the predictive distributions. This is sufficient given a fundamental result of Doob which explained how posteriors can be constructed in the exchangeable case via predictive distributions. We model the predictive distributions using copulas ensuring the c.i.d. structure.

We introduce a class of integral theorems based on cyclic functions and Riemann sums approximating integrals. The Fourier integral theorem, derived as a combination of a transform and inverse transform, arises as a special case. The integral theorems provide natural estimators of density functions via Monte Carlo methods. Assessment of the quality of the density estimators can be used to obtain optimal cyclic functions, alternatives to the sin function, which minimize square integrals. Our proof techniques rely on a variational approach in ordinary differential equations and the Cauchy residue theorem in complex analysis.

This paper is about the modeling of cumulative hazard functions using martingale posterior distributions. The focus is on uncertainty quantification from a nonparametric perspective. The foundational Bayesian model in this case is the beta process and the classic estimator is the Nelson–Aalen. We use a sequence of estimators which form a martingale in order to obtain a random cumulative hazard function from the martingale posterior. The connection with the beta process is established and a number of illustrations is presented.

The family of log-concave density functions contains various kinds of common probability distributions. Due to the shape restriction, it is possible to find the nonparametric estimate of the density, for example, the nonparametric maximum likelihood estimate (NPMLE). However, the associated uncertainty quantification of the NPMLE is less well developed. The current techniques for uncertainty quantification are Bayesian, using a Dirichlet process prior combined with the use of Markov chain Monte Carlo (MCMC) to sample from the posterior. In this paper, we start with the NPMLE and use a version of the martingale posterior distribution to establish uncertainty about the NPMLE. The algorithm can be implemented in parallel and hence is fast. We prove the convergence of the algorithm by constructing suitable submartingales. We also illustrate results with different models and settings and some real data, and compare our method with that within the literature.

This paper proposes a new nonparametric Bayesian bootstrap for a mixture model, by developing the traditional Bayesian bootstrap. We first reinterpret the Bayesian bootstrap, which uses the Pólya-urn scheme, as a gradient ascent algorithm which associated one-step solver. The key then is to use the same basic mechanism as the Bayesian bootstrap with the switch from a point mass kernel to a continuous kernel. Just as the Bayesian bootstrap works solely from the empirical distribution function, so the new Bayesian bootstrap for mixture models works off the nonparametric maximum likelihood estimator for the mixing distribution. From a theoretical perspective, we prove the convergence and exchangeability of the sample sequences from the algorithm and also illustrate our results with different models and settings and some real data.