Downloads & Free Reading Options - Results

We propose a framework for Bayesian non-parametric estimation of the rate at which new infections occur assuming that the epidemic is partially observed. The developed methodology relies on modelling the rate at which new infections occur as a function which only depends on time. Two different types of prior distributions are proposed namely using step-functions and B-splines. The methodology is illustrated using both simulated and real datasets and we show that certain aspects of the epidemic such as seasonality and super-spreading events are picked up without having to explicitly incorporate them into a parametric model.

The parametric bootstrap can be used for the efficient computation of Bayes posterior distributions. Importance sampling formulas take on an easy form relating to the deviance in exponential families and are particularly simple starting from Jeffreys invariant prior. Because of the i.i.d. nature of bootstrap sampling, familiar formulas describe the computational accuracy of the Bayes estimates. Besides computational methods, the theory provides a connection between Bayesian and frequentist analysis. Efficient algorithms for the frequentist accuracy of Bayesian inferences are developed and demonstrated in a model selection example.

This is a textbook on Bayesian Inference, suitable for undergraduate students of statistics and those who want to learn the subject on their own. The author, Ashok Kumar Bansal (1943-2023) was a Professor of Statistics at University of Delhi, India. 

Indirect inference (II) is a methodology for estimating the parameters of an intractable (generative) model on the basis of an alternative parametric (auxiliary) model that is both analytically and computationally easier to deal with. Such an approach has been well explored in the classical literature but has received substantially less attention in the Bayesian paradigm. The purpose of this paper is to compare and contrast a collection of what we call parametric Bayesian indirect inference (pBII) methods. One class of pBII methods uses approximate Bayesian computation (referred to here as ABC II) where the summary statistic is formed on the basis of the auxiliary model, using ideas from II. Another approach proposed in the literature, referred to here as parametric Bayesian indirect likelihood (pBIL), uses the auxiliary likelihood as a replacement to the intractable likelihood. We show that pBIL is a fundamentally different approach to ABC II. We devise new theoretical results for pBIL to give extra insights into its behaviour and also its differences with ABC II. Furthermore, we examine in more detail the assumptions required to use each pBII method. The results, insights and comparisons developed in this paper are illustrated on simple examples and two other substantive applications. The first of the substantive examples involves performing inference for complex quantile distributions based on simulated data while the second is for estimating the parameters of a trivariate stochastic process describing the evolution of macroparasites within a host based on real data. We create a novel framework called Bayesian indirect likelihood (BIL) that encompasses pBII as well as general ABC methods so that the connections between the methods can be established.

Includes bibliographical references and index

Given i.i.d. data from an unknown distribution, we consider the problem of predicting future items. An adaptive way to estimate the probability density is to recursively subdivide the domain to an appropriate data-dependent granularity. A Bayesian would assign a data-independent prior probability to "subdivide", which leads to a prior over infinite(ly many) trees. We derive an exact, fast, and simple inference algorithm for such a prior, for the data evidence, the predictive distribution, the effective model dimension, moments, and other quantities. We prove asymptotic convergence and consistency results, and illustrate the behavior of our model on some prototypical functions.

Given i.i.d. data from an unknown distribution, we consider the problem of predicting future items. An adaptive way to estimate the probability density is to recursively subdivide the domain to an appropriate data-dependent granularity. A Bayesian would assign a data-independent prior probability to "subdivide", which leads to a prior over infinite(ly many) trees. We derive an exact, fast, and simple inference algorithm for such a prior, for the data evidence, the predictive distribution, the effective model dimension, and other quantities.

Given i.i.d. data from an unknown distribution, we consider the problem of predicting future items. An adaptive way to estimate the probability density is to recursively subdivide the domain to an appropriate data-dependent granularity. A Bayesian would assign a data-independent prior probability to "subdivide", which leads to a prior over infinite(ly many) trees. We derive an exact, fast, and simple inference algorithm for such a prior, for the data evidence, the predictive distribution, the effective model dimension, and other quantities.

Many applications, such as intermittent data assimilation, lead to a recursive application of Bayesian inference within a Monte Carlo context. Popular data assimilation algorithms include sequential Monte Carlo methods and ensemble Kalman filters (EnKFs). These methods differ in the way Bayesian inference is implemented. Sequential Monte Carlo methods rely on importance sampling combined with a resampling step while EnKFs utilize a linear transformation of Monte Carlo samples based on the classic Kalman filter. While EnKFs have proven to be quite robust even for small ensemble sizes, they are not consistent since their derivation relies on a linear regression ansatz. In this paper, we propose another transform method, which does not rely on any a prior assumptions on the underlying prior and posterior distributions. The new method is based on solving an optimal transportation problem for discrete random variables.

We investigate Bayesian non-parametric inference of the $\Lambda$-measure of $\Lambda$-coalescent processes with recurrent mutation, parametrised by probability measures on the unit interval. We give verifiable criteria on the prior for posterior consistency when observations form a time series, and prove that any non-trivial prior is inconsistent when all observations are contemporaneous. We then show that the likelihood given a data set of size $n \in \mathbb{N}$ is constant across $\Lambda$-measures whose leading $n - 2$ moments agree, and focus on inferring truncated sequences of moments. We provide a large class of functionals which can be extremised using finite computation given a credible region of posterior truncated moment sequences, and a pseudo-marginal Metropolis-Hastings algorithm for sampling the posterior. Finally, we compare the efficiency of the exact and noisy pseudo-marginal algorithms with and without delayed acceptance acceleration using a simulation study.

The observation model Y sub i = Beta(i/n) + epsilon sub i, 1 or = n, is considered, where the epsilon's are i.i.d. mean zero and variance sigma-sq and beta is an unknown smooth function. A Gaussian prior distribution is specified by assuming beta is the solution of a high order stochastic differential equation. The estimation error delta = beta - beta-average is analyzed, where beta-average is the posterior expectation of beta. Asymptotic posterior and sampling distributional approximations are given for (abs. val del)square when (abs. val)square is one of a family of norms natural to the problem. It is shown that the frequentist coverage probability of a variety of (1 - alpha) posterior probability regions tends to be larger than 1 - alpha, but will be infinitely often less than any epsilon 0 as n approaches infinity with prior probability 1. A related continuous time signal estimation problem is also studied. Keywords: Bayesian inference; Nonparametric regression; Confidence regions; Signal extraction: Smoothing splices.