Accelerating Pseudo-Marginal MCMC using Gaussian Processes

Pseudo-marginal methods such as the grouped independence Metropolis-Hastings (GIMH) and Markov chain within Metropolis (MCWM) algorithms have been introduced in the literature as an approach to perform Bayesian inference in latent variable models. These methods replace intractable likelihood calculations with unbiased estimates within Markov chain Monte Carlo algorithms. The GIMH method has the posterior of interest as its limiting distribution, but suffers from poor mixing if it is too computationally intensive to obtain high-precision likelihood estimates. The MCWM algorithm has better mixing properties, but less theoretical support. In this paper we propose to use Gaussian processes (GP) to accelerate the GIMH method, whilst using a short pilot run of MCWM to train the GP. Our new method, GP-GIMH, is illustrated on simulated data from a stochastic volatility and a gene network model.

Sampling Methods for Exploring Between Subject Variability in Cardiac Electrophysiology Experiments

Between-subject and within-subject variability is ubiquitous in biology and physiology and understanding and dealing with this is one of the biggest challenges in medicine. At the same time it is difficult to investigate this variability by experiments alone. A recent modelling and simulation approach, known as population of models (POM), allows this exploration to take place by building a mathematical model consisting of multiple parameter sets calibrated against experimental data. However, finding such sets within a high-dimensional parameter space of complex electrophysiological models is computationally challenging. By placing the POM approach within a statistical framework, we develop a novel and efficient algorithm based on sequential Monte Carlo (SMC). We compare the SMC approach with Latin hypercube sampling (LHS), a method commonly adopted in the literature for obtaining the POM, in terms of efficiency and output variability in the presence of a drug block through an in-depth investigation via the Beeler-Reuter cardiac electrophysiological model. We show improved efficiency via SMC and that it produces similar responses to LHS when making out-of-sample predictions in the presence of a simulated drug block.

Human error risk management methodology for rail crack incidents

The paper presents an innovative approach to modelling the causal relationships of human errors in rail crack incidents (RCI) from a managerial perspective. A Bayesian belief network is developed to model RCI by considering the human errors of designers, manufactures, operators and maintainers (DMOM) and the causal relationships involved. A set of dependent variables whose combinations express the relevant functions performed by each DMOM participant is used to model the causal relationships. A total of 14 RCI on Hong Kong’s mass transit railway (MTR) from 2008 to 2011 are used to illustrate the application of the model. Bayesian inference is used to conduct an importance analysis to assess the impact of the participants’ errors. Sensitivity analysis is then employed to gauge the effect the increased probability of occurrence of human errors on RCI. Finally, strategies for human error identification and mitigation of RCI are proposed. The identification of ability of maintainer in the case study as the most important factor influencing the probability of RCI implies the priority need to strengthen the maintenance management of the MTR system and that improving the inspection ability of the maintainer is likely to be an effective strategy for RCI risk mitigation.

Bayesian versus frequentist statistical inference for investi gating a one-off cancer cluster reported to a health department

Background The problem of silent multiple comparisons is one of the most difficult statistical problems faced by scientists. It is a particular problem for investigating a one-off cancer cluster reported to a health department because any one of hundreds, or possibly thousands, of neighbourhoods, schools, or workplaces could have reported a cluster, which could have been for any one of several types of cancer or any one of several time periods. Methods This paper contrasts the frequentist approach with a Bayesian approach for dealing with silent multiple comparisons in the context of a one-off cluster reported to a health department. Two published cluster investigations were re-analysed using the Dunn-Sidak method to adjust frequentist p-values and confidence intervals for silent multiple comparisons. Bayesian methods were based on the Gamma distribution. Results Bayesian analysis with non-informative priors produced results similar to the frequentist analysis, and suggested that both clusters represented a statistical excess. In the frequentist framework, the statistical significance of both clusters was extremely sensitive to the number of silent multiple comparisons, which can only ever be a subjective "guesstimate". The Bayesian approach is also subjective: whether there is an apparent statistical excess depends on the specified prior. Conclusion In cluster investigations, the frequentist approach is just as subjective as the Bayesian approach, but the Bayesian approach is less ambitious in that it treats the analysis as a synthesis of data and personal judgements (possibly poor ones), rather than objective reality. Bayesian analysis is (arguably) a useful tool to support complicated decision-making, because it makes the uncertainty associated with silent multiple comparisons explicit.

A fully Bayesian approach to inference for Coxian phase-type distributions with covariate dependent mean

Phase-type distributions represent the time to absorption for a finite state Markov chain in continuous time, generalising the exponential distribution and providing a flexible and useful modelling tool. We present a new reversible jump Markov chain Monte Carlo scheme for performing a fully Bayesian analysis of the popular Coxian subclass of phase-type models; the convenient Coxian representation involves fewer parameters than a more general phase-type model. The key novelty of our approach is that we model covariate dependence in the mean whilst using the Coxian phase-type model as a very general residual distribution. Such incorporation of covariates into the model has not previously been attempted in the Bayesian literature. A further novelty is that we also propose a reversible jump scheme for investigating structural changes to the model brought about by the introduction of Erlang phases. Our approach addresses more questions of inference than previous Bayesian treatments of this model and is automatic in nature. We analyse an example dataset comprising lengths of hospital stays of a sample of patients collected from two Australian hospitals to produce a model for a patient's expected length of stay which incorporates the effects of several covariates. This leads to interesting conclusions about what contributes to length of hospital stay with implications for hospital planning. We compare our results with an alternative classical analysis of these data.

Approximate Bayesian computation using indirect inference

We present a novel approach for developing summary statistics for use in approximate Bayesian computation (ABC) algorithms by using indirect inference. ABC methods are useful for posterior inference in the presence of an intractable likelihood function. In the indirect inference approach to ABC the parameters of an auxiliary model fitted to the data become the summary statistics. Although applicable to any ABC technique, we embed this approach within a sequential Monte Carlo algorithm that is completely adaptive and requires very little tuning. This methodological development was motivated by an application involving data on macroparasite population evolution modelled by a trivariate stochastic process for which there is no tractable likelihood function. The auxiliary model here is based on a beta–binomial distribution. The main objective of the analysis is to determine which parameters of the stochastic model are estimable from the observed data on mature parasite worms.

An intuitive dashboard for Bayesian network inference

Current Bayesian network software packages provide good graphical interface for users who design and develop Bayesian networks for various applications. However, the intended end-users of these networks may not necessarily find such an interface appealing and at times it could be overwhelming, particularly when the number of nodes in the network is large. To circumvent this problem, this paper presents an intuitive dashboard, which provides an additional layer of abstraction, enabling the end-users to easily perform inferences over the Bayesian networks. Unlike most software packages, which display the nodes and arcs of the network, the developed tool organises the nodes based on the cause-and-effect relationship, making the user-interaction more intuitive and friendly. In addition to performing various types of inferences, the users can conveniently use the tool to verify the behaviour of the developed Bayesian network. The tool has been developed using QT and SMILE libraries in C++.

Bayesian indirect inference using a parametric auxiliary model

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), we show to be 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) which encompasses pBII as well as general ABC methods so that the connections between the methods can be established.

Optimal Bayesian experimental design for models with intractable likelihoods using indirect inference applied to biological process models

This paper addresses the problem of determining optimal designs for biological process models with intractable likelihoods, with the goal of parameter inference. The Bayesian approach is to choose a design that maximises the mean of a utility, and the utility is a function of the posterior distribution. Therefore, its estimation requires likelihood evaluations. However, many problems in experimental design involve models with intractable likelihoods, that is, likelihoods that are neither analytic nor can be computed in a reasonable amount of time. We propose a novel solution using indirect inference (II), a well established method in the literature, and the Markov chain Monte Carlo (MCMC) algorithm of Müller et al. (2004). Indirect inference employs an auxiliary model with a tractable likelihood in conjunction with the generative model, the assumed true model of interest, which has an intractable likelihood. Our approach is to estimate a map between the parameters of the generative and auxiliary models, using simulations from the generative model. An II posterior distribution is formed to expedite utility estimation. We also present a modification to the utility that allows the Müller algorithm to sample from a substantially sharpened utility surface, with little computational effort. Unlike competing methods, the II approach can handle complex design problems for models with intractable likelihoods on a continuous design space, with possible extension to many observations. The methodology is demonstrated using two stochastic models; a simple tractable death process used to validate the approach, and a motivating stochastic model for the population evolution of macroparasites.

Bayesian hybrid algorithms and models : implementation and associated issues

This thesis addresses computational challenges arising from Bayesian analysis of complex real-world problems. Many of the models and algorithms designed for such analysis are ‘hybrid’ in nature, in that they are a composition of components for which their individual properties may be easily described but the performance of the model or algorithm as a whole is less well understood. The aim of this research project is to after a better understanding of the performance of hybrid models and algorithms. The goal of this thesis is to analyse the computational aspects of hybrid models and hybrid algorithms in the Bayesian context. The first objective of the research focuses on computational aspects of hybrid models, notably a continuous finite mixture of t-distributions. In the mixture model, an inference of interest is the number of components, as this may relate to both the quality of model fit to data and the computational workload. The analysis of t-mixtures using Markov chain Monte Carlo (MCMC) is described and the model is compared to the Normal case based on the goodness of fit. Through simulation studies, it is demonstrated that the t-mixture model can be more flexible and more parsimonious in terms of number of components, particularly for skewed and heavytailed data. The study also reveals important computational issues associated with the use of t-mixtures, which have not been adequately considered in the literature. The second objective of the research focuses on computational aspects of hybrid algorithms for Bayesian analysis. Two approaches will be considered: a formal comparison of the performance of a range of hybrid algorithms and a theoretical investigation of the performance of one of these algorithms in high dimensions. For the first approach, the delayed rejection algorithm, the pinball sampler, the Metropolis adjusted Langevin algorithm, and the hybrid version of the population Monte Carlo (PMC) algorithm are selected as a set of examples of hybrid algorithms. Statistical literature shows how statistical efficiency is often the only criteria for an efficient algorithm. In this thesis the algorithms are also considered and compared from a more practical perspective. This extends to the study of how individual algorithms contribute to the overall efficiency of hybrid algorithms, and highlights weaknesses that may be introduced by the combination process of these components in a single algorithm. The second approach to considering computational aspects of hybrid algorithms involves an investigation of the performance of the PMC in high dimensions. It is well known that as a model becomes more complex, computation may become increasingly difficult in real time. In particular the importance sampling based algorithms, including the PMC, are known to be unstable in high dimensions. This thesis examines the PMC algorithm in a simplified setting, a single step of the general sampling, and explores a fundamental problem that occurs in applying importance sampling to a high-dimensional problem. The precision of the computed estimate from the simplified setting is measured by the asymptotic variance of the estimate under conditions on the importance function. Additionally, the exponential growth of the asymptotic variance with the dimension is demonstrated and we illustrates that the optimal covariance matrix for the importance function can be estimated in a special case.

Estimation of parameters for macroparasite population evolution using approximate Bayesian computation

We estimate the parameters of a stochastic process model for a macroparasite population within a host using approximate Bayesian computation (ABC). The immunity of the host is an unobserved model variable and only mature macroparasites at sacrifice of the host are counted. With very limited data, process rates are inferred reasonably precisely. Modeling involves a three variable Markov process for which the observed data likelihood is computationally intractable. ABC methods are particularly useful when the likelihood is analytically or computationally intractable. The ABC algorithm we present is based on sequential Monte Carlo, is adaptive in nature, and overcomes some drawbacks of previous approaches to ABC. The algorithm is validated on a test example involving simulated data from an autologistic model before being used to infer parameters of the Markov process model for experimental data. The fitted model explains the observed extra-binomial variation in terms of a zero-one immunity variable, which has a short-lived presence in the host.

Adaptive Bayesian compound designs for dose finding studies

We consider the problem of how to efficiently and safely design dose finding studies. Both current and novel utility functions are explored using Bayesian adaptive design methodology for the estimation of a maximum tolerated dose (MTD). In particular, we explore widely adopted approaches such as the continual reassessment method and minimizing the variance of the estimate of an MTD. New utility functions are constructed in the Bayesian framework and are evaluated against current approaches. To reduce computing time, importance sampling is implemented to re-weight posterior samples thus avoiding the need to draw samples using Markov chain Monte Carlo techniques. Further, as such studies are generally first-in-man, the safety of patients is paramount. We therefore explore methods for the incorporation of safety considerations into utility functions to ensure that only safe and well-predicted doses are administered. The amalgamation of Bayesian methodology, adaptive design and compound utility functions is termed adaptive Bayesian compound design (ABCD). The performance of this amalgamation of methodology is investigated via the simulation of dose finding studies. The paper concludes with a discussion of results and extensions that could be included into our approach.

Bayesian models for the determination of resonant frequencies in a DI diesel engine

A time series method for the determination of combustion chamber resonant frequencies is outlined. This technique employs the use of Markov-chain Monte Carlo (MCMC) to infer parameters in a chosen model of the data. The development of the model is included and the resonant frequency is characterised as a function of time. Potential applications for cycle-by-cycle analysis are discussed and the bulk temperature of the gas and the trapped mass in the combustion chamber are evaluated as a function of time from resonant frequency information.

Using approximate Bayesian computation to estimate transmission rates of nosocomial pathogens

In this paper, we apply a simulation based approach for estimating transmission rates of nosocomial pathogens. In particular, the objective is to infer the transmission rate between colonised health-care practitioners and uncolonised patients (and vice versa) solely from routinely collected incidence data. The method, using approximate Bayesian computation, is substantially less computer intensive and easier to implement than likelihood-based approaches we refer to here. We find through replacing the likelihood with a comparison of an efficient summary statistic between observed and simulated data that little is lost in the precision of estimated transmission rates. Furthermore, we investigate the impact of incorporating uncertainty in previously fixed parameters on the precision of the estimated transmission rates.

Bayesian algorithms with applications

Advances in algorithms for approximate sampling from a multivariable target function have led to solutions to challenging statistical inference problems that would otherwise not be considered by the applied scientist. Such sampling algorithms are particularly relevant to Bayesian statistics, since the target function is the posterior distribution of the unobservables given the observables. In this thesis we develop, adapt and apply Bayesian algorithms, whilst addressing substantive applied problems in biology and medicine as well as other applications. For an increasing number of high-impact research problems, the primary models of interest are often sufficiently complex that the likelihood function is computationally intractable. Rather than discard these models in favour of inferior alternatives, a class of Bayesian "likelihoodfree" techniques (often termed approximate Bayesian computation (ABC)) has emerged in the last few years, which avoids direct likelihood computation through repeated sampling of data from the model and comparing observed and simulated summary statistics. In Part I of this thesis we utilise sequential Monte Carlo (SMC) methodology to develop new algorithms for ABC that are more efficient in terms of the number of model simulations required and are almost black-box since very little algorithmic tuning is required. In addition, we address the issue of deriving appropriate summary statistics to use within ABC via a goodness-of-fit statistic and indirect inference. Another important problem in statistics is the design of experiments. That is, how one should select the values of the controllable variables in order to achieve some design goal. The presences of parameter and/or model uncertainty are computational obstacles when designing experiments but can lead to inefficient designs if not accounted for correctly. The Bayesian framework accommodates such uncertainties in a coherent way. If the amount of uncertainty is substantial, it can be of interest to perform adaptive designs in order to accrue information to make better decisions about future design points. This is of particular interest if the data can be collected sequentially. In a sense, the current posterior distribution becomes the new prior distribution for the next design decision. Part II of this thesis creates new algorithms for Bayesian sequential design to accommodate parameter and model uncertainty using SMC. The algorithms are substantially faster than previous approaches allowing the simulation properties of various design utilities to be investigated in a more timely manner. Furthermore the approach offers convenient estimation of Bayesian utilities and other quantities that are particularly relevant in the presence of model uncertainty. Finally, Part III of this thesis tackles a substantive medical problem. A neurological disorder known as motor neuron disease (MND) progressively causes motor neurons to no longer have the ability to innervate the muscle fibres, causing the muscles to eventually waste away. When this occurs the motor unit effectively ‘dies’. There is no cure for MND, and fatality often results from a lack of muscle strength to breathe. The prognosis for many forms of MND (particularly amyotrophic lateral sclerosis (ALS)) is particularly poor, with patients usually only surviving a small number of years after the initial onset of disease. Measuring the progress of diseases of the motor units, such as ALS, is a challenge for clinical neurologists. Motor unit number estimation (MUNE) is an attempt to directly assess underlying motor unit loss rather than indirect techniques such as muscle strength assessment, which generally is unable to detect progressions due to the body’s natural attempts at compensation. Part III of this thesis builds upon a previous Bayesian technique, which develops a sophisticated statistical model that takes into account physiological information about motor unit activation and various sources of uncertainties. More specifically, we develop a more reliable MUNE method by applying marginalisation over latent variables in order to improve the performance of a previously developed reversible jump Markov chain Monte Carlo sampler. We make other subtle changes to the model and algorithm to improve the robustness of the approach.