Model choice problems using approximate Bayesian computation with applications to pathogen transmission data sets

Analytically or computationally intractable likelihood functions can arise in complex statistical inferential problems making them inaccessible to standard Bayesian inferential methods. Approximate Bayesian computation (ABC) methods address such inferential problems by replacing direct likelihood evaluations with repeated sampling from the model. ABC methods have been predominantly applied to parameter estimation problems and less to model choice problems due to the added difficulty of handling multiple model spaces. The ABC algorithm proposed here addresses model choice problems by extending Fearnhead and Prangle (2012, Journal of the Royal Statistical Society, Series B 74, 1–28) where the posterior mean of the model parameters estimated through regression formed the summary statistics used in the discrepancy measure. An additional stepwise multinomial logistic regression is performed on the model indicator variable in the regression step and the estimated model probabilities are incorporated into the set of summary statistics for model choice purposes. A reversible jump Markov chain Monte Carlo step is also included in the algorithm to increase model diversity for thorough exploration of the model space. This algorithm was applied to a validating example to demonstrate the robustness of the algorithm across a wide range of true model probabilities. Its subsequent use in three pathogen transmission examples of varying complexity illustrates the utility of the algorithm in inferring preference of particular transmission models for the pathogens.

Contributions to Bayesian experimental design

This thesis progresses Bayesian experimental design by developing novel methodologies and extensions to existing algorithms. Through these advancements, this thesis provides solutions to several important and complex experimental design problems, many of which have applications in biology and medicine. This thesis consists of a series of published and submitted papers. In the first paper, we provide a comprehensive literature review on Bayesian design. In the second paper, we discuss methods which may be used to solve design problems in which one is interested in finding a large number of (near) optimal design points. The third paper presents methods for finding fully Bayesian experimental designs for nonlinear mixed effects models, and the fourth paper investigates methods to rapidly approximate the posterior distribution for use in Bayesian utility functions.

Statistical methods for electromyography data and associated problems

This thesis proposes three novel models which extend the statistical methodology for motor unit number estimation, a clinical neurology technique. Motor unit number estimation is important in the treatment of degenerative muscular diseases and, potentially, spinal injury. Additionally, a recent and untested statistic to enable statistical model choice is found to be a practical alternative for larger datasets. The existing methods for dose finding in dual-agent clinical trials are found to be suitable only for designs of modest dimensions. The model choice case-study is the first of its kind containing interesting results using so-called unit information prior distributions.

Multi-modal deformable medical image registration

Non-rigid image registration is an essential tool required for overcoming the inherent local anatomical variations that exist between images acquired from different individuals or atlases. Furthermore, certain applications require this type of registration to operate across images acquired from different imaging modalities. One popular local approach for estimating this registration is a block matching procedure utilising the mutual information criterion. However, previous block matching procedures generate a sparse deformation field containing displacement estimates at uniformly spaced locations. This neglects to make use of the evidence that block matching results are dependent on the amount of local information content. This paper presents a solution to this drawback by proposing the use of a Reversible Jump Markov Chain Monte Carlo statistical procedure to optimally select grid points of interest. Three different methods are then compared to propagate the estimated sparse deformation field to the entire image including a thin-plate spline warp, Gaussian convolution, and a hybrid fluid technique. Results show that non-rigid registration can be improved by using the proposed algorithm to optimally select grid points of interest.

Inter-cycle variability of ignition delay in an ethanol fumigated common rail diesel engine

An experimental study has been performed to investigate the ignition delay of a modern heavy-duty common-rail diesel engine run with fumigated ethanol substitutions up to 40% on an energy basis. The ignition delay was determined through the use of statistical modelling in a Bayesian framework this framework allows for the accurate determination of the start of combustion from single consecutive cycles and does not require any differentiation of the in-cylinder pressure signal. At full load the ignition delay has been shown to decrease with increasing ethanol substitutions and evidence of combustion with high ethanol substitutions prior to diesel injection have also been shown experimentally and by modelling. Whereas, at half load increasing ethanol substitutions have increased the ignition delay. A threshold absolute air to fuel ratio (mole basis) of above ~110 for consistent operation has been determined from the inter-cycle variability of the ignition delay, a result that agrees well with previous research of other in-cylinder parameters and further highlights the correlation between the air to fuel ratio and inter-cycle variability. Numerical modelling to investigate the sensitivity of ethanol combustion has also been performed. It has been shown that ethanol combustion is sensitive to the initial air temperature around the feasible operating conditions of the engine. Moreover, a negative temperature coefficient region of approximately 900{1050 K (the approximate temperature at fuel injection) has been shown with for n-heptane and n-heptane/ethanol blends in the numerical modelling. A consequence of this is that the dominate effect influencing the ignition delay under increasing ethanol substitutions may rather be from an increase in chemical reactions and not from in-cylinder temperature. Further investigation revealed that the chemical reactions at low ethanol substitutions are different compared to the high (> 20%) ethanol substitutions.

A statistical model for combustion resonance from a DI diesel engine with applications

Introduced in this paper is a Bayesian model for isolating the resonant frequency from combustion chamber resonance. The model shown in this paper focused on characterising the initial rise in the resonant frequency to investigate the rise of in-cylinder bulk temperature associated with combustion. By resolving the model parameters, it is possible to determine: the start of pre-mixed combustion, the start of diffusion combustion, the initial resonant frequency, the resonant frequency as a function of crank angle, the in-cylinder bulk temperature as a function of crank angle and the trapped mass as a function of crank angle. The Bayesian method allows for individual cycles to be examined without cycle-averaging|allowing inter-cycle variability studies. Results are shown for a turbo-charged, common-rail compression ignition engine run at 2000 rpm and full load.

Bikeshare's impact on active travel: Evidence from the United States, Great Britain, and Australia

Over 800 cities globally now offer bikeshare programs. One of their purported benefits is increased physical activity. Implicit in this claim is that bikeshare replaces sedentary modes of transport, particularly car use. This paper estimates the median changes in physical activity levels as a result of bikeshare in the cities of Melbourne, Brisbane, Washington, D.C., London, and Minneapolis/St. Paul. This study is the first known multi-city evaluation of the active travel impacts of bikeshare programs. To perform the analysis, data on mode substitution (i.e. the modes that bikeshare replaces) were used to determine the extent of shift from sedentary to active transport modes (e.g. when a car trip is replaced by bikeshare). Potentially offsetting these gains, reductions in physical activity when walking trips are replaced by bikeshare was also estimated. Finally a Markov Chain Monte Carlo analysis was conducted to estimate confidence bounds on estimated impacts on active travel given uncertainties in data sources. The results indicate that on average 60% of bikeshare trips replace sedentary modes of transport (from 42% in Minneapolis/St. Paul to 67% in Brisbane). When bikeshare replaces a walking trip, there is a reduction in active travel time because walking a given distance takes longer than cycling. Considering the active travel balance sheet for the cities included in this analysis, bikeshare activity in 2012 has an overall positive impact on active travel time. This impact ranges from an additional 1.4 million minutes of active travel for the Minneapolis/St. Paul bikeshare program, to just over 74 million minutes of active travel for the London program The analytical approach adopted to estimate bikeshare’s impact on active travel may act as the basis for future bikeshare evaluations or feasibility studies.

Alive SMC^2: Bayesian model selection for low-count time series models with intractable likelihoods

In this paper we present a new method for performing Bayesian parameter inference and model choice for low count time series models with intractable likelihoods. The method involves incorporating an alive particle filter within a sequential Monte Carlo (SMC) algorithm to create a novel pseudo-marginal algorithm, which we refer to as alive SMC^2. The advantages of this approach over competing approaches is that it is naturally adaptive, it does not involve between-model proposals required in reversible jump Markov chain Monte Carlo and does not rely on potentially rough approximations. The algorithm is demonstrated on Markov process and integer autoregressive moving average models applied to real biological datasets of hospital-acquired pathogen incidence, animal health time series and the cumulative number of poison disease cases in mule deer.

Scalable Bayesian Computation for intractable likelihoods in image analysis

The inverse temperature hyperparameter of the hidden Potts model governs the strength of spatial cohesion and therefore has a substantial influence over the resulting model fit. The difficulty arises from the dependence of an intractable normalising constant on the value of the inverse temperature, thus there is no closed form solution for sampling from the distribution directly. We review three computational approaches for addressing this issue, namely pseudolikelihood, path sampling, and the approximate exchange algorithm. We compare the accuracy and scalability of these methods using a simulation study.

Estimation of the time of a linear trend in monitoring survival time

Change point estimation is recognized as an essential tool of root cause analyses within quality control programs as it enables clinical experts to search for potential causes of change in hospital outcomes more effectively. In this paper, we consider estimation of the time when a linear trend disturbance has occurred in survival time following an in-control clinical intervention in the presence of variable patient mix. To model the process and change point, a linear trend in the survival time of patients who underwent cardiac surgery is formulated using hierarchical models in a Bayesian framework. The data are right censored since the monitoring is conducted over a limited follow-up period. We capture the effect of risk factors prior to the surgery using a Weibull accelerated failure time regression model. We use Markov Chain Monte Carlo to obtain posterior distributions of the change point parameters including the location and the slope size of the trend and also corresponding probabilistic intervals and inferences. The performance of the Bayesian estimator is investigated through simulations and the result shows that precise estimates can be obtained when they are used in conjunction with the risk-adjusted survival time cumulative sum control chart (CUSUM) control charts for different trend scenarios. In comparison with the alternatives, step change point model and built-in CUSUM estimator, more accurate and precise estimates are obtained by the proposed Bayesian estimator over linear trends. These superiorities are enhanced when probability quantification, flexibility and generalizability of the Bayesian change point detection model are also considered.

Overfitting Bayesian mixture models with an unknown number of components

This paper proposes solutions to three issues pertaining to the estimation of finite mixture models with an unknown number of components: the non-identifiability induced by overfitting the number of components, the mixing limitations of standard Markov Chain Monte Carlo (MCMC) sampling techniques, and the related label switching problem. An overfitting approach is used to estimate the number of components in a finite mixture model via a Zmix algorithm. Zmix provides a bridge between multidimensional samplers and test based estimation methods, whereby priors are chosen to encourage extra groups to have weights approaching zero. MCMC sampling is made possible by the implementation of prior parallel tempering, an extension of parallel tempering. Zmix can accurately estimate the number of components, posterior parameter estimates and allocation probabilities given a sufficiently large sample size. The results will reflect uncertainty in the final model and will report the range of possible candidate models and their respective estimated probabilities from a single run. Label switching is resolved with a computationally light-weight method, Zswitch, developed for overfitted mixtures by exploiting the intuitiveness of allocation-based relabelling algorithms and the precision of label-invariant loss functions. Four simulation studies are included to illustrate Zmix and Zswitch, as well as three case studies from the literature. All methods are available as part of the R package Zmix, which can currently be applied to univariate Gaussian mixture models.

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.

Estimation of cosmological parameters using adaptive importance sampling

We present a Bayesian sampling algorithm called adaptive importance sampling or population Monte Carlo (PMC), whose computational workload is easily parallelizable and thus has the potential to considerably reduce the wall-clock time required for sampling, along with providing other benefits. To assess the performance of the approach for cosmological problems, we use simulated and actual data consisting of CMB anisotropies, supernovae of type Ia, and weak cosmological lensing, and provide a comparison of results to those obtained using state-of-the-art Markov chain Monte Carlo (MCMC). For both types of data sets, we find comparable parameter estimates for PMC and MCMC, with the advantage of a significantly lower wall-clock time for PMC. In the case of WMAP5 data, for example, the wall-clock time scale reduces from days for MCMC to hours using PMC on a cluster of processors. Other benefits of the PMC approach, along with potential difficulties in using the approach, are analyzed and discussed.

Bayesian mixture model estimation of aerosol particle size distributions

In this paper, we examine approaches to estimate a Bayesian mixture model at both single and multiple time points for a sample of actual and simulated aerosol particle size distribution (PSD) data. For estimation of a mixture model at a single time point, we use Reversible Jump Markov Chain Monte Carlo (RJMCMC) to estimate mixture model parameters including the number of components which is assumed to be unknown. We compare the results of this approach to a commonly used estimation method in the aerosol physics literature. As PSD data is often measured over time, often at small time intervals, we also examine the use of an informative prior for estimation of the mixture parameters which takes into account the correlated nature of the parameters. The Bayesian mixture model offers a promising approach, providing advantages both in estimation and inference.

Bayesian estimation of small effects in exercise and sports science

The aim of this paper is to provide a Bayesian formulation of the so-called magnitude-based inference approach to quantifying and interpreting effects, and in a case study example provide accurate probabilistic statements that correspond to the intended magnitude-based inferences. The model is described in the context of a published small-scale athlete study which employed a magnitude-based inference approach to compare the effect of two altitude training regimens (live high-train low (LHTL), and intermittent hypoxic exposure (IHE)) on running performance and blood measurements of elite triathletes. The posterior distributions, and corresponding point and interval estimates, for the parameters and associated effects and comparisons of interest, were estimated using Markov chain Monte Carlo simulations. The Bayesian analysis was shown to provide more direct probabilistic comparisons of treatments and able to identify small effects of interest. The approach avoided asymptotic assumptions and overcame issues such as multiple testing. Bayesian analysis of unscaled effects showed a probability of 0.96 that LHTL yields a substantially greater increase in hemoglobin mass than IHE, a 0.93 probability of a substantially greater improvement in running economy and a greater than 0.96 probability that both IHE and LHTL yield a substantially greater improvement in maximum blood lactate concentration compared to a Placebo. The conclusions are consistent with those obtained using a ‘magnitude-based inference’ approach that has been promoted in the field. The paper demonstrates that a fully Bayesian analysis is a simple and effective way of analysing small effects, providing a rich set of results that are straightforward to interpret in terms of probabilistic statements.