A Python package for Bayesian estimation using Markov Chain Monte Carlo

Markov chain Monte Carlo (MCMC) estimation provides a solution to the complex integration problems that are faced in the Bayesian analysis of statistical problems. The implementation of MCMC algorithms is, however, code intensive and time consuming. We have developed a Python package, which is called PyMCMC, that aids in the construction of MCMC samplers and helps to substantially reduce the likelihood of coding error, as well as aid in the minimisation of repetitive code. PyMCMC contains classes for Gibbs, Metropolis Hastings, independent Metropolis Hastings, random walk Metropolis Hastings, orientational bias Monte Carlo and slice samplers as well as specific modules for common models such as a module for Bayesian regression analysis. PyMCMC is straightforward to optimise, taking advantage of the Python libraries Numpy and Scipy, as well as being readily extensible with C or Fortran.

Application of electrical resistivity method to detect deep cracks in unsaturated residual soil slope

Crack is a significant influential factor in soil slope that could leads to rainfall-induced slope instability. Existence of cracks at soil surface will decrease the shear strength and increase the hydraulic conductivity of soil slope. Although previous research has shown the effect of surface-cracks in soil stability, the influence of deep-cracks on soil stability is still unknown. The limited availability of deep crack data due to the difficulty of effective investigate methods could be one of the obstacles. Current technology in electrical resistivity can be used to detect deep-cracks in soil. This paper discusses deep cracks in unsaturated residual soil slopes in Indonesia using electrical resistivity method. The field investigation such as bore hole and SPT tests was carried out at multiple locations in the area where the electrical resistivity testing have been conducted. Subsequently, the results from bore-hole and SPT test were used to verify the results of the electrical resistivity test. This study demonstrates the benefits and limitations of the electrical resistivity in detecting deep-cracks in a residual soil slopes.

A Bayesian model predicts the response of axons to molecular gradients

Axon guidance by molecular gradients plays a crucial role in wiring up the nervous system. However, the mechanisms axons use to detect gradients are largely unknown. We first develop a Bayesian “ideal observer” analysis of gradient detection by axons, based on the hypothesis that a principal constraint on gradient detection is intrinsic receptor binding noise. Second, from this model, we derive an equation predicting how the degree of response of an axon to a gradient should vary with gradient steepness and absolute concentration. Third, we confirm this prediction quantitatively by performing the first systematic experimental analysis of how axonal response varies with both these quantities. These experiments demonstrate a degree of sensitivity much higher than previously reported for any chemotacting system. Together, these results reveal both the quantitative constraints that must be satisfied for effective axonal guidance and the computational principles that may be used by the underlying signal transduction pathways, and allow predictions for the degree of response of axons to gradients in a wide variety of in vivo and in vitro settings.

The Bayesian conditional independence model for measurement error: applications in ecology

The measurement error model is a well established statistical method for regression problems in medical sciences, although rarely used in ecological studies. While the situations in which it is appropriate may be less common in ecology, there are instances in which there may be benefits in its use for prediction and estimation of parameters of interest. We have chosen to explore this topic using a conditional independence model in a Bayesian framework using a Gibbs sampler, as this gives a great deal of flexibility, allowing us to analyse a number of different models without losing generality. Using simulations and two examples, we show how the conditional independence model can be used in ecology, and when it is appropriate.

An economic evaluation of the Safe Motherhood programme in Guangxi, China

Maternal and infant mortality is a global health issue with a significant social and economic impact. Each year, over half a million women worldwide die due to complications related to pregnancy or childbirth, four million infants die in the first 28 days of life, and eight million infants die in the first year. Ninety-nine percent of maternal and infant deaths are in developing countries. Reducing maternal and infant mortality is among the key international development goals. In China, the national maternal mortality ratio and infant mortality rate were reduced greatly in the past two decades, yet a large discrepancy remains between urban and rural areas. To address this problem, a large-scale Safe Motherhood Programme was initiated in 2000. The programme was implemented in Guangxi in 2003. Interventions in the programme included both demand-side and supply side-interventions focusing on increasing health service use and improving birth outcomes. Little is known about the effects and economic outcomes of the Safe Motherhood Programme in Guangxi, although it has been implemented for seven years. The aim of this research is to estimate the effectiveness and cost-effectiveness of the interventions in the Safe Motherhood Programme in Guangxi, China. The objectives of this research include: 1. To evaluate whether the changes of health service use and birth outcomes are associated with the interventions in the Safe Motherhood Programme. 2. To estimate the cost-effectiveness of the interventions in the Safe Motherhood Programme and quantify the uncertainty surrounding the decision. 3. To assess the expected value of perfect information associated with both the whole decision and individual parameters, and interpret the findings to inform priority setting in further research and policy making in this area. A quasi-experimental study design was used in this research to assess the effectiveness of the programme in increasing health service use and improving birth outcomes. The study subjects were 51 intervention counties and 30 control counties. Data on the health service use, birth outcomes and socio-economic factors from 2001 to 2007 were collected from the programme database and statistical yearbooks. Based on the profile plots of the data, general linear mixed models were used to evaluate the effectiveness of the programme while controlling for the effects of baseline levels of the response variables, change of socio-economic factors over time and correlations among repeated measurements from the same county. Redundant multicollinear variables were deleted from the mixed model using the results of the multicollinearity diagnoses. For each response variable, the best covariance structure was selected from 15 alternatives according to the fit statistics including Akaike information criterion, Finite-population corrected Akaike information criterion, and Schwarz.s Bayesian information criterion. Residual diagnostics were used to validate the model assumptions. Statistical inferences were made to show the effect of the programme on health service use and birth outcomes. A decision analytic model was developed to evaluate the cost-effectiveness of the programme, quantify the decision uncertainty, and estimate the expected value of perfect information associated with the decision. The model was used to describe the transitions between health states for women and infants and reflect the change of both costs and health benefits associated with implementing the programme. Result gained from the mixed models and other relevant evidence identified were synthesised appropriately to inform the input parameters of the model. Incremental cost-effectiveness ratios of the programme were calculated for the two groups of intervention counties over time. Uncertainty surrounding the parameters was dealt with using probabilistic sensitivity analysis, and uncertainty relating to model assumptions was handled using scenario analysis. Finally the expected value of perfect information for both the whole model and individual parameters in the model were estimated to inform priority setting in further research in this area.The annual change rates of the antenatal care rate and the institutionalised delivery rate were improved significantly in the intervention counties after the programme was implemented. Significant improvements were also found in the annual change rates of the maternal mortality ratio, the infant mortality rate, the incidence rate of neonatal tetanus and the mortality rate of neonatal tetanus in the intervention counties after the implementation of the programme. The annual change rate of the neonatal mortality rate was also improved, although the improvement was only close to statistical significance. The influences of the socio-economic factors on the health service use indicators and birth outcomes were identified. The rural income per capita had a significant positive impact on the health service use indicators, and a significant negative impact on the birth outcomes. The number of beds in healthcare institutions per 1,000 population and the number of rural telephone subscribers per 1,000 were found to be positively significantly related to the institutionalised delivery rate. The length of highway per square kilometre negatively influenced the maternal mortality ratio. The percentage of employed persons in the primary industry had a significant negative impact on the institutionalised delivery rate, and a significant positive impact on the infant mortality rate and neonatal mortality rate. The incremental costs of implementing the programme over the existing practice were US $11.1 million from the societal perspective, and US $13.8 million from the perspective of the Ministry of Health. Overall, 28,711 life years were generated by the programme, producing an overall incremental cost-effectiveness ratio of US $386 from the societal perspective, and US $480 from the perspective of the Ministry of Health, both of which were below the threshold willingness-to-pay ratio of US $675. The expected net monetary benefit generated by the programme was US $8.3 million from the societal perspective, and US $5.5 million from the perspective of the Ministry of Health. The overall probability that the programme was cost-effective was 0.93 and 0.89 from the two perspectives, respectively. The incremental cost-effectiveness ratio of the programme was insensitive to the different estimates of the three parameters relating to the model assumptions. Further research could be conducted to reduce the uncertainty surrounding the decision, in which the upper limit of investment was US $0.6 million from the societal perspective, and US $1.3 million from the perspective of the Ministry of Health. It is also worthwhile to get a more precise estimate of the improvement of infant mortality rate. The population expected value of perfect information for individual parameters associated with this parameter was US $0.99 million from the societal perspective, and US $1.14 million from the perspective of the Ministry of Health. The findings from this study have shown that the interventions in the Safe Motherhood Programme were both effective and cost-effective in increasing health service use and improving birth outcomes in rural areas of Guangxi, China. Therefore, the programme represents a good public health investment and should be adopted and further expanded to an even broader area if possible. This research provides economic evidence to inform efficient decision making in improving maternal and infant health in developing countries.

New variational Bayesian approaches for statistical data mining : with applications to profiling and differentiating habitual consumption behaviour of customers in the wireless telecommunication industry

This thesis investigates profiling and differentiating customers through the use of statistical data mining techniques. The business application of our work centres on examining individuals’ seldomly studied yet critical consumption behaviour over an extensive time period within the context of the wireless telecommunication industry; consumption behaviour (as oppose to purchasing behaviour) is behaviour that has been performed so frequently that it become habitual and involves minimal intentions or decision making. Key variables investigated are the activity initialised timestamp and cell tower location as well as the activity type and usage quantity (e.g., voice call with duration in seconds); and the research focuses are on customers’ spatial and temporal usage behaviour. The main methodological emphasis is on the development of clustering models based on Gaussian mixture models (GMMs) which are fitted with the use of the recently developed variational Bayesian (VB) method. VB is an efficient deterministic alternative to the popular but computationally demandingMarkov chainMonte Carlo (MCMC) methods. The standard VBGMMalgorithm is extended by allowing component splitting such that it is robust to initial parameter choices and can automatically and efficiently determine the number of components. The new algorithm we propose allows more effective modelling of individuals’ highly heterogeneous and spiky spatial usage behaviour, or more generally human mobility patterns; the term spiky describes data patterns with large areas of low probability mixed with small areas of high probability. Customers are then characterised and segmented based on the fitted GMM which corresponds to how each of them uses the products/services spatially in their daily lives; this is essentially their likely lifestyle and occupational traits. Other significant research contributions include fitting GMMs using VB to circular data i.e., the temporal usage behaviour, and developing clustering algorithms suitable for high dimensional data based on the use of VB-GMM.

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.

An approximate Bayesian computation approach for estimating parameters of complex environmental processes in a cellular automata

Modelling an environmental process involves creating a model structure and parameterising the model with appropriate values to accurately represent the process. Determining accurate parameter values for environmental systems can be challenging. Existing methods for parameter estimation typically make assumptions regarding the form of the Likelihood, and will often ignore any uncertainty around estimated values. This can be problematic, however, particularly in complex problems where Likelihoods may be intractable. In this paper we demonstrate an Approximate Bayesian Computational method for the estimation of parameters of a stochastic CA. We use as an example a CA constructed to simulate a range expansion such as might occur after a biological invasion, making parameter estimates using only count data such as could be gathered from field observations. We demonstrate ABC is a highly useful method for parameter estimation, with accurate estimates of parameters that are important for the management of invasive species such as the intrinsic rate of increase and the point in a landscape where a species has invaded. We also show that the method is capable of estimating the probability of long distance dispersal, a characteristic of biological invasions that is very influential in determining spread rates but has until now proved difficult to estimate accurately.

Using Bayesian methods for the estimation of uncertainty in complex statistical models

The research objectives of this thesis were to contribute to Bayesian statistical methodology by contributing to risk assessment statistical methodology, and to spatial and spatio-temporal methodology, by modelling error structures using complex hierarchical models. Specifically, I hoped to consider two applied areas, and use these applications as a springboard for developing new statistical methods as well as undertaking analyses which might give answers to particular applied questions. Thus, this thesis considers a series of models, firstly in the context of risk assessments for recycled water, and secondly in the context of water usage by crops. The research objective was to model error structures using hierarchical models in two problems, namely risk assessment analyses for wastewater, and secondly, in a four dimensional dataset, assessing differences between cropping systems over time and over three spatial dimensions. The aim was to use the simplicity and insight afforded by Bayesian networks to develop appropriate models for risk scenarios, and again to use Bayesian hierarchical models to explore the necessarily complex modelling of four dimensional agricultural data. The specific objectives of the research were to develop a method for the calculation of credible intervals for the point estimates of Bayesian networks; to develop a model structure to incorporate all the experimental uncertainty associated with various constants thereby allowing the calculation of more credible credible intervals for a risk assessment; to model a single day’s data from the agricultural dataset which satisfactorily captured the complexities of the data; to build a model for several days’ data, in order to consider how the full data might be modelled; and finally to build a model for the full four dimensional dataset and to consider the timevarying nature of the contrast of interest, having satisfactorily accounted for possible spatial and temporal autocorrelations. This work forms five papers, two of which have been published, with two submitted, and the final paper still in draft. The first two objectives were met by recasting the risk assessments as directed, acyclic graphs (DAGs). In the first case, we elicited uncertainty for the conditional probabilities needed by the Bayesian net, incorporated these into a corresponding DAG, and used Markov chain Monte Carlo (MCMC) to find credible intervals, for all the scenarios and outcomes of interest. In the second case, we incorporated the experimental data underlying the risk assessment constants into the DAG, and also treated some of that data as needing to be modelled as an ‘errors-invariables’ problem [Fuller, 1987]. This illustrated a simple method for the incorporation of experimental error into risk assessments. In considering one day of the three-dimensional agricultural data, it became clear that geostatistical models or conditional autoregressive (CAR) models over the three dimensions were not the best way to approach the data. Instead CAR models are used with neighbours only in the same depth layer. This gave flexibility to the model, allowing both the spatially structured and non-structured variances to differ at all depths. We call this model the CAR layered model. Given the experimental design, the fixed part of the model could have been modelled as a set of means by treatment and by depth, but doing so allows little insight into how the treatment effects vary with depth. Hence, a number of essentially non-parametric approaches were taken to see the effects of depth on treatment, with the model of choice incorporating an errors-in-variables approach for depth in addition to a non-parametric smooth. The statistical contribution here was the introduction of the CAR layered model, the applied contribution the analysis of moisture over depth and estimation of the contrast of interest together with its credible intervals. These models were fitted using WinBUGS [Lunn et al., 2000]. The work in the fifth paper deals with the fact that with large datasets, the use of WinBUGS becomes more problematic because of its highly correlated term by term updating. In this work, we introduce a Gibbs sampler with block updating for the CAR layered model. The Gibbs sampler was implemented by Chris Strickland using pyMCMC [Strickland, 2010]. This framework is then used to consider five days data, and we show that moisture in the soil for all the various treatments reaches levels particular to each treatment at a depth of 200 cm and thereafter stays constant, albeit with increasing variances with depth. In an analysis across three spatial dimensions and across time, there are many interactions of time and the spatial dimensions to be considered. Hence, we chose to use a daily model and to repeat the analysis at all time points, effectively creating an interaction model of time by the daily model. Such an approach allows great flexibility. However, this approach does not allow insight into the way in which the parameter of interest varies over time. Hence, a two-stage approach was also used, with estimates from the first-stage being analysed as a set of time series. We see this spatio-temporal interaction model as being a useful approach to data measured across three spatial dimensions and time, since it does not assume additivity of the random spatial or temporal effects.

Expert elicitation into Bayesian networks : understanding probability

An introduction to thinking about and understanding probability that highlights the main pits and trapfalls that befall logical reasoning

Expert elicitation into Bayesian networks, Part 1 of 4 : understanding probability

An introduction to elicitation of experts' probabilities, which illustrates common problems with reasoning and how to circumvent them during elicitation.

Expert elicitation into Bayesian networks, part 2 of 4: designing elicitation

An introduction to design of eliciting knowledge from experts.