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.

Asymptotic behaviour of point processes with Markov-dependent intervals

This splitting techniques for MARKOV chains developed by NUMMELIN (1978a) and ATHREYA and NEY (1978b) are used to derive an imbedded renewal process in WOLD's point process with MARKOV-correlated intervals. This leads to a simple proof of renewal theorems for such processes. In particular, a key renewal theorem is proved, from which analogues to both BLACKWELL's and BREIMAN's forms of the renewal theorem can be deduced.

Validation-Based Sparse Gaussian Process Classifier Design

Gaussian processes (GPs) are promising Bayesian methods for classification and regression problems. Design of a GP classifier and making predictions using it is, however, computationally demanding, especially when the training set size is large. Sparse GP classifiers are known to overcome this limitation. In this letter, we propose and study a validation-based method for sparse GP classifier design. The proposed method uses a negative log predictive (NLP) loss measure, which is easy to compute for GP models. We use this measure for both basis vector selection and hyperparameter adaptation. The experimental results on several real-world benchmark data sets show better orcomparable generalization performance over existing methods.

Dynamic Bayesian network inferencing for non-homogeneous complex systems

Dynamic Bayesian Networks (DBNs) provide a versatile platform for predicting and analysing the behaviour of complex systems. As such, they are well suited to the prediction of complex ecosystem population trajectories under anthropogenic disturbances such as the dredging of marine seagrass ecosystems. However, DBNs assume a homogeneous Markov chain whereas a key characteristics of complex ecosystems is the presence of feedback loops, path dependencies and regime changes whereby the behaviour of the system can vary based on past states. This paper develops a method based on the small world structure of complex systems networks to modularise a non-homogeneous DBN and enable the computation of posterior marginal probabilities given evidence in forwards inference. It also provides an approach for an approximate solution for backwards inference as convergence is not guaranteed for a path dependent system. When applied to the seagrass dredging problem, the incorporation of path dependency can implement conditional absorption and allows release from the zero state in line with environmental and ecological observations. As dredging has a marked global impact on seagrass and other marine ecosystems of high environmental and economic value, using such a complex systems model to develop practical ways to meet the needs of conservation and industry through enhancing resistance and/or recovery is of paramount importance.

Hierarchical Bayesian modelling of early detection surveillance for plant pest invasions

Early detection surveillance programs aim to find invasions of exotic plant pests and diseases before they are too widespread to eradicate. However, the value of these programs can be difficult to justify when no positive detections are made. To demonstrate the value of pest absence information provided by these programs, we use a hierarchical Bayesian framework to model estimates of incursion extent with and without surveillance. A model for the latent invasion process provides the baseline against which surveillance data are assessed. Ecological knowledge and pest management criteria are introduced into the model using informative priors for invasion parameters. Observation models assimilate information from spatio-temporal presence/absence data to accommodate imperfect detection and generate posterior estimates of pest extent. When applied to an early detection program operating in Queensland, Australia, the framework demonstrates that this typical surveillance regime provides a modest reduction in the estimate that a surveyed district is infested. More importantly, the model suggests that early detection surveillance programs can provide a dramatic reduction in the putative area of incursion and therefore offer a substantial benefit to incursion management. By mapping spatial estimates of the point probability of infestation, the model identifies where future surveillance resources can be most effectively deployed.

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.

A Bayesian latent class safety performance function for identifying motor vehicle crash blackspots

The current state of the practice in Blackspot Identification (BSI) utilizes safety performance functions based on total crash counts to identify transport system sites with potentially high crash risk. This paper postulates that total crash count variation over a transport network is a result of multiple distinct crash generating processes including geometric characteristics of the road, spatial features of the surrounding environment, and driver behaviour factors. However, these multiple sources are ignored in current modelling methodologies in both trying to explain or predict crash frequencies across sites. Instead, current practice employs models that imply that a single underlying crash generating process exists. The model mis-specification may lead to correlating crashes with the incorrect sources of contributing factors (e.g. concluding a crash is predominately caused by a geometric feature when it is a behavioural issue), which may ultimately lead to inefficient use of public funds and misidentification of true blackspots. This study aims to propose a latent class model consistent with a multiple crash process theory, and to investigate the influence this model has on correctly identifying crash blackspots. We first present the theoretical and corresponding methodological approach in which a Bayesian Latent Class (BLC) model is estimated assuming that crashes arise from two distinct risk generating processes including engineering and unobserved spatial factors. The Bayesian model is used to incorporate prior information about the contribution of each underlying process to the total crash count. The methodology is applied to the state-controlled roads in Queensland, Australia and the results are compared to an Empirical Bayesian Negative Binomial (EB-NB) model. A comparison of goodness of fit measures illustrates significantly improved performance of the proposed model compared to the NB model. The detection of blackspots was also improved when compared to the EB-NB model. In addition, modelling crashes as the result of two fundamentally separate underlying processes reveals more detailed information about unobserved crash causes.

Bayesian analysis of community dynamics

Elucidating the mechanisms responsible for the patterns of species abundance, diversity, and distribution within and across ecological systems is a fundamental research focus in ecology. Species abundance patterns are shaped in a convoluted way by interplays between inter-/intra-specific interactions, environmental forcing, demographic stochasticity, and dispersal. Comprehensive models and suitable inferential and computational tools for teasing out these different factors are quite limited, even though such tools are critically needed to guide the implementation of management and conservation strategies, the efficacy of which rests on a realistic evaluation of the underlying mechanisms. This is even more so in the prevailing context of concerns over climate change progress and its potential impacts on ecosystems. This thesis utilized the flexible hierarchical Bayesian modelling framework in combination with the computer intensive methods known as Markov chain Monte Carlo, to develop methodologies for identifying and evaluating the factors that control the structure and dynamics of ecological communities. These methodologies were used to analyze data from a range of taxa: macro-moths (Lepidoptera), fish, crustaceans, birds, and rodents. Environmental stochasticity emerged as the most important driver of community dynamics, followed by density dependent regulation; the influence of inter-specific interactions on community-level variances was broadly minor. This thesis contributes to the understanding of the mechanisms underlying the structure and dynamics of ecological communities, by showing directly that environmental fluctuations rather than inter-specific competition dominate the dynamics of several systems. This finding emphasizes the need to better understand how species are affected by the environment and acknowledge species differences in their responses to environmental heterogeneity, if we are to effectively model and predict their dynamics (e.g. for management and conservation purposes). The thesis also proposes a model-based approach to integrating the niche and neutral perspectives on community structure and dynamics, making it possible for the relative importance of each category of factors to be evaluated in light of field data.

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.

Accelerator Mass Spectrometry and Bayesian Data Analysis

Accelerator mass spectrometry (AMS) is an ultrasensitive technique for measuring the concentration of a single isotope. The electric and magnetic fields of an electrostatic accelerator system are used to filter out other isotopes from the ion beam. The high velocity means that molecules can be destroyed and removed from the measurement background. As a result, concentrations down to one atom in 10^16 atoms are measurable. This thesis describes the construction of the new AMS system in the Accelerator Laboratory of the University of Helsinki. The system is described in detail along with the relevant ion optics. System performance and some of the 14C measurements done with the system are described. In a second part of the thesis, a novel statistical model for the analysis of AMS data is presented. Bayesian methods are used in order to make the best use of the available information. In the new model, instrumental drift is modelled with a continuous first-order autoregressive process. This enables rigorous normalization to standards measured at different times. The Poisson statistical nature of a 14C measurement is also taken into account properly, so that uncertainty estimates are much more stable. It is shown that, overall, the new model improves both the accuracy and the precision of AMS measurements. In particular, the results can be improved for samples with very low 14C concentrations or measured only a few times.

Communicative and Anticipatory Decision-making Supported by Bayesian Networks

The purpose of this research is to draw up a clear construction of an anticipatory communicative decision-making process and a successful implementation of a Bayesian application that can be used as an anticipatory communicative decision-making support system. This study is a decision-oriented and constructive research project, and it includes examples of simulated situations. As a basis for further methodological discussion about different approaches to management research, in this research, a decision-oriented approach is used, which is based on mathematics and logic, and it is intended to develop problem solving methods. The approach is theoretical and characteristic of normative management science research. Also, the approach of this study is constructive. An essential part of the constructive approach is to tie the problem to its solution with theoretical knowledge. Firstly, the basic definitions and behaviours of an anticipatory management and managerial communication are provided. These descriptions include discussions of the research environment and formed management processes. These issues define and explain the background to further research. Secondly, it is processed to managerial communication and anticipatory decision-making based on preparation, problem solution, and solution search, which are also related to risk management analysis. After that, a solution to the decision-making support application is formed, using four different Bayesian methods, as follows: the Bayesian network, the influence diagram, the qualitative probabilistic network, and the time critical dynamic network. The purpose of the discussion is not to discuss different theories but to explain the theories which are being implemented. Finally, an application of Bayesian networks to the research problem is presented. The usefulness of the prepared model in examining a problem and the represented results of research is shown. The theoretical contribution includes definitions and a model of anticipatory decision-making. The main theoretical contribution of this study has been to develop a process for anticipatory decision-making that includes management with communication, problem-solving, and the improvement of knowledge. The practical contribution includes a Bayesian Decision Support Model, which is based on Bayesian influenced diagrams. The main contributions of this research are two developed processes, one for anticipatory decision-making, and the other to produce a model of a Bayesian network for anticipatory decision-making. In summary, this research contributes to decision-making support by being one of the few publicly available academic descriptions of the anticipatory decision support system, by representing a Bayesian model that is grounded on firm theoretical discussion, by publishing algorithms suitable for decision-making support, and by defining the idea of anticipatory decision-making for a parallel version. Finally, according to the results of research, an analysis of anticipatory management for planned decision-making is presented, which is based on observation of environment, analysis of weak signals, and alternatives to creative problem solving and communication.

Markov random fields and spatial smoothing in improving of forest inventory estimates

Markov random fields (MRF) are popular in image processing applications to describe spatial dependencies between image units. Here, we take a look at the theory and the models of MRFs with an application to improve forest inventory estimates. Typically, autocorrelation between study units is a nuisance in statistical inference, but we take an advantage of the dependencies to smooth noisy measurements by borrowing information from the neighbouring units. We build a stochastic spatial model, which we estimate with a Markov chain Monte Carlo simulation method. The smooth values are validated against another data set increasing our confidence that the estimates are more accurate than the originals.

Semi-Supervised Classification Using Sparse Gaussian Process Regression

Gaussian Processes (GPs) are promising Bayesian methods for classification and regression problems. They have also been used for semi-supervised learning tasks. In this paper, we propose a new algorithm for solving semi-supervised binary classification problem using sparse GP regression (GPR) models. It is closely related to semi-supervised learning based on support vector regression (SVR) and maximum margin clustering. The proposed algorithm is simple and easy to implement. It gives a sparse solution directly unlike the SVR based algorithm. Also, the hyperparameters are estimated easily without resorting to expensive cross-validation technique. Use of sparse GPR model helps in making the proposed algorithm scalable. Preliminary results on synthetic and real-world data sets demonstrate the efficacy of the new algorithm.

On the Milito-Cruz adaptive control scheme for Markov chains

Milito and Cruz have introduced a novel adaptive control scheme for finite Markov chains when a finite parametrized family of possible transition matrices is available. The scheme involves the minimization of a composite functional of the observed history of the process incorporating both control and estimation aspects. We prove the a.s. optimality of a similar scheme when the state space is countable and the parameter space a compact subset ofR.

Structured systems methodology for evaluation of random interruptions in continuous process type manufacturing systems

A structured systems methodology was developed to analyse the problems of production interruptions occurring at random intervals in continuous process type manufacturing systems. At a macro level the methodology focuses on identifying suitable investment policies to reduce interruptions of a total manufacturing system that is a combination of several process plants. An interruption-tree-based simulation model was developed for macroanalysis. At a micro level the methodology focuses on finding the effects of alternative configurations of individual process plants on the overall system performance. A Markov simulation model was developed for microlevel analysis. The methodology was tested with an industry-specific application.