Adaptive Mixture Modelling Metropolis Methods for Bayesian Analysis of Non-linear State-Space Models.

We describe a strategy for Markov chain Monte Carlo analysis of non-linear, non-Gaussian state-space models involving batch analysis for inference on dynamic, latent state variables and fixed model parameters. The key innovation is a Metropolis-Hastings method for the time series of state variables based on sequential approximation of filtering and smoothing densities using normal mixtures. These mixtures are propagated through the non-linearities using an accurate, local mixture approximation method, and we use a regenerating procedure to deal with potential degeneracy of mixture components. This provides accurate, direct approximations to sequential filtering and retrospective smoothing distributions, and hence a useful construction of global Metropolis proposal distributions for simulation of posteriors for the set of states. This analysis is embedded within a Gibbs sampler to include uncertain fixed parameters. We give an example motivated by an application in systems biology. Supplemental materials provide an example based on a stochastic volatility model as well as MATLAB code.

Bayesian inference of the number of factors in gene-expression analysis: application to human virus challenge studies.

BACKGROUND: Nonparametric Bayesian techniques have been developed recently to extend the sophistication of factor models, allowing one to infer the number of appropriate factors from the observed data. We consider such techniques for sparse factor analysis, with application to gene-expression data from three virus challenge studies. Particular attention is placed on employing the Beta Process (BP), the Indian Buffet Process (IBP), and related sparseness-promoting techniques to infer a proper number of factors. The posterior density function on the model parameters is computed using Gibbs sampling and variational Bayesian (VB) analysis. RESULTS: Time-evolving gene-expression data are considered for respiratory syncytial virus (RSV), Rhino virus, and influenza, using blood samples from healthy human subjects. These data were acquired in three challenge studies, each executed after receiving institutional review board (IRB) approval from Duke University. Comparisons are made between several alternative means of per-forming nonparametric factor analysis on these data, with comparisons as well to sparse-PCA and Penalized Matrix Decomposition (PMD), closely related non-Bayesian approaches. CONCLUSIONS: Applying the Beta Process to the factor scores, or to the singular values of a pseudo-SVD construction, the proposed algorithms infer the number of factors in gene-expression data. For real data the "true" number of factors is unknown; in our simulations we consider a range of noise variances, and the proposed Bayesian models inferred the number of factors accurately relative to other methods in the literature, such as sparse-PCA and PMD. We have also identified a "pan-viral" factor of importance for each of the three viruses considered in this study. We have identified a set of genes associated with this pan-viral factor, of interest for early detection of such viruses based upon the host response, as quantified via gene-expression data.

SMERED: A Bayesian Approach to Graphical Record Linkage and De-duplication

We propose a novel unsupervised approach for linking records across arbitrarily many files, while simultaneously detecting duplicate records within files. Our key innovation is to represent the pattern of links between records as a {\em bipartite} graph, in which records are directly linked to latent true individuals, and only indirectly linked to other records. This flexible new representation of the linkage structure naturally allows us to estimate the attributes of the unique observable people in the population, calculate $k$-way posterior probabilities of matches across records, and propagate the uncertainty of record linkage into later analyses. Our linkage structure lends itself to an efficient, linear-time, hybrid Markov chain Monte Carlo algorithm, which overcomes many obstacles encountered by previously proposed methods of record linkage, despite the high dimensional parameter space. We assess our results on real and simulated data.

Generalized Markov branching processes

A generalized Markov Brnching Process (GMBP) is a Markov branching model where the infinitesimal branching rates are modified with an interaction index. It is proved that there always exists only one GMBP. An associated differential-integral equation is derived. The extinction probalility and the mean and conditional mean extinction times are obtained. Ergodicity and stability of GMBP with resurrection are also considered. Easy checking criteria are established for ordinary and strong ergodicty. The equilibrium distribution is given in an elegant closed form. The probability meaning of our results is clear and thus explained.

Uniqueness criteria for continuous-time Markov chains with general transition structures

We derive necessary and sufficient conditions for the existence of bounded or summable solutions to systems of linear equations associated with Markov chains. This substantially extends a famous result of G. E. H. Reuter, which provides a convenient means of checking various uniqueness criteria for birth-death processes. Our result allows chains with much more general transition structures to be accommodated. One application is to give a new proof of an important result of M. F. Chen concerning upwardly skip-free processes. We then use our generalization of Reuter's lemma to prove new results for downwardly skip-free chains, such as the Markov branching process and several of its many generalizations. This permits us to establish uniqueness criteria for several models, including the general birth, death, and catastrophe process, extended branching processes, and asymptotic birth-death processes, the latter being neither upwardly skip-free nor downwardly skip-free.

General Harris regularity criterion for non-linear Markov branching processes

We extend the Harris regularity condition for ordinary Markov branching process to a more general case of non-linear Markov branching process. A regularity criterion which is very easy to check is obtained. In particular, we prove that a super-linear Markov branching process is regular if and only if the per capita offspring mean is less than or equal to I while a sub-linear Markov branching process is regular if the per capita offspring mean is finite. The Harris regularity condition then becomes a special case of our criterion.

Probabilistic approach in weighted Markov branching processes

This note provides a new probabilistic approach in discussing the weighted Markov branching process (WMBP) which is a natural generalisation of the ordinary Markov branching process. Using this approach, some important characteristics regarding the hitting times of such processes can be easily obtained. In particular, the closed forms for the mean extinction time and conditional mean extinction time are presented. The explosion behaviour of the process is investigated and the mean explosion time is derived. The mean global holding time and the mean total survival time are also obtained. The close link between these newly developed processes and the well-known compound Poisson processes is investigated. It is revealed that any weighted Markov branching process (WMBP) is a random time change of a compound Poisson process.

Flexible paleoclimate age-depth models using an autoregressive gamma process

Radiocarbon dating is routinely used in paleoecology to build chronolo- gies of lake and peat sediments, aiming at inferring a model that would relate the sediment depth with its age. We present a new approach for chronology building (called “Bacon”) that has received enthusiastic attention by paleoecologists. Our methodology is based on controlling core accumulation rates using a gamma autoregressive semiparametric model with an arbitrary number of subdivisions along the sediment. Using prior knowledge about accumulation rates is crucial and informative priors are routinely used. Since many sediment cores are currently analyzed, using different data sets and prior distributions, a robust (adaptive) MCMC is very useful. We use the t-walk (Christen and Fox, 2010), a self adjusting, robust MCMC sampling algorithm, that works acceptably well in many situations. Outliers are also addressed using a recent approach that considers a Student-t model for radiocarbon data. Two examples are presented here, that of a peat core and a core from a lake, and our results are compared with other approaches.

EXPERT-SYSTEM SUPPORT USING BAYESIAN BELIEF NETWORKS IN THE DIAGNOSIS OF FINE-NEEDLE ASPIRATION BIOPSY SPECIMENS OF THE BREAST

Aim-To develop an expert system model for the diagnosis of fine needle aspiration cytology (FNAC) of the breast.

Methods-Knowledge and uncertainty were represented in the form of a Bayesian belief network which permitted the combination of diagnostic evidence in a cumulative manner and provided a final probability for the possible diagnostic outcomes. The network comprised 10 cytological features (evidence nodes), each independently linked to the diagnosis (decision node) by a conditional probability matrix. The system was designed to be interactive in that the cytopathologist entered evidence into the network in the form of likelihood ratios for the outcomes at each evidence node.

Results-The efficiency of the network was tested on a series of 40 breast FNAC specimens. The highest diagnostic probability provided by the network agreed with the cytopathologists' diagnosis in 100% of cases for the assessment of discrete, benign, and malignant aspirates. A typical probably benign cases were given probabilities in favour of a benign diagnosis. Suspicious cases tended to have similar probabilities for both diagnostic outcomes and so, correctly, could not be assigned as benign or malignant. A closer examination of cumulative belief graphs for the diagnostic sequence of each case provided insight into the diagnostic process, and quantitative data which improved the identification of suspicious cases.

Conclusion-The further development of such a system will have three important roles in breast cytodiagnosis: (1) to aid the cytologist in making a more consistent and objective diagnosis; (2) to provide a teaching tool on breast cytological diagnosis for the non-expert; and (3) it is the first stage in the development of a system capable of automated diagnosis through the use of expert system machine vision.

Efficient Marginal Likelihood Computation for Gaussian Process Regression

In a Bayesian learning setting, the posterior distribution of a predictive model arises from a trade-off between its prior distribution and the conditional likelihood of observed data. Such distribution functions usually rely on additional hyperparameters which need to be tuned in order to achieve optimum predictive performance; this operation can be efficiently performed in an Empirical Bayes fashion by maximizing the posterior marginal likelihood of the observed data. Since the score function of this optimization problem is in general characterized by the presence of local optima, it is necessary to resort to global optimization strategies, which require a large number of function evaluations. Given that the evaluation is usually computationally intensive and badly scaled with respect to the dataset size, the maximum number of observations that can be treated simultaneously is quite limited. In this paper, we consider the case of hyperparameter tuning in Gaussian process regression. A straightforward implementation of the posterior log-likelihood for this model requires O(N^3) operations for every iteration of the optimization procedure, where N is the number of examples in the input dataset. We derive a novel set of identities that allow, after an initial overhead of O(N^3), the evaluation of the score function, as well as the Jacobian and Hessian matrices, in O(N) operations. We prove how the proposed identities, that follow from the eigendecomposition of the kernel matrix, yield a reduction of several orders of magnitude in the computation time for the hyperparameter optimization problem. Notably, the proposed solution provides computational advantages even with respect to state of the art approximations that rely on sparse kernel matrices.

Métodos probabilísticos e estatísticos na gestão de fogos florestais

Tese de doutoramento, Estatística e Investigação Operacional (Probabilidades e Estatística), Universidade de Lisboa, Faculdade de Ciências, 2014

A review of dynamic Bayesian network techniques with applications in healthcare risk modelling

Coping with an ageing population is a major concern for healthcare organisations around the world. The average cost of hospital care is higher than social care for older and terminally ill patients. Moreover, the average cost of social care increases with the age of the patient. Therefore, it is important to make efficient and fair capacity planning which also incorporates patient centred outcomes. Predictive models can provide predictions which their accuracy can be understood and quantified. Predictive modelling can help patients and carers to get the appropriate support services, and allow clinical decision-makers to improve care quality and reduce the cost of inappropriate hospital and Accident and Emergency admissions. The aim of this study is to provide a review of modelling techniques and frameworks for predictive risk modelling of patients in hospital, based on routinely collected data such as the Hospital Episode Statistics database. A number of sub-problems can be considered such as Length-of-Stay and End-of-Life predictive modelling. The methodologies in the literature are mainly focused on addressing the problems using regression methods and Markov models, and the majority lack generalisability. In some cases, the robustness, accuracy and re-usability of predictive risk models have been shown to be improved using Machine Learning methods. Dynamic Bayesian Network techniques can represent complex correlations models and include small probabilities into the solution. The main focus of this study is to provide a review of major time-varying Dynamic Bayesian Network techniques with applications in healthcare predictive risk modelling.

Time Reversibility of Stationary Regular Finite State Markov Chains

We propose an alternate parameterization of stationary regular finite-state Markov chains, and a decomposition of the parameter into time reversible and time irreversible parts. We demonstrate some useful properties of the decomposition, and propose an index for a certain type of time irreversibility. Two empirical examples illustrate the use of the proposed parameter, decomposition and index. One involves observed states; the other, latent states.

Some Applications of Markov Additive Processes as Models in Insurance and Financial Mathematics

Cette thèse est principalement constituée de trois articles traitant des processus markoviens additifs, des processus de Lévy et d'applications en finance et en assurance. Le premier chapitre est une introduction aux processus markoviens additifs (PMA), et une présentation du problème de ruine et de notions fondamentales des mathématiques financières. Le deuxième chapitre est essentiellement l'article "Lévy Systems and the Time Value of Ruin for Markov Additive Processes" écrit en collaboration avec Manuel Morales et publié dans la revue European Actuarial Journal. Cet article étudie le problème de ruine pour un processus de risque markovien additif. Une identification de systèmes de Lévy est obtenue et utilisée pour donner une expression de l'espérance de la fonction de pénalité actualisée lorsque le PMA est un processus de Lévy avec changement de régimes. Celle-ci est une généralisation des résultats existant dans la littérature pour les processus de risque de Lévy et les processus de risque markoviens additifs avec sauts "phase-type". Le troisième chapitre contient l'article "On a Generalization of the Expected Discounted Penalty Function to Include Deficits at and Beyond Ruin" qui est soumis pour publication. Cet article présente une extension de l'espérance de la fonction de pénalité actualisée pour un processus subordinateur de risque perturbé par un mouvement brownien. Cette extension contient une série de fonctions escomptée éspérée des minima successives dus aux sauts du processus de risque après la ruine. Celle-ci a des applications importantes en gestion de risque et est utilisée pour déterminer la valeur espérée du capital d'injection actualisé. Finallement, le quatrième chapitre contient l'article "The Minimal entropy martingale measure (MEMM) for a Markov-modulated exponential Lévy model" écrit en collaboration avec Romuald Hervé Momeya et publié dans la revue Asia-Pacific Financial Market. Cet article présente de nouveaux résultats en lien avec le problème de l'incomplétude dans un marché financier où le processus de prix de l'actif risqué est décrit par un modèle exponentiel markovien additif. Ces résultats consistent à charactériser la mesure martingale satisfaisant le critère de l'entropie. Cette mesure est utilisée pour calculer le prix d'une option, ainsi que des portefeuilles de couverture dans un modèle exponentiel de Lévy avec changement de régimes.