Paolo Giudici

Efficient construction of reversible jump Markov chain Monte Carlo proposal distributions

The major implementational problem for reversible jump Markov chain Monte Carlo methods is that there is commonly no natural way to choose jump proposals since there is no Euclidean structure in the parameter space to guide our choice. We consider mechanisms for guiding the choice of proposal. The first group of methods is based on an analysis of acceptance probabilities for jumps. Essentially, these methods involve a Taylor series expansion of the acceptance probability around certain canonical jumps and turn out to have close connections to Langevin algorithms. The second group of methods generalizes the reversible jump algorithm by using the so-called saturated space approach. These allow the chain to retain some degree of memory so that, when proposing to move from a smaller to a larger model, information is borrowed from the last time that the reverse move was performed. The main motivation for this paper is that, in complex problems, the probability that the Markov chain moves between such spaces may be prohibitively small, as the probability mass can be very thinly spread across the space. Therefore, finding reasonable jump proposals becomes extremely important. We illustrate the procedure by using several examples of reversible jump Markov chain Monte Carlo applications including the analysis of autoregressive time series, graphical Gaussian modelling and mixture modelling. Copyright 2003 Royal Statistical Society.

Improving Markov Chain Monte Carlo Model Search for Data Mining

The motivation of this paper is the application of MCMC model scoring procedures to data mining problems, involving a large number of competing models and other relevant model choice aspects.To achieve this aim we analyze one of the most popular Markov Chain Monte Carlo methods for structural learning in graphical models, namely, the MC3 algorithm proposed by D. Madigan and J. York (International Statistical Review, 63, 215–232, 1995). Our aim is to improve their algorithm to make it an effective and reliable tool in the field of data mining. In such context, typically highly dimensional in the number of variables, little can be known a priori and, therefore, a good model search algorithm is crucial.We present and describe in detail our implementation of the MC3 algorithm, which provides an efficient general framework for computations with both Directed Acyclic Graphical (DAG) models and Undirected Decomposable Models (UDG). We believe that the possibility of commuting easily between the two classes of models constitutes an important asset in data mining, where an a priori knowledge of causal effects is usually difficult to establish.Furthermore, in order to improve the MC3 method we propose provide several graphical monitors which can help extracting results and assessing the goodness of the Markov chain Monte Carlo approximation to the posterior distribution of interest.We apply our proposed methodology first to the well-known coronary heart disease dataset (D. Edwards &; T. Havránek, Biometrika, 72:2, 339–351, 1985). We then introduce a novel data mining application which concerns market basket analysis.

Data mining of association structures to model consumer behaviour

We describe how statistical association models and, specifically, log linear and graphical models, can be usefully employed to study consumer behaviours. We describe some methodological problems related to the implementation of discrete graphical models for market basket analysis data. In particular, we shall discuss model selection procedures.

Bayesian networks for enterprise risk assessment

According to different typologies of activity and priority, risks can assume diverse meanings and it can be assessed in different ways.

A Bayesian approach to estimate the marginal loss distributions in operational risk management

One of the main problems in operational risk management is the lack of loss data, which affects the parameter estimates of the marginal distributions of the losses. The principal reason is that financial institutions only started to collect operational loss data a few years ago, due to the relatively recent definition of this type of risk. Considering this drawback, the employment of Bayesian methods and simulation tools could be a natural solution to the problem. The use of Bayesian methods allows us to integrate the scarce and, sometimes, inaccurate quantitative data collected by the bank with prior information provided by experts. An original proposal is a Bayesian approach for modelling operational risk and for calculating the capital required to cover the estimated risks. Besides this methodological innovation a computational scheme, based on Markov chain Monte Carlo simulations, is required. In particular, the application of the MCMC method to estimate the parameters of the marginals shows advantages in terms of a reduction of capital charge according to different choices of the marginal loss distributions.

Copulae and Operational Risks

The management of Operational Risks has always been difficult due to the high number of variables to work with and their complex multivariate distribution. A Copula is a statistic tool which has been recently used in finance and engineering to build flexible joint distributions in order to model a high number of variables. The goal of this paper is to propose its use to model Operational Risks, by showing its benefits with an empirical example.

Graphical Network Models for International Financial Flows

The late-2000s financial crisis stressed the need to understand the world financial system as a network of countries, where cross-border financial linkages play a fundamental role in the spread of systemic risks. Financial network models, which take into account the complex interrelationships between countries, seem to be an appropriate tool in this context. To improve the statistical performance of financial network models, we propose to generate them by means of multivariate graphical models. We then introduce Bayesian graphical models, which can take model uncertainty into account, and dynamic Bayesian graphical models, which provide a convenient framework to model temporal cross-border data, decomposing the model into autoregressive and contemporaneous networks. The article shows how the application of the proposed models to the Bank of International Settlements locational banking statistics allows the identification of four distinct groups of countries, that can be considered central in systemic risk contagion.

Association Models for Web Mining

We describe how statistical association models and, specifically, graphical models, can be usefully employed to model web mining data. We describe some methodological problems related to the implementation of discrete graphical models for web mining data. In particular, we discuss model selection procedures.

Mixtures of products of Dirichlet processes for variable selection in survival analysis

A very important problem in survival analysis is the accurate selection of the relevant prognostic explanatory variables. We propose a novel approach, based on mixtures of products of Dirichlet process priors, that provides a formal inferential tool to compare the explanatory power of each covariate, in terms of the marginal likelihood attached to the induced partitions of the observations. Our proposed model is Bayesian nonparametric, and, thus, keeps the amount of model specification to a minimum, increasing robustness of the final inferences.

Nonparametric estimation of survival functions by means of partial exchangeability structures

In the causal analysis of survival data a time-based response is related to a set of explanatory variables. However, selection and proper design of the latter may become a difficult task, particularly in the preliminary stage, when the information is limited. We propose an alternative nonparametric approach to estimate the survival function which allows one to evaluate the relative importance of each potential explanatory variable, in a simple and exploratory fashion. To achieve this aim, each of the explanatory variables is used to partition the observed survival times. The observations are assumed to be partially exchangeable according to such partition. We then consider, conditionally on each partition, a hierarchical nonparametric Bayesian model on the hazard functions. In order to measure the importance of each explanatory variable, we derive the posterior probability of the corresponding partition. Such probabilities are then employed to estimate the hazard functions by averaging the estimated conditional hazard over the set of all entertained partitions.