Bayes Days

How important is the organization of the data?

Leonel Hernandez

To Bayes or Bootstrap - that is the question

Ulrika Sahlin

Probabilistic Numerics

Peter Hristov

Modelling logical statements with credal networks

Diego Estrada-Lugo

Fault Tree Analysis (FTA) is a deductive method to find an undesired event by means of Boolean logic. This methodology has been implemented for decades in reliability analyses of complex engineering systems like the nuclear and oil industries to mention a few. The popularity of this method resides in the logical approach in which the failure event is studied and the simplicity of the relatively small amount of data that is required to model the events and components of the system. In this way, a failure probability of the so-called Top Event can be predicted and studied with relative ease. Despite the FTA success, there are several limitations that restrict the scope of the analysis and modellers had to explore another option. One of them is the Bayesian network (BN) method (or its imprecise probability version Credal networks, CNs), which consist in a probabilistic graphical model to study and understand the dependencies of events with epistemic uncertainty associated. The method allows to perform not only prognostic (like FTA) but also diagnostic analysis of specific components in the system. Furthermore, the CNs can incorporate new information in the form of evidence to update the probabilistic distributions of the model. However, given the significant advantages of CN over FTA, the latter is still very widely implemented. In this presentation, the principles about both methods are revealed and compared by means of didactic examples. and analogies and, to facilitate the discussion, we will ask some questions like why FTA is still widely used even though there are options, like CNs, that can produce generalised analyses? How much is too much in terms of the complexity of a tool that makes it hard to implement in the industry?

Modelling Multiple Longitudanal markers in Large Datasets using Variational Bayes

David Hughes

Data on multiple variables are now collected routinely at repeated visits in many clinical research areas. Mixed models are a popular way of analysing trends in longitudinal data. There is now increasing access to very large datasets, with both large numbers of patients, and large numbers of predictors. In these settings the current modelling approaches for mixed models can be very computationally intensive.
Mean Field Variational Bayes (MFVB) is a technique commonly used in Computer Science, which aims to find approximations to the posterior likelihood for Bayesian Models. In many settings these solutions have been shown to be very accurate and can be calculated in a fraction of the time required for equivalent MCMC models.
In this talk we describe how MFVB approaches can be used to estimate Bayesian Multivariate Mixed Models. I’ll discuss some of the computational obstacles I’ve faced, my current attempts to overcome them, and hopefully give opportunity for the audience to suggest better ones! We compare the time taken to estimate the models, and the accuracy of model parameter estimates through a variety of simulation studies and also through application to a large diabetes dataset aiming to model variables linked to the risk of developing Sight Threatening Diabetic Retinopathy.
The MFVB approach for Bayesian Mixed Models allows fast and approximate inference to be made in a variety of clinical models.

Prior knowledge, proportions and probabilities of probabilities

Alejandro Diaz, Institute for Risk and Uncertainty, University of Liverpool, UK

Bayesian linear regression and hierarchical models

Alfredo Garbuno, Institute for Risk and Uncertainty, University of Liverpool, UK

Quantifying uncertainty using data and experts

Ullrika Sahlin,Centre for Environmental and Climate Research, Lund University, SE

This tutorial introduces some of the basic principles to quantify uncertainty by Bayesian probability. I will demonstrate a way to quantify uncertainty by integrating expert’s knowledge and data. Participants can follow practical examples in R, using existing R packages for expert’s elicitation (SHELF) and sampling from the posterior (rjags requiring JAGS). The first example is a simple risk classification problem under sparse information and several experts with differing judgements. The second example is the familiar task to quantify uncertainty in input parameters of an assessment model using different sources of information and where uncertainty in assessment output matters.

Approximate Bayesian computation (ABC)

Brendan McCabe, Economics, Finance and Accounting, University of Liverpool, UK

This tutorial looks at how to do Bayesian Inference when it is too difficult to calculate the true likelihood and hence the exact posterior. (This is a Bayesian version of frequentist ‘indirect inference’ really.) We use model based summary statistics to match simulations form the assumed (difficult) model with the actual data at hand. Conventional approaches to ABC emphasize the role of parameter estimation but, in time series problems, forecasting is often the focus of attention and so it is to this dimension we direct our efforts. The role of Bayesian consistency is highlighted.

Generating samples from strange probability distributions

Peter Green, Institute for Risk and Uncertainty, University of Liverpool, UK

When conducting a probabilistic analysis, we often end up having to generate samples from a probability distribution. This, for example, is a crucial part of Monte Carlo simulations. For better-known probability distributions (Gaussian, uniform etc.), some simple tricks allow us to generate samples without too much difficulty. For the more ‘strange-looking’ distributions – which commonly arise in a Bayesian analysis – the problem becomes more difficult. This tutorial describes methods which can be used to generate samples from generic probability distributions. They often form an essential part of a Bayesian analysis. The tutorial is aimed at beginners, and will cover basic sampling algorithms before describing Markov chain Monte Carlo (MCMC) and importance sampling algorithms. Sample Matlab code will also be provided.