Marcelo Pereyra

Bayesian inference by convex optimisation: theory, methods, and algorithms.

Abstract:

Convex optimisation has become the main Bayesian computation methodology in many areas of data science such as mathematical imaging and machine learning, where high dimensionality is often addressed by using models that are log-concave and where maximum-a-posteriori (MAP) estimation can be performed efficiently by optimisation. The first part of this talk presents a new decision-theoretic derivation of MAP estimation and shows that, contrary to common belief, under log-concavity MAP estimators are proper Bayesian estimators. A main novelty is that the derivation is based on differential geometry. Following on from this, we establish universal theoretical guarantees for the estimation error involved and show estimation stability in high dimensions. Moreover, the second part of the talk describes a new general methodology for approximating Bayesian high-posterior-density regions in log-concave models. The approximations are derived by using recent concentration of measure results related to information theory, and can be computed very efficiently, even in large-scale problems, by using convex optimisation techniques. The approximations also have favourable theoretical properties, namely they outer-bound the true high-posterior-density credibility regions, and they are stable with respect to model dimension. The proposed methodology is finally illustrated on two high-dimensional imaging inverse problems related to tomographic reconstruction and sparse deconvolution, where they are used to explore the uncertainty about the solutions, and where convex-optimisation-empowered proximal Markov chain Monte Carlo algorithms are used as benchmark to compute exact credible regions and measure the approximation error.

2:00-3:00

Michael Pitt, King's College London

Title: Developments in particle filtering and MCMC

Abstract: In this talk I will give an overview of the developments of sequential Monte Carlo methods and the models to which they can be applied. I will talk about the design and implementation of the auxiliary particle filter. The use of particle filters in pseudo Metropolis Hastings will be introduced. The benefits and limitations of this approach will be highlighted. Optimisation, particularly in the choice of the number of particles, will be covered.

3:30-4:30

Speaker: Panayiota Touloupou (Warwick)

Title: Scalable inference for Markov and semi-Markov epidemic models

Abstract: Epidemiological data from infectious disease studies are very often gathered longitudinally, where a cohort of individuals is sampled through time. Inferences for this type of data are complicated by the fact that the data are usually incomplete, in the sense that the times of acquiring and clearing infection are not directly observed, making the evaluation of the model likelihood intractable. As a result, considerable progress has been made on developing techniques for imputation of the hidden state process, mainly using MCMC methods.

However, as the dimensionality and complexity of the data increases some of these methods become inefficient, either because they produce chains with high autocorrelation or because they become computationally intractable. Motivated by this fact, we develop a novel MCMC algorithm, which is modification of the Forward Filtering Backward Sampling algorithm, that achieves a good balance between computational complexity and mixing properties, and thus can be used to analyse epidemics on large populations. Even though our approach is developed under the assumption of a Markov model, we show how this assumption can be relaxed leading to minor modifications in the algorithm. The performance of our method is assessed on both simulated and real data, considering models with simple structure but also complex dynamics. Joint work with Simon Spencer and Barbel Finkenstadt.

Paul Birrell (MRC Biostatistics Unit, Cambridge)

Towards Computationally Efficient Epidemic Inference

In a pandemic where infection is widespread, there is no direct observation of the infection processes. Instead information comes from a variety of surveillance data schemes that are prone to noise, contamination, bias and sparse sampling. To form an accurate impression of the epidemic and to be able to make forecasts of its evolution, therefore, as many of these data streams as possible need to be assimilated into a single integrated analysis. The result of this is that the transmission model describing the infection process and the linked observation models can become computationally demanding, limiting the capacity for statistical inference in real-time.

I will discuss some of our attempts at making the inferential process more efficient, with particular focus on dynamic emulation, where the computationally expensive epidemic model is replaced by a more readily evaluated proxy, a time-evolving Gaussian process trained on a (relatively) small number of model runs at key input values, training that can be done a priori.