Marcos A. Capistrán

Bayesian Analysis of ODEs: Solver Optimal Accuracy and Bayes Factors

In most cases in the Bayesian analysis of ODE inverse problems, a numerical solver needs to be used. Therefore, we cannot work with the exact theoretical posterior distribution but only with an approximate posterior derived from the error in the numerical solver. To compare an approximate posterior distribution with the theoretical one, we propose using Bayes factors (BFs), considering both of them as models for the data at hand. From a theoretical point of view, we prove that the theoretical vs. numerical posterior BF tends to 1, in the same order as the numerical solver used. In practice, we illustrate the fact that for higher order solvers (e.g., Runge--Kutta) the BF is already nearly 1 for step sizes that would take far less computational effort. Considerable CPU time may be saved by using coarser solvers that nevertheless produce practically error-free posteriors. Two examples are presented where nearly 90% CPU time is saved, with all inference results being identical to those obtained using a solver...

Effective Parameter Dimension via Bayesian Model Selection in the Inverse Acoustic Scattering Problem

We address a prototype inverse scattering problem in the interface of applied mathematics, statistics, and scientific computing. We pose the acoustic inverse scattering problem in a Bayesian inference perspective and simulate from the posterior distribution using MCMC. The PDE forward map is implemented using high performance computing methods. We implement a standard Bayesian model selection method to estimate an effective number of Fourier coefficients that may be retrieved from noisy data within a standard formulation.

Towards uncertainty quantification and inference in the stochastic SIR epidemic model

In this paper we address the problem of estimating the parameters of Markov jump processes modeling epidemics and introduce a novel method to conduct inference when data consists on partial observations in one of the state variables. We take the classical stochastic SIR model as a case study. Using the inverse-size expansion of van Kampen we obtain approximations for the first and second moments of the state variables. These approximate moments are in turn matched to the moments of an inputed Generic Discrete distribution aimed at generating an approximate likelihood that is valid both for low count or high count data. We conduct a full Bayesian inference using informative priors. Estimations and predictions are obtained both in a synthetic data scenario and in two Dengue fever case studies.

On Efficient Numerical Posterior Distribution Error Control in Bayesian Uncertainty Quantification of Inverse Problems

In this paper we explore a strategy towards efficient numerical error control in the Bayesian analysis of inverse problems defined by a system of Ordinary Differential Equations (ODE). We use a simple model from genetic expression as a case study to show the advantage of bounding the Bayes factors in terms of the global error in the numerical method to solve the ODE system. Our findings suggest a wide research avenue ahead of us to quantifying the uncertainty in Bayesian inverse problems in terms of the existing wealth of theory of numerical methods for differential equations.

A Bayesian Outbreak Detection Method for Influenza-Like Illness

Epidemic outbreak detection is an important problem in public health and the development of reliable methods for outbreak detection remains an active research area. In this paper we introduce a Bayesian method to detect outbreaks of influenza-like illness from surveillance data. The rationale is that, during the early phase of the outbreak, surveillance data changes from autoregressive dynamics to a regime of exponential growth. Our method uses Bayesian model selection and Bayesian regression to identify the breakpoint. No free parameters need to be tuned. However, historical information regarding influenza-like illnesses needs to be incorporated into the model. In order to show and discuss the performance of our method we analyze synthetic, seasonal, and pandemic outbreak data.