Bayesian Analysis of ODE's: solver optimal accuracy and Bayes factors

In most relevant 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 deriving from the error in the numerical solver. To compare a numerical and the theoretical posterior distributions we propose to use Bayes Factors (BF), considering both of them as models for the data at hand. We prove that the theoretical vs a numerical posterior BF tends to 1, in the same order (of the step size used) as the numerical forward map solver does. For higher order solvers (eg. Runge-Kutta) the Bayes Factor 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 while all inference results are identical to using a solver with a much finer time step.Comment: 28 pages, 6 figure

Bayesian analysis of 210Pb dating

In many studies of environmental change of the past few centuries, 210Pb dating is used to obtain chronologies for sedimentary sequences. One of the most commonly used approaches to estimate the ages of depths in a sequence is to assume a constant rate of supply (CRS) or influx of `unsupported' 210Pb from the atmosphere, together with a constant or varying amount of `supported' 210Pb. Current 210Pb dating models do not use a proper statistical framework and thus provide poor estimates of errors. Here we develop a new model for 210Pb dating, where both ages and values of supported and unsupported 210Pb form part of the parameters. We apply our model to a case study from Canada as well as to some simulated examples. Our model can extend beyond the current CRS approach, deal with asymmetric errors and mix 210Pb with other types of dating, thus obtaining more robust, realistic and statistically better defined estimates.Comment: 22 Pages, 4 Figure