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

Numerical Solution of the Cauchy Problem for the Laplace Equation: A Deterministic and Bayesian Approach

We consider a statistical inversion computational model with Gaussian distributions for the numerical solution of the Cauchy problem for the Laplace equation. The a priori model is built up from Gaussian Markov random fields. Different precision matrices for the Cauchy problem are introduced. We take advantage of the relationship between the a priori distribution and traditional Tikhonov regularization to propose different models where smooth and non-smooth regularization is possible. A low range analysis allow us to estimate the optimal dimension of data and its relation to the the unknown