53 research outputs found
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
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