A methodology for sensitivity analysis in inverse problems—application to a palaeoclimate study

We present a methodology for implementing sensitivity analysis in the case of inverse problems. This is motivated by a problem in palaeoclimatology, where interest lies in reconstruction of past climates from fossil data. In such situations fossil data are modelled as functions of climate variables, since in general ecological terms, variations in climate drive variations in fossil. However, prediction of climate variables are of interest, indicating the inverse nature of the problem. Technically, given a data set { x i , y i }; i = 1, … , n , and a probability distribution p ( y i | x i , θ), where θ is the set of model parameters, the problem is ‘inverse’ if, given a new observed y , prediction of the corresponding unknown x is required. On the other hand, the more usual forward problem considers prediction of y for given x . Sensitivity analysis in forward problems is about examining if the results are robust with respect to changes of the prior on θ. This has received a lot of attention in the statistical literature. However, in the case of inverse problems, a prior on x is required, in addition to the prior on θ. This complicates sensitivity analysis, at least operationally, and, to our knowledge, there exists no work in the literature that addresses this. In this paper, we propose a new methodology for sensitivity analysis in inverse problems, and discuss its application to a real, complex, palaeoclimate problem.