## Credal Model Averaging For dealing with Uncertainty in Environmental Modelling

Acronym CREDO

Abstract In this project we address the problem of dealing with uncertainty in models of environmental systems. Such systems are often modelled by complex sets of equations, which tend to be overparametrised. Different models can therefore be calibrated to describe the same phenomenon and the model parameters display a complex interaction structure. A well-founded statistical framework to deal with model uncertainty is Bayesian model averaging (BMA), which prescribes to statistically aggregate the predictions issued by a set of several models, rather than relying on a single model. Yet, an open problem of BMA is how to set the prior distribution, which represents our beliefs about the models. A methodological solution to this problem would require to model prior ignorance, i.e., to regard as feasible all the priors that satisfy some general conditions, rather than ask the investigator to specify a unique, precise prior distribution. To this purpose, we propose to develop CMA (credal model averaging), i.e., to extend BMA to imprecise probabilities (or credal sets). CMA will output a set of posterior distributions (derived from the set or priors) for the variable of interest, rather than a single posterior distribution (derived from a single prior) as in BMA. The developed methodology will be validated by two applications, where model uncertainty plays a key role and nevertheless only little work has been done using BMA. The first one regards epidemiological studies of time series of air pollution and human health; US and European studies report indeed a difference of as much as 50-100% in the estimated air pollution effects, depending on the modelling approach adopted, and point out that there is no valid rule to choose from among them. The second application regards demographic models of animal populations; they are typically characterized by small data sets (for instance, 20 data). Usually, a single model is chosen to issue the predictions; moreover, its structure is also deeply analyzed, in order to point out how the different factors affect the population dynamics. In this case, not accounting for model uncertainty might lead to sub-optimal predictions, and also to unsafe conclusions about the population regulation.

SUPSI Organisations Involved Dalle Molle Institute for Artificial Intelligence

People involved Marco Zaffalon, Giorgio Corani

Managers Andrea Emilio Rizzoli

Date of Project Start 1 October 2008

Date of Project Completion 30 September 2010