Probability density functions, ensembles and limits to statistical predictability

Ensemble predictions are an integral part of routine weather and climate prediction because of the sensitivity of such projections to the specification of the initial state. In many discussions it is tacitly assumed that ensembles are equivalent to probability distribution functions (p.d.fs) of the random variables of interest. In general for vector valued random variables this is not the case (not even approximately) since practical ensembles do not adequately sample the high dimensional state spaces of dynamical systems of practical relevance. In this talk these ideas are placed on a rigorous footing using concepts derived from Bayesian analysis and information theory. In particular it is shown that ensembles must imply a coarse graining of state space and that this coarse graining implies loss of information relative to the converged p.d.f. To cope with the needed coarse graining in the context of practical applications, a heirarchy of entropic functionals is introduced. These measure the information content of multivariate marginal distributions of increasing order. For fully converged distributions (i.e. p.d.f.s) these functionals form a strictly ordered heirarchy. As one proceeds up the heirarchy however, increasingly coarser partitions are required by the functionals which implies that the strict ordering of the p.d.f. based functionals breaks down. This breakdown is symptomatic of the poor sampling by ensembles of high dimensional state spaces and is unavoidable for most practical applications.

In the second part of the talk the theoretical machinery developed above is applied to the practical problem of mid-latitude weather prediction. It is shown that the functionals derived in the first part all decline essentially linearly with time and there appears in fact to be a fairly well defined cut off time (roughly 45 days for the model analyzed) beyond which initial condition information is unimportant to statistical prediction.