Identification of likelihood and prior dependence structures for hydrologic uncertainty processor

A multivariate hydrologic uncertainty processor (HUP) is required by the Bayesian forecasting system that will produce a short-term probabilistic stage transition forecast. The HUP must supply a family of posterior joint distributions of the actual river stage process, conditional on the model river stage process. A posterior joint distribution can be factorized into posterior one-step transition distributions, each of which being obtained via Bayes theorem from a likelihood function and a prior distribution. A systematic procedure is presented for identifying the dependence structure of the model–actual river stage process. The procedure seeks to reduce the conditioning of the likelihood functions and the prior distributions to the smallest dimension that is necessary to capture the empirical dependence structures. Tests for removing extraneous variates are derived following the Bayesian inference principle. For the likelihood functions, there are tests for causal, acausal, instantaneous, and conditional Markov dependence. For the prior distributions, there is a test for conditional Markov dependence. The identification procedure is applied to daily river stage processes at four forecast points. The main conclusion is twofold. First, the HUP must have a two-branch structure that allows for (i) the conditioning on the occurrence or nonoccurrence of precipitation and (ii) the nonstationarity of both the prior distributions and the likelihood functions. Second, in each of the branches, (i) the first-order Markov dependence structure is sufficient for modeling the prior one-step transition distributions, and (ii) the second-order conditional dependence structure is sufficient for modeling the likelihood functions.