Probabilities for a period and its subperiods: Theoretical relations for forecasting

Abstract

Consider an event definable in terms of two subevents as, for example, the occurrence of precipitation within a 24-h period is definable in terms of the occurrence of precipitation within each of the 12-h subperiods. A complete forecast must specify three probabilities; these may be marginal probabilities, one for the period and two for subperiods. Theoretical relations between these probabilities are investigated and solutions are presented to three problems encountered in operational forecasting: (i) guaranteeing that the marginal probabilities jointly obey the laws of probability, (ii) structuring admissible procedures for adjusting the initial (guidance) probabilities by forecasters, and (iii) formulating optimal estimators of the probability for period in terms of the probabilities for subperiods.