Mathematics and Statistics

On the predictive use of insufficient statistics: an intriguing family of distributions

Frank Lad and Romano Schozzafava

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Abstract

We derive the most general relation that coherency requires of the simultaneous assertion of a probability mass function for the sum of N+1 ordered events, and a conditional probability function for the final event given each possible value of the sum of the first N events. We then use this relation to characterise the family of all distributions on N+1 events that support the well-known sequential forecasting equations (conditioning only on a sum) that are motivated by exchangeable assessments. Surprisingly, this intriguing family is much larger than the family of exchangeable distributions, but is included within the family of all pairwise exchangeable (and thus equiprobable) distributions. Of course the sum is not generally a sufficient statistic for this family. We display some small numerical examples, and we discuss the implication of this discovery for applied sequential forecasting.

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