Fiducial Inference and Belief Functions

One aspect of Fisher’s work which has puzzled a great many statisticians is the idea of Fiducial Inference. Using “pivotal” variables, Fisher moves from logical statements about restrictions of parameter spaces to probabilistic statements about parameters. The method works, but with a lot of caveats: the pivotal variables must be sufficient statistics, must be continuous, &c. Dempster’s explorations of the fiducial method as applied to the binomial distribution in 1966 through 1968 lead to the formulation of an upper and lower probability model later extended and named “belief functions” by Shafer. Dempster revisits the ideas in a 1989 paper and provides a philosophical synthesis of the ideas of Bayesian, Belief Function and Fiducial Inference. In particular, the theory of belief functions contains representations for logical statements about membership of parameters in sets and probabilistic statments about parameters. It moves between the two representations through subjective statments about the state of knowledge in the form of upper and lower probabilities. This paper explores the belief function and fiducial arguments using a “state-of-knowledge” argument tie the twomethods together. In this light the paper reviews fiducial arguments for the binomial and normal distribution. It also compares the fiducial and belief function arguments for the Poisson process and arrives at an interesting paradox. Key Concepts: Fiducial Inference, Belief Functions, Normal Inference, Binomial distribution, Poisson process, Pivotal variables