Upper Probabilities Based Only on the Likelihood Function

In the problem of parametric statistical inference with a ®nite parameter space, we propose some simple rules for de®ning posterior upper and lower probabilities directly from the observed likelihood function, without using any prior information. The rules satisfy the likelihood principle and a basic consistency principle (`avoiding sure loss'), they produce vacuous inferences when the likelihood function is constant, and they have other symmetry, monotonicity and continuity properties. One of the rules also satis®es fundamental frequentist principles. The rules can be used to eliminate nuisance parameters, and to interpret the likelihood function and to use it in making decisions. To compare the rules, they are applied to the problem of sampling from a ®nite population. Our results indicate that there are objective statistical methods which can reconcile three general approaches to statistical inference: likelihood inference, coherent inference and frequentist inference.