Upper Probabilities Based Only on theLikelihood

In the problem of parametric statistical inference with a nite parameter space, we study some simple rules for deening posterior upper and lower probabilities directly from the observed likelihood function, without using any prior probabilities. 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. The rules can be used to eliminate nuisance parameters , and to interpret the likelihood function and 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 two general approaches to statistical inference: likelihood inference and coherent inference.