In the problem of parametric statistical inference with a finite parameter
space, we propose some simple rules far defining posterior upper and lower
probabilities directly from the observed likelihood function, without using
any prior information. The rules satisfy the likelihood principle and a ba
sic consistency principle ('avoiding sure loss'), they produce vacuous infe
rences when the likelihood function is constant, and they have other symmet
ry, monotonicity and continuity properties. One of the rules also satisfies
fundamental frequentist principles. The rules can be used to eliminate nui
sance 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 finite population. Our results indicate that there are obj
ective statistical methods which can reconcile three general approaches to
statistical inference: likelihood inference, coherent inference and frequen
tist inference.