Tighter estimates for the posteriors of imprecise prior and conditional probabilities

White (1986) and Snow (1991) have presented approximate characterizations for the posterior probabilities when the priors and conditionals are specified through linear constraint systems. This paper extends their results by developing alternative linear inequality approximations for posterior probabilities, with a particular focus on partial information expressed in terms of (i) bounds on the components of probability vectors and (ii) bounds on ratios of probabilities.