We present theoretical foundations and computational procedures of a theory for analysing decisions under risk when the available information is vague and imprecise. The impreciseness is expressed by a set of global distributions T over a space S, where the latter represents the classes of all probability and utility measures over a set of discrete outcomes. We show how local distributions, i.e. distributions over projections of S on various subspaces of S, can be derived from T and investigate in which extent user-asserted local distributions can be used for defining T. We also study invariants under local projections. The approach allows a decision maker to be as deliberately imprecise as she feels natural, as well as provides her with the means for expressing varying degrees of imprecision in the input sentences.