Relative coarse-graining

Abstract

The problem of statistical inference based on a partial measurement (“coarse-graining”) requires the specification of an a priori distribution. We reformulate the ordinary theory such as to encompass systematically a wide range of a priori distributions (“relative coarse-graining”). This is done in a mathematical setting which admits an interpretation in both classical probability and quantum mechanics. The formalism is illustrated in a few simple examples, such as the die whose geometrical shape is known, the spin in thermal equilibrium with an unknown reservoir, and the position measurement of a one-dimensional particle. It is shown that some of the limitations of the usual theory are a consequence of the fact that it is restricted to “equidistributed” (symmetric) a priori states.