  We consider two different ways of representing stochastic matrices by
bi-stochastic ones acting on a larger probability space, referred to as
``dilation by uniform coarse graining" and ``environmental dilation". The
latter is motivated by analogy to the dilation of operations in quantum theory.
Both types of dilation can be viewed as special cases of a general ``dilation
by coarse graining". We also discuss the entropy balance and illustrate our
results, among others, by an example of a stochastic $4\times 4$-matrix, which
serves as a simplified model of the conditional action of Maxwell's demon.
