Given an initial probability distribution for a stochastic process one can use the transition matrix to compute a change in the probability distribution. If one tries to go backwards one finds that the inverse matrix has numbers outside the interval [0,1] and so cannot be interpreted as a probability matrix. Bayes' Theorem provides a solution to this problem. Suppose we start with a simple transition diagram.
Focusing on the states marked in red we see that both initial states contribute to the final state s'0. We can view the transition matrix as mixing events from the two sources. Suppose N0 events come from s0 and N1 come from s1 making a total for s'0 of N'0=N0+N1. Bayes' theorem tell us the probability that s0 was responsible for some event in s'0 is P'0,0=N0/N'0=P0,0 p0/p'0 and likewise the probability that s1 was responsible is P'0,1=N1/N'0=P0,0 p0/p'0. Doing this for all the paths in the diagram gives a set of inverse probabilities, P'j,k. Factoring out the components of p and p' into separate matrices and generalizing to n states gives a matrix expression for P' which can be generalized for n states .
D(p) is a matrix whose non-zero components along the diagonal are the components of p. We can show how this works with a sample calculation. A check shows that P' has the properties of a probability matrix. The sum of each column is 1 and we see that p=P'p'.
Given two sets of data for p and p' we see from the argument above that P=p'p-1 and a check of the column vectors in p and p' shows p'=Pp. One needs to be careful about the inverse probabilities P' since they are dependent on both p and p' so the P' for the first columns of p and p' is not the same as that for the second columns. The inverse probabilities are not determined by P alone.