Lumping the States of a Finite Markov Chain through Stochastic Factorization

In this work we show how the lumping of states of a finite Markov chain can be regarded as a special decomposition of its transition matrix called stochastic factorization. The idea is simple: when a transition matrix is factored into the product of two stochastic matrices, one can swap the factors of the multiplication to obtain another model, potentially much smaller than the original one. We prove in the paper that the smaller Markov chain has the same reducibility and the same number of closed sets as the original model. Additionally, the stationary distributions of both chains are related through a linear transformation. By interpreting the lumping of states as a particular case of stochastic factorization, we discuss in which circumstances the lumped transition matrix can be used in place of the original one to compute its stationary distribution. To illustrate our ideas we use the computation of Google’s PageRank as an example.