Lie geometry of 2x2 Markov matrices

In recent work discussing model choice for continuous-time Markov chains, we have argued that it is important that the Markov matrices that define the model are closed under matrix multiplication (Sumner 2012a, 2012b). The primary requirement is then that the associated set of rate matrices form a Lie algebra. For the generic case, this connection to Lie theory seems to have first been made by Johnson (1985), with applications for specific models given in Bashford (2004) and House (2012). Here we take a different perspective: given a model that forms a Lie algebra, we apply existing Lie theory to gain additional insight into the geometry of the associated Markov matrices. In this short note, we present the simplest case possible of 2x2 Markov matrices. The main result is a novel decomposition of 2x2 Markov matrices that parameterises the general Markov model as a perturbation away from the binary-symmetric model. This alternative parameterisation provides a useful tool for visualising the binary-symmetric model as a submodel of the general Markov model.