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In recent work discussing model choice for continuous-time Markov chains, wehave argued that it is important that the Markov matrices that define the modelare closed under matrix multiplication (Sumner 2012a, 2012b). The primaryrequirement is then that the associated set of rate matrices form a Liealgebra. For the generic case, this connection to Lie theory seems to havefirst been made by Johnson (1985), with applications for specific models givenin 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 gainadditional insight into the geometry of the associated Markov matrices. In thisshort note, we present the simplest case possible of 2x2 Markov matrices. Themain result is a novel decomposition of 2x2 Markov matrices that parameterisesthe general Markov model as a perturbation away from the binary-symmetricmodel. This alternative parameterisation provides a useful tool for visualisingthe binary-symmetric model as a submodel of the general Markov model.

Lie geometry of 2x2 Markov matrices