Using the QR factorization and group inversion to compute, differentiate, and estimate the sensitivity of stationary probabilities for Markov chains

For an $n$-state finite, homogeneous, ergodic Markov chain, with transition matrix ${\bf P}$ and stationary distribution ${\boldsymbol \pi} $ we assume that the entries of ${\bf P}$ are differentiable functions of a parameter $t$ and we obtain an expression for $d{\boldsymbol \pi} /dt$. This expression is given in terms of the group inverse of ${\bf I} - {\bf P}$ and is used in a sensitivity analysis of ${\boldsymbol \pi}$. Finally, it is demonstrated how a ${\boldsymbol QR}$ factorization can be used to simultaneously compute the stationary distribution of an ergodic chain along with estimates which gauge the sensitivity of the stationary distribution to perturbations in the transition probabilities.