 Marginal posterior weights of a Poisson count change point

Integrating the two rates out of the Poisson count change-point model with gamma priors gives these weights up to a common factor, by the defining Gamma function integral. Normalizing across the finitely many possible indices gives the exact marginal posterior distribution of $m$, providing a check on Markov chain Monte Carlo estimates. One can also draw $m$ from these weights and then draw both rates from their conditional Gamma distributions, producing independent exact joint posterior samples.