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The event (m=N) represents no change within the observed sequence: every observation belongs to the first-rate segment, and the second-rate segment is empty. It is not the event $\lambda =\varphi$; under continuous independent rate priors that exact equality has posterior probability zero. The endpoint is instead a distinct index value with positive prior mass.

For retained Markov chain Monte Carlo draws $({\lambda }^{(b)},{\varphi }^{(b)},m^{(b)})$, the direct estimate isIts consistency follows from the ergodic theorem for a positive Harris recurrent Markov chain. The draws are correlated, so an independent-binomial standard error would generally be incorrect; uncertainty should account for the Markov chain Monte Carlo asymptotic variance.

A conditional Monte Carlo alternative averages the conditional probability from the preceding solution:The tower property of conditional expectation shows that its posterior expectation is the same target. This replaces an indicator by its conditional expectation; for correlated chains one should assess efficiency rather than automatically assume its entire asymptotic variance is smaller.

An exact finite-sum check is available by integrating out both rates. The marginal posterior weights of a Poisson count change point areThese follow from the Gamma function integral ${\int }_0^{\infty }{z}^{a-1}{e}^{-bz}dz=\Gamma (a)/b^a$. All omitted prior-normalization and factorial factors are independent of $j$, so they cancel from this ratio. This exact calculation provides a useful independent check on either simulation estimate.