Why exposure-time transformations do not generally homogenize Markov-modulated Poisson processes: A common-clock criterion and corrected calendar-time inference

Avanzi et al. (2021) introduced an exposure-modulated Markov-modulated Poisson process in which a homogeneous continuous-time Markov chain $M$ with generator $Q$ modulates a counting process with conditional intensity $\gamma(t)\lambda_{M(t-)}$. They asserted that the exposure clock $\rho(t)=\int_0^t\gamma(s)ds$ converts this model into a homogeneous MMPP while leaving $Q$ unchanged, and they used that representation in an expectation--maximization calibration scheme. The published argument contains several mathematical errors. We show that its central homogenization theorem cannot be correct under the stated model when $\gamma$ is nonconstant. Time-changing only the count leaves it modulated by $M(\rho^{-1}(u))$, not by $M(u)$; time-changing the whole pair instead produces the time-dependent hidden generator $Q/\gamma(\rho^{-1}(u))$. More generally, we prove a necessary and sufficient common-clock criterion: if the hidden generator and event rates are multiplied by $a(t)$ and $b(t)$, respectively, one deterministic clock makes both constant if and only if $a(t)/b(t)$ is constant almost everywhere. We also explain why the supporting Markov and homogeneity lemmas do not establish the published result and identify consequential inconsistencies in the printed EM construction, including an incorrect hidden-transition statistic, reversal of event and exposure-change kernels, and recursions that treat deterministic exposure breakpoints as though they incremented the observed count. The exposure-modulated model itself remains well defined. We therefore derive its exact calendar-time likelihood, a corrected EM algorithm, and a stable piecewise-constant implementation, together with an $L^1$ perturbation bound for exposure approximation. Thus the published homogeneous representation and the inferential formulas built upon it are invalid as stated, while direct calendar-time inference provides a coherent replacement.