Panel-observed multi-state likelihood

When a multi-state process is observed only at clinic times, an observed interval from state $r$ to state $s$ over duration $u$ contributes the transition probability $P_{rs}(u)$. An exactly observed transition $r\to s$ at the interval endpoint additionally contributes its transition intensity $q_{rs}$.

A clinic-observed interval ending in a recorded state contributes a transition probability. If the next observation is exact entry into an absorbing state but the preceding state is unobserved, its statistical probability density is ${\sum }_{s\text{ living}}{p}_{rs}(t)q_{sD}$, not $p_{rD}(t)$. Multiplying these conditional contributions yields a likelihood for mixed panel and exact-event observations.