Panel-observed multi-state likelihood ($P(u)=e^{Qu}$)

For a time-homogeneous continuous-time Markov chain with generator matrix $Q=(q_{rs})$, let $P(u)=e^{Qu}$ be its transition matrix. Conditional on a known state $r$ at time zero, a panel visit recording state $s$ at duration $u$ contributes $P_{rs}(u)$ to the likelihood. This transition probability sums over all intervening paths, including unobserved transitions.

An exact endpoint transition is a different observation. If the state immediately before the transition is known to be $k$ and the destination is $s$, its time-density contribution isFor a small interval $du$, this follows by multiplying the probability of being in $k$ just before $u$ by the transition probability $q_{ks}du+o(du)$. The relevant occupancy is the pre-event state $k$, rather than the destination $s$. If the record additionally establishes no transition at all before a first jump $r\to s$ at $u$, the contribution instead is $e^{q_{rr}u}{q}_{rs}$.

For exact entry into an absorbing state $D$ when the immediately preceding state is hidden, sum over possible living states:This is the mixed panel and exact-death likelihood contribution. Multiplying a panel destination probability $P_{rs}(u)$ by $q_{rs}$ would generally confuse observation of the endpoint state with observation of a jump into it.

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.