= Panel-observed multi-state likelihood
{title2=$P(u)=e^{Qu}$}
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 is
$$
P_{rk}(u)q_{ks},\qquad k\ne s.
$$
For 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:
$$
g_{rD}(u)=\sum_{k\text{ living}}P_{rk}(u)q_{kD}.
$$
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.
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