$P(o \mid i) = P(o \mid o_{i-1})$

For iteration $i$,

$P(o \mid i) = P(o \mid i=0) \cdot T^i$

The animation overlays $P(i \mid o)$ on a 2D UMAP embedding of the data (Cerletti et. al. 2020) Since we are interested in modelling the dynamics in a smaller latent state space, we factorise the MSM simulation,

$P(o \mid i) = \sum\limits_{s \in S} P(o \mid s,i) P(s \mid i)$

Assuming Markovian dynamics in the latent space aswell,

$P(o \mid i) = \sum\limits_{s_{i} \in S} P(o \mid s_{i}) \sum\limits_{s_{i-1} \in S} P(s_{i} \mid s_{i-1})$

Multiple independent chains in a common latent space can be modelled using conditional latent TPMs (Ghahramani & Jordan 1997),

$P(o \mid i) = \sum\limits_{s_{i} \in S} P(o \mid s_{i}) \sum\limits_{l \in L} P(l) \sum\limits_{s_{i-1} \in S} P(s_{i} \mid s_{i-1}, l)$