The MR-HMM likelihood function is a complete
statistical model of the concatenated input time-series data .
By expanding, we can write
where is the set of all possible segmentations,
is the a priori probability of that segmentation,
are the MR-HMM parameters.
A segmentation defines not only the
segment sizes , but also the sub-class identities ,
as ranges over all the segments in .
Due to the conditional independence of the segments, we may write
where is a segment within segmentation ,
is the subclass identity, and is the
time-series data in the segment.