MR-HMM Likelihood function

The MR-HMM likelihood function is a complete statistical model of the concatenated input time-series data ${\bf x}$. By expanding, we can write

$$L({\bf x}; \mbox{\boldmath$\Lambda$} ) = \sum_{{\bf q}\in {\cal Q}} \; p({\bf x}\vert {\bf q}) \; p({\bf q}),$$ (14.1)

where ${\cal Q}$ is the set of all possible segmentations, and $p({\bf q})$ is the a priori probability of that segmentation, and $\Lambda$ are the MR-HMM parameters. A segmentation ${\bf q}$ defines not only the segment sizes $k_s$, but also the sub-class identities $m_s$, as $s$ ranges over all the segments in ${\bf q}$. Due to the conditional independence of the segments, we may write

$$p({\bf x}\vert {\bf q})=\prod_{s\in {\bf q}} \; p({\bf x}_s \vert m_s,k_s),$$ (14.2)

where $s$ is a segment within segmentation ${\bf q}$, $m_s$ is the subclass identity, and ${\bf x}_s$ is the time-series data in the segment.