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

$\displaystyle L({\bf x};$   $\displaystyle \mbox{\boldmath$\Lambda$}$$\displaystyle ) = \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

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

Baggenstoss 2017-05-19