## Computation of likelihood function using Proxy HMM

To implement (14.1), (14.2) brute-force is obviously impractical because of the combinatorial number of state sequences , but we can use the proxy HMM to do it efficiently. Note that each segmentation corresponds to a distinct path through the proxy HMM state trellis, which has probability given by equation (14) in [57]. If we had the likelihood functions of each base-segment of the proxy HMM, the proxy likelihood function would be where stands for p(r)oxy", and where where is the -th base segment and is the proxy HMM state at time step corresponding to segmentation . The well-known HMM forward procedure [57] calculates the total proxy likelihood function efficiently using dynamic programming and without enumerating the segmentations.

But the proxy log likelihood functions are defined for a single base segment, whereas the MR-HMM segment log-likelihood functions are defined for segments spanning base segments. This problem is solved by writing as a sum of equal parts: , where each part is assumed to apply to just one base segment, so can be used in place of the proxy segment log-likelihood functions. This substitution forces the proxy HMM forward procedure to compute for the the MR-HMM. This substitution can be more precicely written as

 (14.3)

where for segmentation , base segment time lies within segment of length . The classical forward procedure applied to the proxy HMM then produces . Furthermore, by calculating the backward procedure on the proxy HMM and combining with the forward procedure, we obtain the gamma probability , which is the a posteriori probability that the system is in proxy state at base segment given all the available data. This is illustrated in Figure 14.1 as the filled-in circles in the proxy state trellis. These filled-in circles correspond to when has a high value. In the gap between the two pulses, is a case when probability is shared between more than one candidate path. If we sum up all the gamma probabilities for a given sub-class, we get an indication of the probability of each sub-class (illustrated at very bottom of Figure 14.1).

Baggenstoss 2017-05-19