The $ M$-state HMM model parameters consist of $ \Lambda$$ =\left[ \{\pi_m\}, \{A_{i,j}\}, \{b_i({\bf y})\}\right],$ where $ \pi_m, \; 1\leq m \leq M$ are the prior probabilities, $ A_{i,j}, \; 1\leq i \leq M,\; 1\leq j \leq M$ are the state transition probabilities, and $ b_i({\bf y}), \; 1\leq i \leq M$ are the state observation probability densities. The well-known forward procedure [57] computes the likelihood function or joint probability density function (PDF) $ L_y({\bf Y}) = p({\bf y}_2 , {\bf y}_3 \ldots {\bf y}_{\mbox{\tiny $T$}} ; \mbox{\boldmath $\Lambda$}).$ To convert $ L_y({\bf Y})$ into a PDF on $ {\bf X}$, we require the integral $ K=\int_{{\bf X}} \; L_y({\bf D}({\bf X})) \; {\rm d}{\bf X},$ where $ {\bf Y}={\bf D}({\bf X})$ is the DAF transformation that implements (15.1).

Baggenstoss 2017-05-19