Given a data sequence of noisy observations of a 3-state Markov chain $X$ -- $y_1$,$y_2$,...$y_n$, with two transition matrices $A_1$ and $A_2$ corresponding to different regions (**) in the (unit) belief simplex (a triangle for 3-state). I need to fit an observation matrix $B_y$ for updating the belief (over the states) of the Hidden Markov Model filter :
$T(\pi,y)$ = $\frac{B_yA_1'\pi}{1^TB_yA_1'\pi}$ if $c_1' \pi < c_2' \pi$
$T(\pi,y)$ = $\frac{B_yA_2'\pi}{1^TB_yA_2'\pi}$ if $c_2' \pi < c_1' \pi$
Here $c_1' \pi$ and $c_2' \pi$ define hyperplanes, with the inequalities specifying (**).
