Feasible HMM's

From a slightly more general point of view the HMM can be defined by the initial and transition probabilities

$$p(s_1)$$

$$p(s_t x_t\vert s_{t-1})$$

and the forward/backward recursion still hold (although the factorization in the M-step of the EM-algorithm fails). The defining probabilities are feasible for the recursions if $\alpha$ and $\beta$ are stable under the updating equations:

$$\alpha(s) \sim \int ds^\prime \alpha(s^\prime) \, p(x_t\,s_t=s\vert s_{t-1}=s^\prime)\quad\mathrm{and}$$

$$\beta(s) \sim \int ds^\prime \beta(s^\prime) \, p(x_t\,s_t=s^\prime\vert s_{t-1}=s)\quad\mathrm{and}$$

where $\sim$ means that left-hand side and right-hand side are of the same familiy of distributions.