#alpha-recursion

For an HMM $p(h_{1:d}, v_{1:d}) = p(h_1)p(v_1 \mid h_1)\prod_{i=2}^d p(h_i \mid h_{i-1})p(v_i \mid h_i)$ you observe only $v_{1:u}$ with $u < d$. After filtering up to time $u$, how do you continue to obtain $\alpha(h_s) \propto p(h_s \mid v_{1:u})$ for $s = u+1, \ldots, t$, for some $t \le d$?