The likelihood of a hidden Markov model (HMM) for states $x_0, \dots, x_N$ and observations $y_1, \dots, y_N$ can be written as
$$ L = f(x_0) \prod_{i=1}^N f(y_i | x_i) f(x_i | x_{i-1} )$$
where we use $f(\cdot)$ to represent a pdf.
If the states $x_0, \dots, x_N$ can only take values from a discrete set, then we can find the most likely sequence of states using the Viterbi algorithm.
My question is then, what is this the best way to go about the optimisation in the continuous case?
Note
If the states take values on a continuous set, then we could use gradient descent for example to try to estimate the most likely sequence of states.
However if $N$ is very large, it could take a long time to calculate the gradient of $L$. We may be able to perform stochastic gradient descent but I feel we would not want to re-order the observations $i$, as you would between batches of standard gradient descent. Therefore I am unsure how well this would work.