Update Stan users guide chapter on HMMs

[charlesm93]

Summary:

The section on HMMs (under Time-Series models: https://mc-stan.org/docs/stan-users-guide/time-series.html#hmms.section) in the Stan users guide currently doesn't use Stan's dedicated functions for HMMs (described here: https://mc-stan.org/docs/functions-reference/hidden_markov_models.html).

I'll update the section and point users towards the convenient functions we have.

[charlesm93]

I've taken a closer look at the user guide's section on HMMs, and I find that it treats specific class of HMMs, rather than provide general guidance.

First, the section focuses on the case where the likelihood (observational distribution) is categorical. Then, it considers:

• the case where the hidden states are observed ("supervised parameter estimation"), including naive coding, coding with sufficient statistics, and coding the posterior analytically
• the case where some of the hidden states are observed ("semisupervised estimation")

I'm not sure that the supervised case is particularly interesting, since we can compute the posterior analytically. Its main interest is setting up the semi-supervised example.

If I look at the HMM suite we developed, its use-case seems a bit orthogonal, in that:

• there are no restrictions on the observational model
• none of the hidden states are observed ("unsupervised" case).

So I think it can make sense to keep the existing example and add an example which uses the HMM suite. I'll base myself on the case study @bbbales2 put together after we released the functions (https://mc-stan.org/users/documentation/case-studies/hmm-example.html).

Two organizational questions:

• does it make sense to have the general HMM example first and then the specific example with supervised and semi-supervised learning?
• should we harmonize notation between the examples, and between the user and functions guide?

My answer to both questions is yes. But I'm leaving it up for discussion.

@bob-carpenter