Issue/838 hmm

[charlesm93]

Submission Checklist

• Builds locally
• New functions marked with <<{ since VERSION }>>
• Declare copyright holder and open-source license: see below

Summary

Addresses issue #838 and updates user-doc on HMMs.

Copyright and Licensing

Please list the copyright holder for the work you are submitting (this will be you or your assignee, such as a university or company): Charles Margossian, Simons Foundation

By submitting this pull request, the copyright holder is agreeing to license the submitted work under the following licenses:

• Code: BSD 3-clause (https://opensource.org/licenses/BSD-3-Clause)
• Documentation: CC BY-ND 4.0 (https://creativecommons.org/licenses/by-nd/4.0/)

[bob-carpenter]

I took a look through this and while I'm in favor of minimal examples, this one's a bit too minimal. I would really like to see the model laid out in more detail than the conditional distributions. Here are some concrete suggestions:

1. Use z for latent discrete parameters. We already use x for covariates, so it's confusing to use x. I've used z elsewhere in the user manual for this in the latent discrete parameter chapter.

2. Define the complete data likelihood $p(y, z | \phi) = p(z \mid \phi) \cdot p(y \mid z, \phi)$ and then the marginalization $p(y \mid \phi)$ that we actually fit.

3. Mention that $\phi$ is actually several parameters.

4. I find the constrained version of the transition matrix where you insert zeros makes this first example too challenging. If you want to discuss that case, I'd suggest another section after this one where you talk about imposing structural zeros. Otherwise it makes the simple case seem too complicated. And then you can just define

array[3] simplex[3] gamma_arr;

matrix[3, 3] gamma;
for (n in 1:3) gamma[n] = gamma_arr[n];

5. For the doc, those mu and sigma are not just the measurement model---that's all the error terms.

6. Wherever you have repetition, use loops. It's less error prone and more clear that it's a homogeneous operation:

for (n in 1:N) {
  for (k in 1:3) {
    log_omega[k, n] = normal_lpdf(y[n] | mu[k], sigma);
  }
}

7. Given that you're tying the parameter sigma across outputs, you need to mention that. I'd recommend just keeping this simple with a vector of sigma values.

8. You can't just say "computes the relevant log marginal distribution"---you have to say what that is. I don't mean including the marginalization algorithm, I just mean as I wrote it above.

9. For more details, since -> For more details, see. Also, I wouldn't say "corresponding case study", I'd just say it's a case study on HMMs. And then you should put it in the bibtex file and cite it properly with reference to Ben (the author). And you should cite that you "borrowed" the example from Ben Bales's case study.

[charlesm93]

I'm rewriting the code to make the transition matrix less constrained (per comment 4) and I wanted to check: we don't have a stochastic matrix type, right?

[WardBrian]

We have row and column but not double yet https://mc-stan.org/docs/reference-manual/types.html#stochastic-matrices

[charlesm93]

@bob-carpenter I implemented your feedback.

Some questions:

5. What exactly is the difference between a measurement and an error model? I'm ok to use either term, and I know they have different conceptual implication, but I'm wondering if they have a formal definition.

6. I'm keeping sigma a scalar. I'm not sure it's simpler conceptually to pass it as a vector.

[bob-carpenter]

For (5), I think the idea's that there are three sources of error: measurement error, modeling error, and sampling error. For example, sampling error arises when you subsample a population and use that for estimation. You get modeling error if you use a linear regression for a relationship that's not linear or use normal errors when the errors are skewed, and so on. If you're weighing things with a scale and you know the scale's biased to the high side, you can correct that measurement error. You can explicitly add a measurement error model if you know your measurement model (e.g., gravitational lensing is part of the measurement error model; your work with Bruno et al. on deconvolving galactic dust is part of the measurement model for the CMB, etc.).

[bob-carpenter]

There's a ton of little things to fix, but nothing major.

[bob-carpenter]

Why is this PR touching negative binomial? Are you up to date with the main branch?

[charlesm93]

hmmm... that's odd. Let me check.

[charlesm93]

As far as I can tell, I am up to date with the main branch, so I'm not sure how this error occurred. I suppose I could simply delete the extra lines.

[bob-carpenter]

Isn't z_{1:N} a subset of $\{ 1, ..., K \}^N$ when $K$ is the number of states? I don't think it can be continuous.

Did you perhaps mean to say that the output $y_1, \ldots, y_T$ can be continuous? It can also be mixed discrete and continuous, it can be multivariate, etc.---it can really be anything.

[charlesm93]

z_{1:N} denotes the latent variables. In general, the domain of z can be continuous, say $\mathbb R$, or discrete ${1, \cdots, K}^N$.

[bob-carpenter]

Eliminate the comma---English doesn't use commas between conjunctions unless there are more than two. So it's just "A and B", but it's either "A, B, and C" (Oxford style) or "A, B and C" (defective American style).

[charlesm93]

Ok. Although I was under the impression they didn't use the Oxford comma in the UK. I remember noticing this during StanCon in Cambridge and I figured out the Cambridge folks were just taking a piss. But when I asked some of the locals, it turned out they were familiar with the Oxford comma.

[bob-carpenter]

You don't typically need draws from $z$. If you do need statistics involving $z$, it's often easier to compute them in expectation (i.e., take advantage of the Rao-Blackwellization you get from marginalizing). For example, if there are low probability but non-zero $z$, they're very hard to evaluate with sampling. There are examples in the latent discrete parameters chapter.

You can also use the forward-backward algorithm to work out marginally $p(z_n \mid y)$ and you can use Viterbi with backtracking to find the most probable z given y.

[charlesm93]

I added "If we desire samples from the posterior distribution of $z$..."

[bob-carpenter]

Typically this phi is marginalized out and we look at $p(z_n = k | y)$.

[charlesm93]

You get $p(z_n = k \mid y)$ by averaging all the MCMC draws. I'll specify this in the doc.

[bob-carpenter]

Although not necessary, it's nice to have examples of how to do that. Like:

generated quantities {
   array[N] int<lower=1, upper=K> z = hmm_latent_rng(...fill-in params here to match example...);
}

I know this feels obvious, but it can be hard for the users to put the arguments and result types together.

[charlesm93]

@bob-carpenter I completely missed your review. Sorry about that. Hopefully we'll merge this in soon.