Substantial updates to tutorial 01_gaussian-mixture-model

[JasonPekos]

First PR! Hope everything is ok.

As discussed in the slack, this PR adds the following significant changes to this tutorial

• Covers the use of ordered() from Bijectors.jl in making the model identifiable (currently it is multimodal, and the seed is just lucky.)
• Introduces a (MUCH faster) version of the model with assignments marginalized out with Turing.@addlogprob!
• A very simple version of the marginalized model using only ~ MixtureModel(dists, weights)
• Example of recovering marginalized assignment draws with generated_quantities()

There are also a few minor changes:

• Changed chain plots from grouping chains together to grouping parameters together. I think this makes the multimodality discussion much clearer.
• Added a tiny amount of burnin to the MCMC sampler, as really far-out initialization was making the plots hard to read
• Added one more chain to n-chains to make multimodality more likely
• Added and reworked tests for the models (now tests rhat to make sure the bijector discussions are all correct)

[devmotion]

Looks good, I made a few comments 🙂

[devmotion]

IMO we should not recommend the use of Turing.@addlogprob! in it's so easy to misuse and to get (silently) wrong results because it operates completely outside of the ~ logic in Turing/DynamicPPL. Instead I think usually one should use ~ with a (possibly custom) distribution.

[JasonPekos]

Sounds good! I initially wasn't going to include that section for basically the reasons you bring up, but I ended up including it (even though I don't actually sample from that model) to motivate what's going on with the MixtureModel lpdf.

I can replace it with a custom distribution (although this might be a little long for a model that's really just exposition), or omit it entirely.

[devmotion]

Maybe let's keep the tutorial simple and avoid surprising warnings in singlethreaded environments:

Suggested change

	chains = sample(model, sampler, MCMCThreads(), nsamples, nchains; discard_initial = burn);
	chains = sample(model, sampler, nsamples, nchains; discard_initial = burn);

[JasonPekos]

Actually un-resolving this because I don't think it works? As it is right now, I'm not sure if Turing allows multiple chains without specifying a type of parallelism.

The documentation, if it's current, seems to suggest I should do something like:

chains = mapreduce(c -> sample(model_fun, sampler, 1000), chainscat, 1:num_chains)

I'm not sure if that's worth it just to get rid of the warning — let me know what you think though.

[devmotion]

This should be defined outside of the model and probably use softmax or softmax! directly.

[JasonPekos]

I think maybe this should be closed and revisited when #441 is done?

fwiw the current thing that's keeping this frozen is the multithreading stuff. If we want to stay away from:

chains = sample(model, sampler, MCMCThreads(), nsamples, nchains; discard_initial = burn);

to avoid warnings in single threaded environments, we'll need to update a bunch of tutorials, because this is pretty common across all the tutorials.

[yebai]

Thanks, @JasonPekos, for the PR. Would you like to migrate your changes here to #441?

[JasonPekos]

Thanks, @JasonPekos, for the PR. Would you like to migrate your changes here to #441?

Yup, will do.