Explicitly track constrained and unconstrained versions of transformed variables

[davmre]

This is a proposed fix to #807 (Automated transformations break log_joint computation). It:

• modifies the auto_transform section of inference.initialize to maintain two dicts with clear semantics: latent_vars maps the original zs to qzs with matching support (possibly constructed as a transformation of an unconstrained qz), while latent_vars_unconstrained guarantees to contain versions of the zs and qzs supported over an unconstrained space. This ensures that any inference method that looks at latent_vars to compute a log joint will correctly substitute zs with qzs in the same space.
• Updates HMC to wrap the log_joint computation by explicitly transforming back into the original (potentially constrained) latent space. I believe HMC needs special attention to work in the unconstrained space because it directly accesses qz.params; other inference methods that just treat qz as a Distribution (eg variational methods such as KLqp) should automatically do the right thing when given a qz having constrained support matching z.
• Adds test cases for a simple beta-bernoulli model with automated transformations using HMC and KLqp.

[dustinvtran]

Thanks Dave! Merging.

There's an interesting implication of this PR. Namely, ADVI works by transforming prior variables to the unconstrained space, then minimizing trainable Gaussians to the unconstrained posterior. What this PR does is transform trainable Gaussians to the constrained space, then minimize over the original space to the posterior. Parameterization of course matters—though I'd be hard-pressed to point to an example where one approach is preferred over the other (?).

[davmre]

Thanks Dustin! Your point about parametrization is interesting. For HMC this PR still runs the dynamics in the unconstrained space. For VI, are you sure that parametrization makes a difference? We're now computing KL[q|p] (or rather, the ELBO which is equivalent up to a constant) in the original rather than transformed parametrization, but since KL is an information theoretic quantity I want to say that this shouldn't change the solution. On Jan 6, 2018 7:54 PM, "Dustin Tran" <notifications@github.com> wrote: Merged #808 <#808>. — You are receiving this because you authored the thread. Reply to this email directly, view it on GitHub <#808 (comment)>, or mute the thread <https://github.com/notifications/unsubscribe-auth/AA8iiiIXv9Ps_8vRQ6fWJ1ycnyHpclVJks5tICPggaJpZM4RELcN> .

[davmre]

I realized I misstated this slightly, because KL involves a ratio of densities it's invariant to parametrization (https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence#Properties) even though other information theoretic quantities like differential entropy aren't.