MCMC Sampling of partially-observed multivariate normal nodes

[danielturek]

Addresses NCT issue 296.

This PR adds support for MCMC sampling of partially-observed multivariate normal nodes. That is, multivariate-distributed nodes, where some of the dimensions are observed data, and some of the dimensions are unobserved (latent). The partial_mvn sampler essentially operates by assigning univariate RW samplers to the (individual) unobserved dimensions of partially-observed (and therefore also partially-unobserved) MVN-distributed nodes.

More specifically, several cases for sampling the unobserved dimensions are handled:

• Any latent dimensions which have no stochastic dependencies are assigned posterior_predictive samplers.
• If multivariateNodesAsScalars = TRUE, then all remaining unobserved dimensions are assigned univariate RW samplers.
• If multivariateNodesAsScalars = FALSE, then all remaining unobserved dimensions are assigned a single multivariate RW_block.

A variety of test cases are also added.

[danielturek]

Testing passed. I welcome any thoughts or review from @paciorek or @perrydv

[paciorek]

So do we think the most common use case would be that y[1:n] is a leaf node containing some data and some non-data elements and therefore the non-data are predictive nodes? In that case it looks like we assign a RW_block sampler. So @danielturek I'm not sure what you meant by "Any latent dimensions which have no stochastic dependencies are assigned posterior_predictive samplers." More substantively, should we work out the true MVN conditional distribution in that situation and sample from the true predictive? If we do think this is a common case, then I think it might be worth it. (And I would be happy to lend a hand if we decide to do it.)

[danielturek]

@paciorek Please see the addition of the partial_mvn_pp sampler ("partial MVN posterior predictive" sampler), which is used internally in the partial_mvn sampler.

When one or more dimensions (from among the unobserved dimensions) of a partially-observed MVN node themselves have no data dependencies, then a partial_mvn_pp sampler is now assigned to these dimension(s) - instead of either RW or RW_block samplers, as before. This new partial_mvn_pp generates draws directly from the conditional MVN distribution.

@paciorek A second set of eyes would be appreciated. There might also be some efficiency improvements with the linear algebra within the partial_mvn_pp sampler, which could be improved upon from the present implementation.