Quadrature/Monte Carlo

[thangbui]

An alternative to the currently implemented Gaussian projection and the linearisation technique in #5 is a quadrature/Monte Carlo/particle based approach.

Need code to implement this and comparison to moment matching and #5.

[thangbui]

Notes:

• use logsumexp

• use the reparameterisation trick/store random numbers/only change them after t iterations

• high-dimensional quadratures?

[thangbui]

Notes:

• Simple Monte Carlo for the latent variables should work for GPLVMs, GPSSMs and deep GPs with hidden variables. This should be enough because, conditioned on a latent sample, the distro of the outputs when the latent function is marginalised out for all alpha values is tractable (as a Gaussian) or can computed using quadrature. This commit is an initial attempt for GPLVMs.

• For deep GPs (as in the ICML version), simple Monte Carlo for the inducing points is needed for alpha values other than 1.