Add SparseFiniteGP type and associated functionality

[ElOceanografo]

This addresses https://github.com/willtebbutt/Stheno.jl/issues/134, adding a SparseFiniteGP type that wraps two GPs for the observation and inducing points and can be plugged into Turing. The PR contains constructors and basic methods for the new type, mostly falling through to the underlying FiniteGP for the observation points. It provides a logpdf function that wraps the preexisting elbo method for sparse GPs, and also provides a convenience constructor for PseudoObs so that the following "just works":

x = 0:10
xu = 0:2:10
f = GP(Matern32(), GPC())
fx = SparseFiniteGP(f, x, xu)
y = rand(fx)
fpost = f | (fx ← y)

There are a couple of remaining issues/questions:

• I thought of defining a convenience constructor so you could define a SparseFiniteGP by doing fx = f(x, xu) (using the variables from the example above). However, this conflicts with this constructor for FiniteGP, where the second AbstractArray in the function signature is interpreted as the diagonal of the observation covariance matrix. I like that compact definition, but using it would likely mean a breaking change (i.e. requiring users write f(x, Diagonal(σ²)) instead of f(x, σ²)).
• I decided to make cov(fx::SparseFiniteGP) throw an error instead of calculating the covariance of the observation GP. This follows behavior in base for e.g. inv(A::SparseArrays.AbstractSparseMatrixCSC). The use case for a sparse approximation is by default one where you don't want to or can't use the full covariance matrix, so I think this is the safe and fast option for this behavior. If the user really wants the dense covariance matrix, she can do cov(f.fobs).

Comments on code, tests, organization, or style welcome...

[codecov[bot]]

Codecov Report

Merging #136 into master will increase coverage by 0.06%. The diff coverage is 85.71%.

@@            Coverage Diff             @@
##           master     #136      +/-   ##
==========================================
+ Coverage   82.37%   82.43%   +0.06%     
==========================================
  Files          26       27       +1     
  Lines         749      763      +14     
==========================================
+ Hits          617      629      +12     
- Misses        132      134       +2     

Impacted Files	Coverage Δ
src/Stheno.jl	100.00% <ø> (ø)
src/approximate_inference.jl	85.71% <85.71%> (ø)

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[ElOceanografo]

To clarify, do you mean I should update the version number in Project.toml?

[willtebbutt]

To clarify, do you mean I should update the version number in Project.toml?

Yes please :)