GaussHermiteQuadrature(num_points::Int)

Specify that the expectation required for the reconstruction term is to be computed using Gauss-Hermite quadrature with num_points quadrature points.

UnivariateFactorisedLikelihood(build_lik)

A likelihood function which factorises across inputs. Applying this likelihood to any vector f::AbstractVector{<:Real} yields a Product distribution. Exposing this structure is useful when working with quadrature methods.

elbo(
    f::AbstractGP,
    x::AbstractVector,
    η1::AbstractVector{<:Real},
    η2::AbstractVector{<:Real},
    r,
)

Compute the evidence lower bound associated with the GP f, with reconstruction term r, variational parameters η1 and η2, and collection of inputs x.

approx_posterior(
    f::AbstractGP,
    x::AbstractVector,
    η1::AbstractVector{<:Real},
    η2::AbstractVector{<:Real},
)

Compute the approximate posterior. This is just a posterior GP some kind, which is another AbstractGP.

batch_quadrature(
    fs::AbstractVector,
    ms::AbstractVector{<:Real},
    σs::AbstractVector{<:Real},
    num_points::Integer,
)

Approximate the integrals

∫ fs[n](x) pdf(Normal(ms[n], σs[n]), x) dx

for all n in eachindex(fs) using Gauss-Hermite quadrature with num_points.

canonical_from_expectation(m1::AbstractVector{<:Real}, m2::AbstractVector{<:Real})

Compute canonical parameters m and σ² from expectation parameters m1 and m2. Inverse of expectation_from_canonical.

canonical_from_natural(η1::AbstractVector{<:Real}, η2::AbstractVector{<:Real})

Compute canonical parameters y and σ² from natural parameters η1 and η2. Inverse of natural_from_canonical.

expectation_from_canonical(m::AbstractVector{<:Real}, σ²::AbstractVector{<:Real})

Compute expectation parameters m1 and m2 from canonical parameters m and σ². Inverse of expectation_from_canonical.

gaussian_reconstruction_term(
    y::AbstractVector{<:Real},
    σ²::AbstractVector{<:Real},
    m̃::AbstractVector{<:Real},
    σ̃²::AbstractVector{<:Real},
)

Computes

∫ logpdf(Normal(f, σ²[n]), y[n]) pdf(Normal(m̃[n], σ̃²[n]), f) df

for each n ∈ eachindex(y).

natural_from_canonical(y::AbstractVector{<:Real}, σ²::AbstractVector{<:Real})

Compute natural parameters η1 and η2 from canonical paramters y and σ². Inverse of canonical_from_natural.

optimise_approx_posterior(
    f::AbstractGP,
    x::AbstractVector,
    η1::AbstractVector{<:Real},
    η2::AbstractVector{<:Real},
    ∇r,
    ρ::Real;
    max_iterations=1_000,
    gtol=1e-8,
)

optimize_elbo(
    build_latent_gp,
    integrater::AbstractIntegrater,
    x::AbstractVector,
    y::AbstractVector,
    θ0::AbstractVector{<:Real},
    optimiser,
    options,
)

Optimise the elbo w.r.t. model parameters.

Arguments

• build_latent_gp: a unary function accepting an AbstractVector{<:Real} returning a LatentGP
• integrater::AbstractIntegrater: an object specifying how to perform inference
• x::AbstractVector: a collection of inputs
• y::AbstractVector: a collection of outputs
• θ0::AbstractVector{<:Real}: initial model parameter values
• optimiser: an optimiser that can be passed to Optim.optimize
• options: options to be passed to Optim.optimize

Returns

• AbstractGP: the approximate posterior at the optimum
• NamedTuple: results summary containing various useful bits of information

update_approx_posterior(
    f::AbstractGP,
    x::AbstractVector,
    η1::AbstractVector{<:Real},
    η2::AbstractVector{<:Real},
    r,
    ρ::Real,
)