Hyperparameter optimization

The hyperparameter vector η = (λ, ρ)^⊤ will be calibrated by posterior optimization. Following [25] and [24], the hyperparameter vector can be approximated as follows:

Approximation (6) can be written extensively as:

where the K/2 power of λ comes from the determinant |Qλ-1|-1/2=|λP|1/2∝λK/2. As δϕ2+aδ-1exp(-δ(ϕλ2+bδ)) is the kernel of a Gamma distribution for the dispersion parameter δ, the following integral can be analytically solved:

Using the transformation of variables (ensuring numerical stability during optimization) w = log(ρ), v = log(λ), one can show that p˜(η|D) can be written as follows after using the multivariate transformation method:

where η˜=(w,v)⊤. The approximated log-posterior becomes:

Eq (7) is numerically optimized and yields η˜*=argmaxη˜logp˜(η˜|D). Plugging the latter vector into the Laplace approximation (5), we obtain the estimate θ^=θ*(η˜*) of the spline vector. The latter can be seen as a MAP estimate of θ. Thus, the approximated (conditional) posterior of the spline vector is:

and can be used to construct credible intervals for functions that depend on θ, such as Rt as shown in the following section.