Can we combine multiple GP's into a single additive model?

[kaijennissen]

I'd like to reproduce the birthdays case study.
The key idea is to model the relative number of births given per day $(y)$ using, amongst others, the following model:

$log(y)=intercept + f_{1}(x)+f_{2}(x)+exp(f_{3})\beta_{Day Of Week}+\beta_{Day Of Year}+\varepsilon$
$\varepsilon\sim Normal(0,\sigma)$
$intercept \sim Normal(0,1)$
$f_{1}~\mathcal{GP}(0,K_{1})$
$f_{2}~\mathcal{GP}(0,K_{2})$
$f_{3}~\mathcal{GP}(0,K_{3})$
$\beta_{Day Of Week} = 0, if, Monday$
$\beta_{Day Of Week} \sim Normal(0,1)$
$\beta_{Day Of Year} \sim Normal(0,1)$

Is it possible to build such a model with GPFlow?

[st--]

In principle, yes; in practice, you might not want to:) You can build "simple" additive models where each component is a GP (including "parametric GPs" such as the intercept, which you can model with a Constant() kernel, or day-of-year and other special day effects for which you can build custom kernels that reflect the basis functions) by constructing a sum of kernels for each term in the sum. You can still build models that include non-linear effects such as the $\exp(f_3)$ but that becomes more involved (you have to construct an appropriate likelihood that combines all the GPs in whichever nonlinear way you want). You can do MCMC using the GPMC and SGPMC model classes together with tensorflow_probability (see notebook). That said, if you want to quickly iterate with different complex hierarchical models, Stan is probably more suitable!