Design Principles
GPJax is designed to be a Gaussian process package that provides an accurate
representation of the underlying maths. Variable names are chosen to closely
match the notation in [@rasmussen2006gaussian]. We here list the notation used
in GPJax with its corresponding mathematical quantity.
Gaussian process notation
| On paper | GPJax code | Description |
|---|---|---|
| \(n\) | n | Number of train inputs |
| \(\boldsymbol{x} = (x_1,\dotsc,x_{n})\) | x | Train inputs |
| \(\boldsymbol{y} = (y_1,\dotsc,y_{n})\) | y | Train labels |
| \(\boldsymbol{t}\) | t | Test inputs |
| \(f(\cdot)\) | f | Latent function modelled as a GP |
| \(f({\boldsymbol{x}})\) | fx | Latent function at inputs \(\boldsymbol{x}\) |
| \(\boldsymbol{\mu}_{\boldsymbol{x}}\) | mux | Prior mean at inputs \(\boldsymbol{x}\) |
| \(\mathbf{K}_{\boldsymbol{x}\boldsymbol{x}}\) | Kxx | Kernel Gram matrix at inputs \(\boldsymbol{x}\) |
| \(\mathbf{L}_{\boldsymbol{x}}\) | Lx | Lower Cholesky decomposition of \(\boldsymbol{K}_{\boldsymbol{x}\boldsymbol{x}}\) |
| \(\mathbf{K}_{\boldsymbol{t}\boldsymbol{x}}\) | Ktx | Cross-covariance between inputs \(\boldsymbol{t}\) and \(\boldsymbol{x}\) |
Sparse Gaussian process notation
| On paper | GPJax code | Description |
|---|---|---|
| \(m\) | m | Number of inducing inputs |
| \(\boldsymbol{z} = (z_1,\dotsc,z_{m})\) | z | Inducing inputs |
| \(\boldsymbol{u} = (u_1,\dotsc,u_{m})\) | u | Inducing outputs |
Package style
Prior to building GPJax, the developers of GPJax have benefited greatly from the GPFlow and GPyTorch packages. As such, many of the design principles in GPJax are inspired by the excellent precursory packages. Documentation designs have been greatly inspired by the exceptional Flax docs.