# 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](https://github.com/GPflow/GPflow) and [GPyTorch](https://github.com/cornellius-gp/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](https://flax.readthedocs.io/en/latest/index.html).