# Models

Models play an essential role in Bayesian Optimization (BO). A model is used as a
surrogate function for the actual underlying black box function to be optimized.
In BoTorch, a `Model`

maps a set of design points to a posterior probability
distribution of its output(s) over the design points.

In BO, the model used is traditionally a Gaussian Process (GP),
in which case the posterior distribution, by definition is a multivariate
normal. However, with the exception of some of the analytic acquisition
functions in the
`botorch.acquisition.analytic`

module, **BoTorch makes no assumption on the model being a GP**, or on the
posterior being a multivariate normal. The only requirement for using
BoTorch's Monte-Carlo based acquisition functions is that the model returns a
`Posterior`

object that implements an
`rsample()`

method for sampling from the posterior of the model. If you wish to
use gradient-based optimization algorithms, the model should allow
back-propagating gradients through the samples to the model input.

## The BoTorch Model Interface

BoTorch models are PyTorch modules that implement the light-weight
`Model`

interface. A BoTorch `Model`

requires only
a single `posterior()`

method that takes in a Tensor `X`

of design points,
and returns a `Posterior`

object describing
the (joint) probability distribution of the model output(s) over the design
points in `X`

.

When working with GPs, `GPyTorchModel`

provides a base class for conveniently wrapping GPyTorch models.

## Terminology

Models may have multiple outputs, multiple inputs, and may exploit correlation between different inputs. BoTorch uses the following terminology to distinguish these model types:

*Multi-Output Model*: a`Model`

(as in the BoTorch object) with multiple outputs.*Multi-Task Model*: a`Model`

making use of a logical grouping of inputs/observations (as in the underlying process). For example, there could be multiple tasks where each task has a different fidelity.

Note the following:

- A multi-task (MT) model may or may not be a multi-output model.
- Conversely, a multi-output (MO) model may or may not be a multi-task model.
- If a model is both, we refer to it as a multi-task-multi-output (MTMO) model.

## Standard BoTorch Models

BoTorch provides several GPyTorch models to cover most standard BO use cases:

### Single-Task GPs

These models use the same training data for all outputs and assume conditional
independence of the outputs given the input. If different training data is
required for each output, use a `ModelListGP`

instead.

`SingleTaskGP`

: a single-task exact GP that infers a homoskedastic noise level (no noise observations).`FixedNoiseGP`

: a single-task exact GP that uses fixed observation noise levels (requires noise observations).`HeteroskedasticSingleTaskGP`

: a single-task exact GP that models heteroskedastic noise using an additional internal GP model (requires noise observations).

### Model List of Single-Task GPs

`ModelListGP`

: A multi-output model in which outcomes are modeled independently, given a list of any type of single-task GP. This model should be used when the same training data is not used for all outputs.

### Multi-Task GPs

`MultiTaskGP`

: a Hadamard multi-task, multi-output GP using an ICM kernel, inferring the noise level (does not require noise observations).`FixedNoiseMultiTaskGP`

: a Hadamard multi-task, multi-output GP using an ICM kernel, with fixed observation noise levels (requires noise observations).

All of the above models use Matérn 5/2 kernels with Automatic Relevance
Discovery (ARD), and have reasonable priors on hyperparameters that make them
work well in settings where the **input features are normalized to the unit
cube** and the **observations are standardized** (zero mean, unit variance).

## Implementing Custom Models

The configurability of the above models is limited (for instance, it is not straightforward to use a different kernel). Doing so is an intentional design decision -- we believe that having a few simple and easy-to-understand models for basic use cases is more valuable than having a highly complex and configurable model class whose implementation is difficult to understand.

Instead, we advocate that users implement their own models to cover more specialized use cases. The light-weight nature of BoTorch's Model API makes this easy to do. See the Using a custom BoTorch model in Ax tutorial for an example.

If you happen to implement a model that would be useful for other researchers as well (and involves more than just swapping out the Matérn kernel for an RBF kernel), please consider contributing this model to BoTorch.