elbo

gpjax.objectives.elbo(variational_family, data)

Compute the evidence lower bound of a variational approximation.

Compute the evidence lower bound under this model. In short, this requires evaluating the expectation of the model’s log-likelihood under the variational approximation. To this, we sum the KL divergence from the variational posterior to the prior. When batching occurs, the result is scaled by the batch size relative to the full dataset size.

Example

>>> import gpjax as gpx
>>> import jax.numpy as jnp

>>> xtrain = jnp.linspace(0, 1).reshape(-1, 1)
>>> ytrain = jnp.sin(xtrain)
>>> D = gpx.Dataset(X=xtrain, y=ytrain)

>>> meanf = gpx.mean_functions.Constant()
>>> kernel = gpx.kernels.RBF()
>>> likelihood = gpx.likelihoods.Bernoulli()
>>> prior = gpx.gps.Prior(mean_function=meanf, kernel=kernel)
>>> posterior = prior * likelihood

>>> z = jnp.linspace(0, 1, 10).reshape(-1, 1)
>>> q = gpx.variational_families.VariationalGaussian(
...     model=posterior, inducing_inputs=z
... )

>>> gpx.objectives.elbo(q, D)

Parameters:

• variational_family (VF) – The variational approximation for whose parameters we should maximise the ELBO with respect to.

• data (Dataset) – The training data for which we should maximise the ELBO with respect to.

Returns:

The evidence lower bound of the variational approximation.

Return type:

ScalarFloat

Expand for references to gpjax.objectives.elbo

Dual Parameterisation of Sparse GPs (t-SVGP)

Natural Gradients

fit_natgrads