src.models.tvgp

Module for the t-VGP model class

Module Contents

class t_VGP(data: gpflow.models.model.RegressionData, kernel: gpflow.kernels.Kernel, likelihood: gpflow.likelihoods.Likelihood, mean_function: Optional[gpflow.mean_functions.MeanFunction] = None, num_latent: Optional[int] = 1)

Bases: gpflow.models.model.GPModel, gpflow.models.training_mixins.InternalDataTrainingLossMixin

This method approximates the Gaussian process posterior using a multivariate Gaussian.

The idea is that the posterior over the function-value vector F is approximated by a Gaussian, and the KL divergence is minimised between the approximation and the posterior.

The key reference is:

Khan, M., & Lin, W. (2017). Conjugate-Computation Variational Inference: Converting Variational Inference in Non-Conjugate Models to Inferences in Conjugate Models. In Artificial Intelligence and Statistics (pp. 878-887).

X is a data matrix, size [N, D] Y is a data matrix, size [N, R] kernel, likelihood, mean_function are appropriate GPflow objects

property lambda_1(self)

first natural parameter

property lambda_2(self)

second natural parameter

maximum_log_likelihood_objective(self, *args, **kwargs) → tf.Tensor

Objective for maximum likelihood estimation. Should be maximized. E.g. log-marginal likelihood (hyperparameter likelihood) for GPR, or lower bound to the log-marginal likelihood (ELBO) for sparse and variational GPs.

elbo(self) → tf.Tensor

This gives a variational bound (the evidence lower bound or ELBO) on the log marginal likelihood of the model.

update_variational_parameters(self, beta=0.05) → tf.Tensor

Takes natural gradient step in Variational parameters in the local parameters λₜ = rₜ▽[Var_exp] + (1-rₜ)λₜ₋₁ Input: :param: X : N x D :param: Y: N x 1 :param: lr: Scalar

Output: Updates the params

predict_f(self, Xnew: gpflow.models.model.InputData, full_cov: bool = False, full_output_cov: bool = False) → gpflow.models.model.MeanAndVariance

The posterior variance of F is given by

q(f) = N(f | K alpha + mean, [K⁻¹ + diag(lambda²)]⁻¹)

Here we project this to F*, the values of the GP at Xnew which is given by

q(F*) = N ( F* | K_{F} alpha + mean, K_{*} - K_{*f}[K_{ff} +

diag(lambda⁻²)]⁻¹ K_{f*} )