sgpmc.py
# Copyright 2016 James Hensman, alexggmatthews
#
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#
# Unless required by applicable law or agreed to in writing, software
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and

import numpy as np
import tensorflow as tf
import tensorflow_probability as tfp

from gpflow.base import Parameter
from gpflow.features import InducingPoints
from ..conditionals import conditional
from ..models.model import GPModelOLD, MeanAndVariance

class SGPMC(GPModelOLD):
"""
This is the Sparse Variational GP using MCMC (SGPMC). The key reference is

::

@inproceedings{hensman2015mcmc,
title={MCMC for Variatinoally Sparse Gaussian Processes},
author={Hensman, James and Matthews, Alexander G. de G.
and Filippone, Maurizio and Ghahramani, Zoubin},
booktitle={Proceedings of NIPS},
year={2015}
}

The latent function values are represented by centered
(whitened) variables, so

.. math::
:nowrap:

\\begin{align}
\\mathbf v & \\sim N(0, \\mathbf I) \\\\
\\mathbf u &= \\mathbf L\\mathbf v
\\end{align}

with

.. math::
\\mathbf L \\mathbf L^\\top = \\mathbf K

"""

def __init__(self,
X,
Y,
kernel,
likelihood,
mean_function=None,
num_latent=None,
features=None,
**kwargs):
"""
X is a data matrix, size [N, D]
Y is a data matrix, size [N, R]
Z is a data matrix, of inducing inputs, size [M, D]
kernel, likelihood, mean_function are appropriate GPflow objects
"""
GPModelOLD.__init__(self,
X,
Y,
kernel,
likelihood,
mean_function,
num_latent=num_latent,
**kwargs)
self.num_data = X.shape[0]
self.feature = InducingPoints(features)
self.V = Parameter(np.zeros((len(self.feature), self.num_latent)))
self.V.prior = tfp.distributions.Normal(loc=0., scale=1.)

def log_likelihood(self, *args, **kwargs) -> tf.Tensor:
"""
This function computes the optimal density for v, q*(v), up to a constant
"""
# get the (marginals of) q(f): exactly predicting!
fmean, fvar = self.predict_f(self.X, full_cov=False)
return tf.reduce_sum(
self.likelihood.variational_expectations(fmean, fvar, self.Y))

def predict_f(self, X: tf.Tensor, full_cov=False,
full_output_cov=False) -> MeanAndVariance:
"""
Xnew is a data matrix, point at which we want to predict

This method computes

p(F* | (U=LV) )

where F* are points on the GP at Xnew, F=LV are points on the GP at Z,

"""
mu, var = conditional(X,
self.feature,
self.kernel,
self.V,
full_cov=full_cov,
q_sqrt=None,
white=True,
full_output_cov=full_output_cov)
return mu + self.mean_function(X), var