sgpmc.py
# Copyright 2016 James Hensman, alexggmatthews
#
# Licensed under the Apache License, Version 2.0 (the "License");
# 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
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.

import numpy as np
import tensorflow as tf

from ..models.model import GPModel
from ..conditionals import conditional
from ..features import inducingpoint_wrapper
from ..params import Parameter, DataHolder
from ..priors import Gaussian
from ..decors import params_as_tensors

class SGPMC(GPModel):
r"""
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, kern, likelihood, feat=None,
mean_function=None,
num_latent=None,
Z=None,
**kwargs):
"""
X is a data matrix, size N x D
Y is a data matrix, size N x R
Z is a data matrix, of inducing inputs, size M x D
kern, likelihood, mean_function are appropriate GPflow objects

"""
X = DataHolder(X)
Y = DataHolder(Y)
GPModel.__init__(self, X, Y, kern, likelihood, mean_function, num_latent=num_latent, **kwargs)
self.num_data = X.shape[0]
self.feature = inducingpoint_wrapper(feat, Z)
self.V = Parameter(np.zeros((len(self.feature), self.num_latent)))
self.V.prior = Gaussian(0., 1.)

@params_as_tensors
def _build_likelihood(self):
"""
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._build_predict(self.X, full_cov=False)
return tf.reduce_sum(self.likelihood.variational_expectations(fmean, fvar, self.Y))

@params_as_tensors
def _build_predict(self, Xnew, full_cov=False, full_output_cov=False):
"""
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(Xnew, self.feature, self.kern, self.V, full_cov=full_cov, q_sqrt=None,
white=True, full_output_cov=full_output_cov)
return mu + self.mean_function(Xnew), var