# Copyright (c) 2015, James Hensman
# Licensed under the BSD 3-clause license (see LICENSE.txt)
import numpy as np
from ...util.linalg import pdinv
from .posterior import Posterior
from . import LatentFunctionInference
log_2_pi = np.log(2*np.pi)
[docs]class VarGauss(LatentFunctionInference):
"""
The Variational Gaussian Approximation revisited
@article{Opper:2009,
title = {The Variational Gaussian Approximation Revisited},
author = {Opper, Manfred and Archambeau, C{\'e}dric},
journal = {Neural Comput.},
year = {2009},
pages = {786--792},
}
"""
def __init__(self, alpha, beta):
"""
:param alpha: GPy.core.Param varational parameter
:param beta: GPy.core.Param varational parameter
"""
self.alpha, self.beta = alpha, beta
[docs] def inference(self, kern, X, likelihood, Y, mean_function=None, Y_metadata=None, Z=None):
if mean_function is not None:
raise NotImplementedError
num_data, output_dim = Y.shape
assert output_dim ==1, "Only one output supported"
K = kern.K(X)
m = K.dot(self.alpha)
KB = K*self.beta[:, None]
BKB = KB*self.beta[None, :]
A = np.eye(num_data) + BKB
Ai, LA, _, Alogdet = pdinv(A)
Sigma = np.diag(self.beta**-2) - Ai/self.beta[:, None]/self.beta[None, :] # posterior coavairance: need full matrix for gradients
var = np.diag(Sigma).reshape(-1,1)
F, dF_dm, dF_dv, dF_dthetaL = likelihood.variational_expectations(Y, m, var, Y_metadata=Y_metadata)
if dF_dthetaL is not None:
dL_dthetaL = dF_dthetaL.sum(1).sum(1)
else:
dL_dthetaL = np.array([])
dF_da = np.dot(K, dF_dm)
SigmaB = Sigma*self.beta
#dF_db_ = -np.diag(Sigma.dot(np.diag(dF_dv.flatten())).dot(SigmaB))*2
dF_db = -2*np.sum(Sigma**2 * (dF_dv * self.beta), 0)
#assert np.allclose(dF_db, dF_db_)
KL = 0.5*(Alogdet + np.trace(Ai) - num_data + np.sum(m*self.alpha))
dKL_da = m
A_A2 = Ai - Ai.dot(Ai)
dKL_db = np.diag(np.dot(KB.T, A_A2))
log_marginal = F.sum() - KL
self.alpha.gradient = dF_da - dKL_da
self.beta.gradient = dF_db - dKL_db
# K-gradients
dKL_dK = 0.5*(self.alpha*self.alpha.T + self.beta[:, None]*self.beta[None, :]*A_A2)
tmp = Ai*self.beta[:, None]/self.beta[None, :]
dF_dK = self.alpha*dF_dm.T + np.dot(tmp*dF_dv, tmp.T)
return Posterior(mean=m, cov=Sigma ,K=K),\
log_marginal,\
{'dL_dK':dF_dK-dKL_dK, 'dL_dthetaL':dL_dthetaL}