diff --git a/GPy/inference/latent_function_inference/var_gauss.py b/GPy/inference/latent_function_inference/var_gauss.py new file mode 100644 index 00000000..e71416b4 --- /dev/null +++ b/GPy/inference/latent_function_inference/var_gauss.py @@ -0,0 +1,66 @@ +# 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) + +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 + + 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 + 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}