adding var_gauss.py

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}