def inference(self,
kern,
X,
W,
likelihood,
Y,
mean_function=None,
Y_metadata=None,
K=None,
variance=None,
Z_tilde=None):
"""
Returns a Posterior class containing essential quantities of the posterior
"""
if mean_function is None:
m = 0
else:
m = mean_function.f(X)
if variance is None:
variance = likelihood.gaussian_variance(Y_metadata)
YYT_factor = Y - m
if K is None:
K = kern.K(X)
Ky = K.copy()
diag.add(Ky, variance + 1e-8)
Wi, LW, LWi, W_logdet = pdinv(Ky)
alpha, _ = dpotrs(LW, YYT_factor, lower=1)
log_marginal = 0.5 * (-Y.size * log_2_pi - Y.shape[1] * W_logdet -
np.sum(alpha * YYT_factor))
if Z_tilde is not None:
# This is a correction term for the log marginal likelihood
# In EP this is log Z_tilde, which is the difference between the
# Gaussian marginal and Z_EP
log_marginal += Z_tilde
dL_dK = 0.5 * (tdot(alpha) - Y.shape[1] * Wi)
dL_dthetaL = likelihood.exact_inference_gradients(
np.diag(dL_dK), Y_metadata)
posterior_ = Posterior(woodbury_chol=LW, woodbury_vector=alpha, K=K)
return posterior_, log_marginal, {
'dL_dK': dL_dK,
'dL_dthetaL': dL_dthetaL,
'dL_dm': alpha
}, W_logdet
def my_predictive_gradient(Xnew, GPmodel, test_point, kern=None):
"""
Returns gradients of predictive mean, variance, and covariance
"""
if kern is None:
kern = GPmodel.kern
mean_jac = np.empty((Xnew.shape[0],Xnew.shape[1],GPmodel.output_dim))
for i in range(GPmodel.output_dim):
mean_jac[:,:,i] = kern.gradients_X(GPmodel.posterior.woodbury_vector[:,i:i+1].T, Xnew, GPmodel._predictive_variable)
# gradients wrt the diagonal part k_{xx}
dv_dX = kern.gradients_X(np.eye(Xnew.shape[0]), Xnew)
#grads wrt 'Schur' part K_{xf}K_{ff}^{-1}K_{fx}
alpha = -2.*np.dot(kern.K(Xnew, GPmodel._predictive_variable), GPmodel.posterior.woodbury_inv)
dv_dX += kern.gradients_X(alpha, Xnew, GPmodel._predictive_variable)
# Gradient w.r.t. covariance
dC_dX = kern.gradients_X(np.eye(Xnew.shape[0]), Xnew, test_point.reshape(1,test_point.size))
alpha2 = -np.dot(kern.K(test_point.reshape(1,test_point.size), GPmodel._predictive_variable), GPmodel.posterior.woodbury_inv)
dC_dX += kern.gradients_X(alpha2, Xnew, GPmodel._predictive_variable)
return mean_jac, dv_dX - 2.*dC_dX
# return mean_jac, dv_dX
def latent_funs_cov(Z, kernel_list):
"""
Description: Builds the full-covariance cov[u(z),u(z)] of a Multi-output GP for a Sparse approximation
:param Z: Inducing Points
:param kernel_list: Kernels of u_q functions priors
:return: Kuu
"""
Q = len(kernel_list)
M,Dz = Z.shape
Xdim = int(Dz/Q)
Kuu = np.empty((Q, M, M))
Luu = np.empty((Q, M, M))
Kuui = np.empty((Q, M, M))
for q, kern in enumerate(kernel_list):
Kuu[q, :, :] = kern.K(Z[:,q*Xdim:q*Xdim+Xdim],Z[:,q*Xdim:q*Xdim+Xdim])
Luu[q, :, :] = linalg.jitchol(Kuu[q, :, :])
Kuui[q, :, :], _ = linalg.dpotri(np.asfortranarray(Luu[q, :, :]))
return Kuu, Luu, Kuui
def latent_funs_cov(Z, kernel_list):
"""
Builds the full-covariance cov[u(z),u(z)] of a Multi-output GP
for a Sparse approximation
:param Z: Inducing Points
:param kernel_list: Kernels of u_q functions priors
:return: Kuu
"""
Q = len(kernel_list)
M, Dz = Z.shape
Xdim = int(Dz / Q)
#Kuu = np.zeros([Q*M,Q*M])
Kuu = np.empty((Q, M, M))
Luu = np.empty((Q, M, M))
Kuui = np.empty((Q, M, M))
for q, kern in enumerate(kernel_list):
Kuu[q, :, :] = kern.K(Z[:, q * Xdim:q * Xdim + Xdim],
Z[:, q * Xdim:q * Xdim + Xdim])
Kuu[q, :, :] = Kuu[
q, :, :] #+ 1.0e-6*np.eye(*Kuu[q, :, :].shape) #This line included by Juan for numerical stability
Luu[q, :, :] = linalg.jitchol(Kuu[q, :, :], maxtries=10)
Kuui[q, :, :], _ = linalg.dpotri(np.asfortranarray(Luu[q, :, :]))
return Kuu, Luu, Kuui