# Copyright (c) 2012-2014, GPy authors (see AUTHORS.txt).
# Licensed under the BSD 3-clause license (see LICENSE.txt)
from .posterior import PosteriorExact as Posterior
from ...util.linalg import pdinv, dpotrs, tdot
from ...util import diag
import numpy as np
from . import LatentFunctionInference
log_2_pi = np.log(2*np.pi)
[docs]class ExactGaussianInference(LatentFunctionInference):
"""
An object for inference when the likelihood is Gaussian.
The function self.inference returns a Posterior object, which summarizes
the posterior.
For efficiency, we sometimes work with the cholesky of Y*Y.T. To save repeatedly recomputing this, we cache it.
"""
def __init__(self):
pass#self._YYTfactor_cache = caching.cache()
[docs] def to_dict(self):
"""
Convert the object into a json serializable dictionary.
Note: It uses the private method _save_to_input_dict of the parent.
:return dict: json serializable dictionary containing the needed information to instantiate the object
"""
input_dict = super(ExactGaussianInference, self)._save_to_input_dict()
input_dict["class"] = "GPy.inference.latent_function_inference.exact_gaussian_inference.ExactGaussianInference"
return input_dict
[docs] def inference(self, kern, X, 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)
return Posterior(woodbury_chol=LW, woodbury_vector=alpha, K=K), log_marginal, {'dL_dK':dL_dK, 'dL_dthetaL':dL_dthetaL, 'dL_dm':alpha}
[docs] def LOO(self, kern, X, Y, likelihood, posterior, Y_metadata=None, K=None):
"""
Leave one out error as found in
"Bayesian leave-one-out cross-validation approximations for Gaussian latent variable models"
Vehtari et al. 2014.
"""
g = posterior.woodbury_vector
c = posterior.woodbury_inv
c_diag = np.diag(c)[:, None]
neg_log_marginal_LOO = 0.5*np.log(2*np.pi) - 0.5*np.log(c_diag) + 0.5*(g**2)/c_diag
#believe from Predictive Approaches for Choosing Hyperparameters in Gaussian Processes
#this is the negative marginal LOO
return -neg_log_marginal_LOO