# Source code for GPy.inference.latent_function_inference.pep

from .posterior import Posterior
from ...util.linalg import jitchol, tdot, dtrtrs, dtrtri, pdinv
from ...util import diag
import numpy as np
from . import LatentFunctionInference
log_2_pi = np.log(2*np.pi)

[docs]class PEP(LatentFunctionInference):
'''
Sparse Gaussian processes using Power-Expectation Propagation
for regression: alpha \approx 0 gives VarDTC and alpha = 1 gives FITC

Reference: A Unifying Framework for Sparse Gaussian Process Approximation using
Power Expectation Propagation, https://arxiv.org/abs/1605.07066

'''
const_jitter = 1e-6

def __init__(self, alpha):
super(PEP, self).__init__()
self.alpha = alpha

[docs]    def inference(self, kern, X, Z, likelihood, Y, mean_function=None, Y_metadata=None):
assert mean_function is None, "inference with a mean function not implemented"

num_inducing, _ = Z.shape
num_data, output_dim = Y.shape

#make sure the noise is not hetero
if sigma_n.size >1:
raise NotImplementedError("no hetero noise with this implementation of PEP")

Kmm = kern.K(Z)
Knn = kern.Kdiag(X)
Knm = kern.K(X, Z)
U = Knm

#factor Kmm
Kmmi, L, Li, _ = pdinv(Kmm)

#compute beta_star, the effective noise precision
LiUT = np.dot(Li, U.T)
sigma_star = sigma_n + self.alpha * (Knn - np.sum(np.square(LiUT),0))
beta_star = 1./sigma_star

# Compute and factor A
A = tdot(LiUT*np.sqrt(beta_star)) + np.eye(num_inducing)
LA = jitchol(A)

# back substitute to get b, P, v
URiy = np.dot(U.T*beta_star,Y)
tmp, _ = dtrtrs(L, URiy, lower=1)
b, _ = dtrtrs(LA, tmp, lower=1)
tmp, _ = dtrtrs(LA, b, lower=1, trans=1)
v, _ = dtrtrs(L, tmp, lower=1, trans=1)
tmp, _ = dtrtrs(LA, Li, lower=1, trans=0)
P = tdot(tmp.T)

alpha_const_term = (1.0-self.alpha) / self.alpha

#compute log marginal
log_marginal = -0.5*num_data*output_dim*np.log(2*np.pi) + \
-np.sum(np.log(np.diag(LA)))*output_dim + \
0.5*output_dim*(1+alpha_const_term)*np.sum(np.log(beta_star)) + \
-0.5*np.sum(np.square(Y.T*np.sqrt(beta_star))) + \
0.5*np.sum(np.square(b)) + 0.5*alpha_const_term*num_data*np.log(sigma_n)
#compute dL_dR
Uv = np.dot(U, v)
dL_dR = 0.5*(np.sum(U*np.dot(U,P), 1) - (1.0+alpha_const_term)/beta_star + np.sum(np.square(Y), 1) - 2.*np.sum(Uv*Y, 1) \
+ np.sum(np.square(Uv), 1))*beta_star**2

# Compute dL_dKmm
vvT_P = tdot(v.reshape(-1,1)) + P
dL_dK = 0.5*(Kmmi - vvT_P)
KiU = np.dot(Kmmi, U.T)
dL_dK += self.alpha * np.dot(KiU*dL_dR, KiU.T)

# Compute dL_dU
vY = np.dot(v.reshape(-1,1),Y.T)
dL_dU = vY - np.dot(vvT_P, U.T)
dL_dU *= beta_star
dL_dU -= self.alpha * 2.*KiU*dL_dR