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new FITC... again :(
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Ricardo
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237
GPy/models/FITC.py
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237
GPy/models/FITC.py
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# Copyright (c) 2012, GPy authors (see AUTHORS.txt).
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# Licensed under the BSD 3-clause license (see LICENSE.txt)
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import numpy as np
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import pylab as pb
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from ..util.linalg import mdot, jitchol, tdot, symmetrify,pdinv
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#from ..util.linalg import mdot, jitchol, chol_inv, pdinv, trace_dot
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from ..util.plot import gpplot
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from .. import kern
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from scipy import stats, linalg
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from sparse_GP import sparse_GP
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def backsub_both_sides(L,X):
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""" Return L^-T * X * L^-1, assumuing X is symmetrical and L is lower cholesky"""
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tmp,_ = linalg.lapack.flapack.dtrtrs(L,np.asfortranarray(X),lower=1,trans=1)
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return linalg.lapack.flapack.dtrtrs(L,np.asfortranarray(tmp.T),lower=1,trans=1)[0].T
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class FITC(sparse_GP):
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def __init__(self, X, likelihood, kernel, Z, X_variance=None, normalize_X=False):
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self.Z = Z
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self.M = self.Z.shape[0]
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self.true_precision = likelihood.precision
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sparse_GP.__init__(self, X, likelihood, kernel=kernel, Z=self.Z, X_variance=None, normalize_X=False)
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def _set_params(self, p):
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self.Z = p[:self.M*self.Q].reshape(self.M, self.Q)
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self.kern._set_params(p[self.Z.size:self.Z.size+self.kern.Nparam])
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self.likelihood._set_params(p[self.Z.size+self.kern.Nparam:])
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self._compute_kernel_matrices()
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self.scale_factor = 1.
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self._computations()
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def update_likelihood_approximation(self):
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"""
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Approximates a non-gaussian likelihood using Expectation Propagation
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For a Gaussian (or direct: TODO) likelihood, no iteration is required:
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this function does nothing
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Diag(Knn - Qnn) is added to the noise term to use the tools already implemented in sparse_GP.
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The true precison is now 'true_precision' not 'precision'.
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"""
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if self.has_uncertain_inputs:
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raise NotImplementedError, "FITC approximation not implemented for uncertain inputs"
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else:
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self.likelihood.fit_FITC(self.Kmm,self.psi1,self.psi0)
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self._set_params(self._get_params()) # update the GP
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def _computations(self):
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#factor Kmm
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self.Lm = jitchol(self.Kmm)
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self.Lmi,info = linalg.lapack.flapack.dtrtrs(self.Lm,np.eye(self.M),lower=1)
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Lmipsi1 = np.dot(self.Lmi,self.psi1)
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self.Qnn = np.dot(Lmipsi1.T,Lmipsi1)
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self.Diag0 = self.psi0 - np.diag(self.Qnn)
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self.beta_star = self.likelihood.precision/(1. + self.likelihood.precision*self.Diag0[:,None]) #Includes Diag0 in the precision
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self.V_star = self.beta_star * self.likelihood.Y
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# The rather complex computations of self.A
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if self.has_uncertain_inputs:
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raise NotImplementedError
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else:
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if self.likelihood.is_heteroscedastic:
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assert self.likelihood.D == 1 # TODO: what if the likelihood is heterscedatic and there are multiple independent outputs?
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tmp = self.psi1 * (np.sqrt(self.beta_star.flatten().reshape(1, self.N)))
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tmp, _ = linalg.lapack.flapack.dtrtrs(self.Lm, np.asfortranarray(tmp), lower=1)
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self.A = tdot(tmp)
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# factor B
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self.B = np.eye(self.M) + self.A
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self.LB = jitchol(self.B)
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self.psi1V = np.dot(self.psi1, self.V_star)
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# back substutue C into psi1V
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tmp, info1 = linalg.lapack.flapack.dtrtrs(self.Lm, np.asfortranarray(self.psi1V), lower=1, trans=0)
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self._LBi_Lmi_psi1V, _ = linalg.lapack.flapack.dtrtrs(self.LB, np.asfortranarray(tmp), lower=1, trans=0)
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# dlogbeta_dtheta
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Kmmipsi1 = np.dot(self.Lmi.T,Lmipsi1)
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b_psi1_Ki = self.beta_star * Kmmipsi1.T
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Ki_pbp_Ki = np.dot(Kmmipsi1,b_psi1_Ki)
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dlogB_dpsi0 = -.5*self.kern.dKdiag_dtheta(self.beta_star,X=self.X)
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dlogB_dpsi1 = self.kern.dK_dtheta(b_psi1_Ki,self.X,self.Z)
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dlogB_dKmm = -.5*self.kern.dK_dtheta(Ki_pbp_Ki,X=self.Z)
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self.dlogB_dtheta = dlogB_dpsi0 + dlogB_dpsi1 + dlogB_dKmm
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# dyby_dtheta
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Kmmi = np.dot(self.Lmi.T,self.Lmi)
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VVT = np.outer(self.V_star,self.V_star)
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VV_p_Ki = np.dot(VVT,Kmmipsi1.T)
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Ki_pVVp_Ki = np.dot(Kmmipsi1,VV_p_Ki)
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dyby_dpsi0 = .5 * self.kern.dKdiag_dtheta(self.V_star**2,self.X)
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dyby_dpsi1 = 0
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dyby_dKmm = 0
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dyby_dtheta = dyby_dpsi0
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for psi1_n,V_n,X_n in zip(self.psi1.T,self.V_star,self.X):
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dyby_dpsi1 = -V_n**2 * np.dot(psi1_n[None,:],Kmmi)
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dyby_dtheta += self.kern.dK_dtheta(dyby_dpsi1,X_n[:,None],self.Z)
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for psi1_n,V_n,X_n in zip(self.psi1.T,self.V_star,self.X):
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psin_K = np.dot(psi1_n[None,:],Kmmi)
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tmp = np.dot(psin_K.T,psin_K)
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dyby_dKmm = .5*V_n**2 * tmp
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dyby_dtheta += self.kern.dK_dtheta(dyby_dKmm,self.Z)
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#self.dyby_dtheta = dyby_dpsi0 #+ dyby_dpsi1 + dyby_dKmm
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self.dyby_dtheta = dyby_dtheta
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# the partial derivative vector for the likelihood
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if self.likelihood.Nparams == 0:
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# save computation here.
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self.partial_for_likelihood = None
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elif self.likelihood.is_heteroscedastic:
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raise NotImplementedError, "heteroscedatic derivates not implemented"
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else:
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# likelihood is not heterscedatic
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self.partial_for_likelihood = 0
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#self.partial_for_likelihood = -0.5 * self.N * self.D * self.likelihood.precision + 0.5 * self.likelihood.trYYT * self.likelihood.precision ** 2
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#self.partial_for_likelihood += 0.5 * self.D * (self.psi0.sum() * self.likelihood.precision ** 2 - np.trace(self.A) * self.likelihood.precision)
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#self.partial_for_likelihood += self.likelihood.precision * (0.5 * np.sum(self.A * self.DBi_plus_BiPBi) - np.sum(np.square(self._LBi_Lmi_psi1V)))
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"""
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tmp, info2 = linalg.lapack.flapack.dpotrs(self.LB, tmp, lower=1)
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self.Cpsi1V, info3 = linalg.lapack.flapack.dtrtrs(self.Lm, tmp, lower=1, trans=1)
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# Compute dL_dKmm
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tmp = tdot(self._LBi_Lmi_psi1V)
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self.DBi_plus_BiPBi = backsub_both_sides(self.LB, self.D * np.eye(self.M) + tmp)
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tmp = -0.5 * self.DBi_plus_BiPBi
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tmp += -0.5 * self.B * self.D
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tmp += self.D * np.eye(self.M)
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self.dL_dKmm = backsub_both_sides(self.Lm, tmp)
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# Compute dL_dpsi # FIXME: this is untested for the heterscedastic + uncertain inputs case
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self.dL_dpsi0 = -0.5 * self.D * (self.likelihood.precision * np.ones([self.N, 1])).flatten()
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self.dL_dpsi1 = np.dot(self.Cpsi1V, self.likelihood.V.T)
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dL_dpsi2_beta = 0.5 * backsub_both_sides(self.Lm, self.D * np.eye(self.M) - self.DBi_plus_BiPBi)
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if self.likelihood.is_heteroscedastic:
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if self.has_uncertain_inputs:
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self.dL_dpsi2 = self.likelihood.precision[:, None, None] * dL_dpsi2_beta[None, :, :]
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else:
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self.dL_dpsi1 += 2.*np.dot(dL_dpsi2_beta, self.psi1 * self.likelihood.precision.reshape(1, self.N))
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self.dL_dpsi2 = None
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else:
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dL_dpsi2 = self.likelihood.precision * dL_dpsi2_beta
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if self.has_uncertain_inputs:
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# repeat for each of the N psi_2 matrices
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self.dL_dpsi2 = np.repeat(dL_dpsi2[None, :, :], self.N, axis=0)
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else:
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# subsume back into psi1 (==Kmn)
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self.dL_dpsi1 += 2.*np.dot(dL_dpsi2, self.psi1)
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self.dL_dpsi2 = None
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# the partial derivative vector for the likelihood
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if self.likelihood.Nparams == 0:
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# save computation here.
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self.partial_for_likelihood = None
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elif self.likelihood.is_heteroscedastic:
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raise NotImplementedError, "heteroscedatic derivates not implemented"
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else:
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# likelihood is not heterscedatic
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self.partial_for_likelihood = -0.5 * self.N * self.D * self.likelihood.precision + 0.5 * self.likelihood.trYYT * self.likelihood.precision ** 2
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self.partial_for_likelihood += 0.5 * self.D * (self.psi0.sum() * self.likelihood.precision ** 2 - np.trace(self.A) * self.likelihood.precision)
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self.partial_for_likelihood += self.likelihood.precision * (0.5 * np.sum(self.A * self.DBi_plus_BiPBi) - np.sum(np.square(self._LBi_Lmi_psi1V)))
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"""
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def log_likelihood(self):
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""" Compute the (lower bound on the) log marginal likelihood """
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#A = -0.5 * self.N * self.D * np.log(2.*np.pi) + 0.5 * np.sum(np.log(self.beta_star))
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A = - 0.5 * np.sum(self.V_star * self.likelihood.Y)
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"""
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A = -0.5 * self.N * self.D * np.log(2.*np.pi) + 0.5 * np.sum(np.log(self.beta_star)) - 0.5 * np.sum(self.V_star * self.likelihood.Y)
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#B = -0.5 * self.D * (np.sum(self.likelihood.precision.flatten() * self.psi0) - np.trace(self.A))
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C = -self.D * (np.sum(np.log(np.diag(self.LB))))
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D = 0.5 * np.sum(np.square(self._LBi_Lmi_psi1V))
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return A + C + D # +B
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"""
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return A
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def _log_likelihood_gradients(self):
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pass
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return np.hstack((self.dL_dZ().flatten(), self.dL_dtheta(), self.likelihood._gradients(partial=self.partial_for_likelihood)))
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def dL_dtheta(self):
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#dL_dtheta = self.dlogB_dtheta
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dL_dtheta = self.dyby_dtheta
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"""
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dL_dtheta = self.kern.dK_dtheta(self.dL_dKmm, self.Z)
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if self.has_uncertain_inputs:
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dL_dtheta += self.kern.dpsi0_dtheta(self.dL_dpsi0, self.Z, self.X, self.X_variance)
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dL_dtheta += self.kern.dpsi1_dtheta(self.dL_dpsi1.T, self.Z, self.X, self.X_variance)
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dL_dtheta += self.kern.dpsi2_dtheta(self.dL_dpsi2, self.Z, self.X, self.X_variance)
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else:
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dL_dtheta += self.kern.dK_dtheta(self.dL_dpsi1, self.Z, self.X)
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dL_dtheta += self.kern.dKdiag_dtheta(self.dL_dpsi0, self.X)
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"""
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return dL_dtheta
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def dL_dZ(self):
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dL_dZ = np.zeros(self.M)
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"""
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dL_dZ = 2.*self.kern.dK_dX(self.dL_dKmm, self.Z) # factor of two becase of vertical and horizontal 'stripes' in dKmm_dZ
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if self.has_uncertain_inputs:
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dL_dZ += self.kern.dpsi1_dZ(self.dL_dpsi1, self.Z, self.X, self.X_variance)
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dL_dZ += self.kern.dpsi2_dZ(self.dL_dpsi2, self.Z, self.X, self.X_variance)
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else:
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dL_dZ += self.kern.dK_dX(self.dL_dpsi1, self.Z, self.X)
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"""
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return dL_dZ
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