def _prepare(self):
D, Mr, Mc = self.Xr.shape[0], self.Zr.shape[
0], self.LcInvMLrInvT.shape[0]
psi2_r_n = self.kern_r.psi2n(self.Zr, self.Xr)
psi0_r = self.kern_r.psi0(self.Zr, self.Xr)
psi1_r = self.kern_r.psi1(self.Zr, self.Xr)
LrInvPsi1_rT = dtrtrs(self.Lr, psi1_r.T)[0]
self.woodbury_vector = self.LcInvMLrInvT.dot(LrInvPsi1_rT)
LrInvPsi2_r_nLrInvT = dtrtrs(
self.Lr,
np.swapaxes(
(dtrtrs(self.Lr,
psi2_r_n.reshape(D * Mr,
Mr).T)[0].T).reshape(D, Mr, Mr), 1,
2).reshape(D * Mr, Mr).T)[0].T.reshape(D, Mr, Mr)
tr_LrInvPsi2_r_nLrInvT = LrInvPsi2_r_nLrInvT.reshape(D, Mr * Mr).sum(1)
tr_LrInvPsi2_r_nLrInvT_LrInvSrLrInvT = LrInvPsi2_r_nLrInvT.reshape(
D, Mr * mr).dot(self.LrInvSrLrInvT.flat)
tmp = LrInvPsi2_r_nLrInvT - LrInvPsi1_rT.T[:, :,
None] * LrInvPsi1_rT.T[:,
None, :]
tmp = np.swapaxes(
tmp.reshape(D * Mr,
Mr).dot(self.LcInvMLrInvT.T).reshape(D, Mr, Mc), 1,
2).reshape(D * Mc, Mr).dot(self.LcInvMLrInvT.T).reshape(D, Mc, Mc)
def _compute_dL_dR(likelihood, het_noise, uncertain_inputs, LB, _LBi_Lmi_psi1Vf, DBi_plus_BiPBi, Lm, A, psi0, psi1, beta, data_fit, num_data, output_dim, trYYT, Y, VVT_factr=None):
# the partial derivative vector for the likelihood
if likelihood.size == 0:
# save computation here.
dL_dR = None
elif het_noise:
if uncertain_inputs:
raise(NotImplementedError, "heteroscedatic derivates with uncertain inputs not implemented")
else:
#from ...util.linalg import chol_inv
#LBi = chol_inv(LB)
LBi, _ = dtrtrs(LB,np.eye(LB.shape[0]))
Lmi_psi1, nil = dtrtrs(Lm, psi1.T, lower=1, trans=0)
_LBi_Lmi_psi1, _ = dtrtrs(LB, Lmi_psi1, lower=1, trans=0)
dL_dR = -0.5 * beta + 0.5 * VVT_factr**2
dL_dR += 0.5 * output_dim * (psi0 - np.sum(Lmi_psi1**2,0))[:,None] * beta**2
dL_dR += 0.5*np.sum(mdot(LBi.T,LBi,Lmi_psi1)*Lmi_psi1,0)[:,None]*beta**2
dL_dR += -np.dot(_LBi_Lmi_psi1Vf.T,_LBi_Lmi_psi1).T * Y * beta**2
dL_dR += 0.5*np.dot(_LBi_Lmi_psi1Vf.T,_LBi_Lmi_psi1).T**2 * beta**2
else:
# likelihood is not heteroscedatic
dL_dR = -0.5 * num_data * output_dim * beta + 0.5 * trYYT * beta ** 2
dL_dR += 0.5 * output_dim * (psi0.sum() * beta ** 2 - np.trace(A) * beta)
dL_dR += beta * (0.5 * np.sum(A * DBi_plus_BiPBi) - data_fit)
return dL_dR
def parameters_changed(self):
N, D = self.Y.shape
Kss = self.kern.K(self.X)
Ksu = self.kern.K(self.X, self.Z)
wv = self.posterior.woodbury_vector
wi = self.posterior.woodbury_inv
a = self.Y - Ksu.dot(wv)
C = Kss + np.eye(N)*self.likelihood.variance - Ksu.dot(wi).dot(Ksu.T)
Lc = jitchol(C)
LcInva = dtrtrs(Lc, a)[0]
LcInv = dtrtri(Lc)
CInva = dtrtrs(Lc, LcInva,trans=1)[0]
self._log_marginal_likelihood = -N*D/2.*np.log(2*np.pi) - D*np.log(np.diag(Lc)).sum() - np.square(LcInva).sum()/2.
dKsu = CInva.dot(wv.T)
dKss = tdot(CInva)/2. -D* tdot(LcInv.T)/2.
dKsu += -2. * dKss.dot(Ksu).dot(wi)
X_grad = self.kern.gradients_X(dKss, self.X)
X_grad += self.kern.gradients_X(dKsu, self.X, self.Z)
self.X.gradient = X_grad
if self.uncertain_input:
# Update Log-likelihood
KL_div = self.variational_prior.KL_divergence(self.X)
# update for the KL divergence
self.variational_prior.update_gradients_KL(self.X)
self._log_marginal_likelihood += -KL_div
def _raw_predict(self, kern, Xnew, pred_var, full_cov=False):
N = Xnew.shape[0]
psi1_c = kern.K(Xnew, pred_var)
psi0_c = kern.Kdiag(Xnew)
LcInvPsi1_cT = dtrtrs(self.Lc, psi1_c.T)[0]
D, Mr, Mc = self.Xr.shape[0], self.Zr.shape[
0], self.LcInvMLrInvT.shape[0]
psi2_r_n = self.kern_r.psi2n(self.Zr, self.Xr)
psi0_r = self.kern_r.psi0(self.Zr, self.Xr)
psi1_r = self.kern_r.psi1(self.Zr, self.Xr)
LrInvPsi1_rT = dtrtrs(self.Lr, psi1_r.T)[0]
woodbury_vector = self.LcInvMLrInvT.dot(LrInvPsi1_rT)
mu = np.dot(LcInvPsi1_cT.T, woodbury_vector)
LrInvPsi2_r_nLrInvT = dtrtrs(
self.Lr,
np.swapaxes(
(dtrtrs(self.Lr,
psi2_r_n.reshape(D * Mr,
Mr).T)[0].T).reshape(D, Mr, Mr), 1,
2).reshape(D * Mr, Mr).T)[0].T.reshape(D, Mr, Mr)
tr_LrInvPsi2_r_nLrInvT = np.diagonal(LrInvPsi2_r_nLrInvT,
axis1=1,
axis2=2).sum(1)
tr_LrInvPsi2_r_nLrInvT_LrInvSrLrInvT = LrInvPsi2_r_nLrInvT.reshape(
D, Mr * Mr).dot(self.LrInvSrLrInvT.flat)
tmp = LrInvPsi2_r_nLrInvT - LrInvPsi1_rT.T[:, :,
None] * LrInvPsi1_rT.T[:,
None, :]
tmp = np.swapaxes(
tmp.reshape(D * Mr,
Mr).dot(self.LcInvMLrInvT.T).reshape(D, Mr, Mc), 1,
2).reshape(D * Mc, Mr).dot(self.LcInvMLrInvT.T).reshape(D, Mc, Mc)
var1 = (tmp.reshape(D * Mc, Mc).dot(LcInvPsi1_cT).reshape(D, Mc, N) *
LcInvPsi1_cT[None, :, :]).sum(1).T
var2 = psi0_c[:, None] * psi0_r[None, :]
var3 = tr_LrInvPsi2_r_nLrInvT[None, :] * np.square(LcInvPsi1_cT).sum(
0)[:, None]
var4 = tr_LrInvPsi2_r_nLrInvT_LrInvSrLrInvT[None, :] * (
self.LcInvScLcInvT.dot(LcInvPsi1_cT) * LcInvPsi1_cT).sum(0)[:,
None]
var = var1 + var2 - var3 + var4
return mu, var
def do_computations(self):
"""
Here we do all the computations that are required whenever the kernels
or the variational parameters are changed.
"""
# sufficient stats.
self.ybark = np.dot(self.phi.T, self.Y).T
# compute posterior variances of each cluster (lambda_inv)
tmp = backsub_both_sides(self.Sy_chol, self.Sf, transpose="right")
self.Cs = [np.eye(self.D) + tmp * phi_hat_i for phi_hat_i in self.phi_hat]
self._C_chols = [jitchol(C) for C in self.Cs]
self.log_det_diff = np.array([2.0 * np.sum(np.log(np.diag(L))) for L in self._C_chols])
tmp = [dtrtrs(L, self.Sy_chol.T, lower=1)[0] for L in self._C_chols]
self.Lambda_inv = np.array(
[
(self.Sy - np.dot(tmp_i.T, tmp_i)) / phi_hat_i if (phi_hat_i > 1e-6) else self.Sf
for phi_hat_i, tmp_i in zip(self.phi_hat, tmp)
]
)
# posterior mean and other useful quantities
self.Syi_ybark, _ = dpotrs(self.Sy_chol, self.ybark, lower=1)
self.Syi_ybarkybarkT_Syi = self.Syi_ybark.T[:, None, :] * self.Syi_ybark.T[:, :, None]
self.muk = (self.Lambda_inv * self.Syi_ybark.T[:, :, None]).sum(1).T
def update_kern_grads(self):
"""
Set the derivative of the lower bound wrt the (kernel) parameters
"""
tmp = [dtrtrs(L, self.Sy_chol_inv, lower=1)[0] for L in self._C_chols]
B_invs = [phi_hat_i*np.dot(tmp_i.T, tmp_i) for phi_hat_i, tmp_i in zip(self.phi_hat, tmp)]
#B_invs = [phi_hat_i*mdot(self.Sy_chol_inv.T,Ci,self.Sy_chol_inv) for phi_hat_i, Ci in zip(self.phi_hat,self.C_invs)]
#heres the mukmukT*Lambda term
LiSfi = [np.eye(self.D)-np.dot(self.Sf,Bi) for Bi in B_invs]#seems okay
tmp1 = [np.dot(LiSfik.T,Sy_inv_ybark_k) for LiSfik, Sy_inv_ybark_k in zip(LiSfi,self.Syi_ybark.T)]
tmp = 0.5*sum([np.dot(tmpi[:,None],tmpi[None,:]) for tmpi in tmp1])
#here's the difference in log determinants term
tmp += -0.5*sum(B_invs)
#kernF_grads = np.array([np.sum(tmp*g) for g in self.kernF.extract_gradients()]) # OKAY!
self.kernF.update_gradients_full(dL_dK=tmp,X=self.X)
#gradient wrt Sigma_Y
ybarkybarkT = self.ybark.T[:,None,:]*self.ybark.T[:,:,None]
Byks = [np.dot(Bi,yk) for Bi,yk in zip(B_invs,self.ybark.T)]
tmp = sum([np.dot(Byk[:,None],Byk[None,:])/np.power(ph_k,3)\
-Syi_ybarkybarkT_Syi/ph_k -Bi/ph_k for Bi, Byk, yyT, ph_k, Syi_ybarkybarkT_Syi in zip(B_invs, Byks, ybarkybarkT, self.phi_hat, self.Syi_ybarkybarkT_Syi) if ph_k >1e-6])
tmp += (self.K-self.N)*self.Sy_inv
tmp += mdot(self.Sy_inv,self.YTY,self.Sy_inv)
tmp /= 2.
#kernY_grads = np.array([np.sum(tmp*g) for g in self.kernY.extract_gradients()])
self.kernY.update_gradients_full(dL_dK=tmp, X=self.X)
def do_computations(self):
"""
Here we do all the computations that are required whenever the kernels
or the variational parameters are changed.
"""
#sufficient stats.
self.ybark = np.dot(self.phi.T, self.Y).T
# compute posterior variances of each cluster (lambda_inv)
tmp = backsub_both_sides(self.Sy_chol, self.Sf, transpose='right')
self.Cs = [
np.eye(self.D) + tmp * phi_hat_i for phi_hat_i in self.phi_hat
]
self._C_chols = [jitchol(C) for C in self.Cs]
self.log_det_diff = np.array(
[2. * np.sum(np.log(np.diag(L))) for L in self._C_chols])
tmp = [dtrtrs(L, self.Sy_chol.T, lower=1)[0] for L in self._C_chols]
self.Lambda_inv = np.array([
(self.Sy - np.dot(tmp_i.T, tmp_i)) / phi_hat_i if
(phi_hat_i > 1e-6) else self.Sf
for phi_hat_i, tmp_i in zip(self.phi_hat, tmp)
])
#posterior mean and other useful quantities
self.Syi_ybark, _ = dpotrs(self.Sy_chol, self.ybark, lower=1)
self.Syi_ybarkybarkT_Syi = self.Syi_ybark.T[:,
None, :] * self.Syi_ybark.T[:, :,
None]
self.muk = (self.Lambda_inv * self.Syi_ybark.T[:, :, None]).sum(1).T
def predict_components(self, Xnew):
"""The predictive density under each component"""
tmp = [dtrtrs(L, self.Sy_chol_inv, lower=1)[0] for L in self._C_chols]
B_invs = [
phi_hat_i * np.dot(tmp_i.T, tmp_i)
for phi_hat_i, tmp_i in zip(self.phi_hat, tmp)
]
kx = self.kernF.K(self.X, Xnew)
try:
kxx = self.kernF.K(Xnew) + self.kernY.K(Xnew)
except TypeError:
#kernY has a hierarchical structure that we should deal with
con = np.ones((Xnew.shape[0], self.kernY.connections.shape[1]))
kxx = self.kernF.K(Xnew) + self.kernY.K(Xnew, con)
#prediction as per my notes
tmp = [np.eye(self.D) - np.dot(Bi, self.Sf) for Bi in B_invs]
mu = [
mdot(kx.T, tmpi, self.Sy_inv, ybark)
for tmpi, ybark in zip(tmp, self.ybark.T)
]
var = [kxx - mdot(kx.T, Bi, kx) for Bi in B_invs]
return mu, var
def _inference(K: np.ndarray, ga_approx: GaussianApproximation, cav_params: CavityParams, Z_tilde: float,
y: List[Tuple[int, float]], yc: List[List[Tuple[int, int]]]) -> Tuple[Posterior, int, Dict]:
"""
Compute the posterior approximation
:param K: prior covariance matrix
:param ga_approx: Gaussian approximation of the batches
:param cav_params: Cavity parameters of the posterior
:param Z_tilde: Log marginal likelihood
:param y: Direct observations as a list of tuples telling location index (row in X) and observation value.
:param yc: Batch comparisons in a list of lists of tuples. Each batch is a list and tuples tell the comparisons (winner index, loser index)
:return: A tuple consisting of the posterior approximation, log marginal likelihood and gradient dictionary
"""
log_marginal, post_params = _ep_marginal(K, ga_approx, Z_tilde,y,yc)
tau_tilde_root = sqrtm_block(ga_approx.tau, y,yc)
Sroot_tilde_K = np.dot(tau_tilde_root, K)
aux_alpha , _ = dpotrs(post_params.L, np.dot(Sroot_tilde_K, ga_approx.v), lower=1)
alpha = (ga_approx.v - np.dot(tau_tilde_root, aux_alpha))[:,None] #(K + Sigma^(\tilde))^(-1) /mu^(/tilde)
LWi, _ = dtrtrs(post_params.L, tau_tilde_root, lower=1)
Wi = np.dot(LWi.T,LWi)
symmetrify(Wi) #(K + Sigma^(\tilde))^(-1)
dL_dK = 0.5 * (tdot(alpha) - Wi)
dL_dthetaL = 0
return Posterior(woodbury_inv=np.asfortranarray(Wi), woodbury_vector=alpha, K=K), log_marginal, {'dL_dK':dL_dK, 'dL_dthetaL':dL_dthetaL, 'dL_dm':alpha}
def _raw_predict(self, kern, Xnew, A_ast, pred_var, full_cov=False):
"""
pred_var: _predictive_variable, X_t (all X up to round t)
"""
# print('PosteriorExactGroup _raw_predict')
# NOTE: change Kx to AKx and add .dot(A_ast.T)
# NOTE: 20210827 confirm mu and var (for self._woodbury_chol.ndim == 2 case)
Kx = self.A.dot(kern.K(pred_var, Xnew)).dot(A_ast.T) # A_t k(X_t, X_\ast) A_\ast
mu = np.dot(Kx.T, self.woodbury_vector)
# mu = A_\ast k(X_t, X_\ast)^T A_t^T (A_t k(X_t, X_t)A_t^T + sigma^2 I)^{-1} (Y_t - m)
if len(mu.shape) == 1:
mu = mu.reshape(-1, 1)
if full_cov:
Kxx = kern.K(Xnew) # k(X_ast, X_ast)
# self._woodbury_chol Cholesky decomposition of A_t k(X_t, X_t)A_t^T + sigma^2 I
if self._woodbury_chol.ndim == 2:
# DTRTRS solves a triangular system of the form A * X = B or A**T * X = B, where A is a triangular matrix of order N, and B is an N-by-NRHS matrix. A check is made to verify that A is nonsingular.
tmp = dtrtrs(self._woodbury_chol, Kx)[0] # (A_t k(X_t, X_t)A_t^T + sigma^2 I)^{-1} k(X_ast, X_ast) -> v
# tdot: returns np.dot(mat, mat.T), but faster for large 2D arrays of doubles.
var = A_ast.dot(Kxx - tdot(tmp.T)).dot(A_ast.T)
elif self._woodbury_chol.ndim == 3: # Missing data
raise NotImplementedError('Need to be extended to group case!')
var = np.empty((Kxx.shape[0], Kxx.shape[1], self._woodbury_chol.shape[2]))
for i in range(var.shape[2]):
tmp = dtrtrs(self._woodbury_chol[:, :, i], Kx)[0]
var[:, :, i] = (Kxx - tdot(tmp.T))
var = var
else:
Kxx = np.diag(A_ast.dot(kern.K(Xnew, Xnew)).dot(A_ast.T))
if self._woodbury_chol.ndim == 2:
tmp = dtrtrs(self._woodbury_chol, Kx)[0]
var = (Kxx - np.square(tmp).sum(0))[:, None]
elif self._woodbury_chol.ndim == 3: # Missing data
raise NotImplementedError('Need to be extended to group case!')
var = np.empty((Kxx.shape[0], self._woodbury_chol.shape[2]))
for i in range(var.shape[1]):
tmp = dtrtrs(self._woodbury_chol[:, :, i], Kx)[0]
var[:, i] = (Kxx - np.square(tmp).sum(0))
var = var
return mu, var
def _compute_B_statistics(K, W):
if np.any(np.isnan(W)):
raise ValueError('One or more element(s) of W is NaN')
W_12 = np.sqrt(W)
B = np.eye(K.shape[0]) + W_12*K*W_12.T
L = jitchol(B)
LiW12, _ = dtrtrs(L, np.diagflat(W_12), lower=1, trans=0)
K_Wi_i = np.dot(LiW12.T, LiW12) # R = W12BiW12, in R&W p 126, eq 5.25
C = np.dot(LiW12, K)
Ki_W_i = K - C.T.dot(C)
I_KW_i = np.eye(K.shape[0]) - np.dot(K, K_Wi_i)
logdet_I_KW = 2*np.sum(np.log(np.diag(L)))
return K_Wi_i, logdet_I_KW, I_KW_i, Ki_W_i
def compute_dl_dK(posterior, K, eta, theta, prior_mean = 0):
tau, v = theta, eta
tau_tilde_root = np.sqrt(tau)
Sroot_tilde_K = tau_tilde_root[:,None] * K
aux_alpha , _ = dpotrs(posterior.L, np.dot(Sroot_tilde_K, v), lower=1)
alpha = (v - tau_tilde_root * aux_alpha)[:,None] #(K + Sigma^(\tilde))^(-1) /mu^(/tilde)
LWi, _ = dtrtrs(posterior.L, np.diag(tau_tilde_root), lower=1)
Wi = np.dot(LWi.T, LWi)
symmetrify(Wi) #(K + Sigma^(\tilde))^(-1)
dL_dK = 0.5 * (tdot(alpha) - Wi)
return dL_dK
def _inference(K, ga_approx, cav_params, likelihood, Z_tilde, Y_metadata=None):
log_marginal, post_params = _ep_marginal(K, ga_approx, Z_tilde)
tau_tilde_root = np.sqrt(ga_approx.tau)
Sroot_tilde_K = tau_tilde_root[:,None] * K
aux_alpha , _ = dpotrs(post_params.L, np.dot(Sroot_tilde_K, ga_approx.v), lower=1)
alpha = (ga_approx.v - tau_tilde_root * aux_alpha)[:,None] #(K + Sigma^(\tilde))^(-1) /mu^(/tilde)
LWi, _ = dtrtrs(post_params.L, np.diag(tau_tilde_root), lower=1)
Wi = np.dot(LWi.T,LWi)
symmetrify(Wi) #(K + Sigma^(\tilde))^(-1)
dL_dK = 0.5 * (tdot(alpha) - Wi)
dL_dthetaL = 0 #likelihood.ep_gradients(Y, cav_params.tau, cav_params.v, np.diag(dL_dK), Y_metadata=Y_metadata, quad_mode='gh')
#temp2 = likelihood.ep_gradients(Y, cav_params.tau, cav_params.v, np.diag(dL_dK), Y_metadata=Y_metadata, quad_mode='naive')
#temp = likelihood.exact_inference_gradients(np.diag(dL_dK), Y_metadata = Y_metadata)
#print("exact: {}, approx: {}, Ztilde: {}, naive: {}".format(temp, dL_dthetaL, Z_tilde, temp2))
return Posterior(woodbury_inv=Wi, woodbury_vector=alpha, K=K), log_marginal, {'dL_dK':dL_dK, 'dL_dthetaL':dL_dthetaL, 'dL_dm':alpha}
def predict_components(self,Xnew):
"""The predictive density under each component"""
tmp = [dtrtrs(L, self.Sy_chol_inv, lower=1)[0] for L in self._C_chols]
B_invs = [phi_hat_i*np.dot(tmp_i.T, tmp_i) for phi_hat_i, tmp_i in zip(self.phi_hat, tmp)]
kx= self.kernF.K(self.X,Xnew)
try:
kxx = self.kernF.K(Xnew) + self.kernY.K(Xnew)
except TypeError:
#kernY has a hierarchical structure that we should deal with
con = np.ones((Xnew.shape[0],self.kernY.connections.shape[1]))
kxx = self.kernF.K(Xnew) + self.kernY.K(Xnew,con)
#prediction as per my notes
tmp = [np.eye(self.D) - np.dot(Bi,self.Sf) for Bi in B_invs]
mu = [mdot(kx.T,tmpi,self.Sy_inv,ybark) for tmpi,ybark in zip(tmp,self.ybark.T)]
var = [kxx - mdot(kx.T,Bi,kx) for Bi in B_invs]
return mu,var
def comp_KL_qU(self, qU_mean ,qU_var):
M,D = qU_mean.shape[0], qU_mean.shape[1]
qU_L = self.mid['qU_L']
L = self.mid['L']
Linvmu = self.mid['Linvmu']
LinvLu = self.mid['LinvLu']
KuuInv = dpotri(L, lower=1)[0]
Lu = qU_L
LuInv = dtrtri(Lu)
KL = D*M/-2. - np.log(np.diag(Lu)).sum()*D +np.log(np.diag(L)).sum()*D + np.square(LinvLu).sum()/2.*D + np.square(Linvmu).sum()/2.
dKL_dqU_mean = dtrtrs(L, Linvmu, trans=True)[0]
dKL_dqU_var = (tdot(LuInv.T)/-2. + KuuInv/2.)*D
dKL_dKuu = KuuInv*D/2. -KuuInv.dot( tdot(qU_mean)+qU_var*D).dot(KuuInv)/2.
return float(KL), dKL_dqU_mean, dKL_dqU_var, dKL_dKuu
def _log_likelihood_gradients(self):
"""
The derivative of the lower bound wrt the (kernel) parameters
"""
tmp = [dtrtrs(L, self.Sy_chol_inv, lower=1)[0] for L in self._C_chols]
B_invs = [
phi_hat_i * np.dot(tmp_i.T, tmp_i)
for phi_hat_i, tmp_i in zip(self.phi_hat, tmp)
]
#B_invs = [phi_hat_i*mdot(self.Sy_chol_inv.T,Ci,self.Sy_chol_inv) for phi_hat_i, Ci in zip(self.phi_hat,self.C_invs)]
#heres the mukmukT*Lambda term
LiSfi = [np.eye(self.D) - np.dot(self.Sf, Bi)
for Bi in B_invs] #seems okay
tmp1 = [
np.dot(LiSfik.T, Sy_inv_ybark_k)
for LiSfik, Sy_inv_ybark_k in zip(LiSfi, self.Syi_ybark.T)
]
tmp = 0.5 * sum(
[np.dot(tmpi[:, None], tmpi[None, :]) for tmpi in tmp1])
#here's the difference in log determinants term
tmp += -0.5 * sum(B_invs)
#kernF_grads = np.array([np.sum(tmp*g) for g in self.kernF.extract_gradients()]) # OKAY!
kernF_grads = self.kernF.dK_dtheta(tmp, self.X)
#gradient wrt Sigma_Y
ybarkybarkT = self.ybark.T[:, None, :] * self.ybark.T[:, :, None]
Byks = [np.dot(Bi, yk) for Bi, yk in zip(B_invs, self.ybark.T)]
tmp = sum([np.dot(Byk[:,None],Byk[None,:])/np.power(ph_k,3)\
-Syi_ybarkybarkT_Syi/ph_k -Bi/ph_k for Bi, Byk, yyT, ph_k, Syi_ybarkybarkT_Syi in zip(B_invs, Byks, ybarkybarkT, self.phi_hat, self.Syi_ybarkybarkT_Syi) if ph_k >1e-6])
tmp += (self.K - self.N) * self.Sy_inv
tmp += mdot(self.Sy_inv, self.YTY, self.Sy_inv)
tmp /= 2.
#kernY_grads = np.array([np.sum(tmp*g) for g in self.kernY.extract_gradients()])
kernY_grads = self.kernY.dK_dtheta(tmp, self.X)
return np.hstack((kernF_grads, kernY_grads))
def pdinv_inc(L_old, A_inc, A_inc2, Ai_old, *args):
"""
similar to pdinv, but uses old choleski decompositon to compute new
as proposed in https://github.com/SheffieldML/GPy/issues/464#issuecomment-285500122
:rval Ai: the inverse of A
:rtype Ai: np.ndarray
:rval L: the Cholesky decomposition of A
:rtype L: np.ndarray
:rval Li: the Cholesky decomposition of Ai
:rtype Li: np.ndarray (set to None for now, because not needed)
:rval logdet: the log of the determinant of A
:rtype logdet: float64
"""
"""
"""
# A_inc = A_inc.reshape(-1)
u = dtrtrs(L_old, A_inc, lower=1)[0]
v = np.sqrt(A_inc2 - np.sum(u * u)).reshape(1)
z = np.zeros((A_inc.shape[0], 1))
L = np.asfortranarray(np.block([[L_old, z], [u, v]]))
logdet = 2. * np.sum(np.log(np.diag(L)))
# _Ai, _ = dpotri(L, lower=1) # consider also incrementally updating this
# incrementally update the inverse
alpha = Ai_old.dot(A_inc).reshape((-1, 1))
# print('alpha', alpha)
gamma = alpha.dot(alpha.T)
# print('gamma', gamma)
beta = 1 / (A_inc2 - alpha.T.dot(A_inc)).reshape(-1)
beta_alpha = -beta * alpha
# print('beta', beta)
Ai = np.block([[Ai_old + beta * gamma, beta_alpha], [beta_alpha.T, beta]])
# print(Ai -_Ai)
symmetrify(Ai)
return Ai, L, None, logdet # could also return Li as in pdinv, but we don't need this right now
def comp_KL_qU(self, qU_mean, qU_var):
M, D = qU_mean.shape[0], qU_mean.shape[1]
qU_L = self.mid['qU_L']
L = self.mid['L']
Linvmu = self.mid['Linvmu']
LinvLu = self.mid['LinvLu']
KuuInv = dpotri(L, lower=1)[0]
Lu = qU_L
LuInv = dtrtri(Lu)
KL = D * M / -2. - np.log(np.diag(Lu)).sum() * D + np.log(
np.diag(L)).sum() * D + np.square(
LinvLu).sum() / 2. * D + np.square(Linvmu).sum() / 2.
dKL_dqU_mean = dtrtrs(L, Linvmu, trans=True)[0]
dKL_dqU_var = (tdot(LuInv.T) / -2. + KuuInv / 2.) * D
dKL_dKuu = KuuInv * D / 2. - KuuInv.dot(tdot(qU_mean) +
qU_var * D).dot(KuuInv) / 2.
return float(KL), dKL_dqU_mean, dKL_dqU_var, dKL_dKuu
def inference(self, kern, X, Z, likelihood, Y, Y_metadata=None, Lm=None, dL_dKmm=None, Kuu_sigma=None):
"""
The first phase of inference:
Compute: log-likelihood, dL_dKmm
Cached intermediate results: Kmm, KmmInv,
"""
num_data, output_dim = Y.shape
input_dim = Z.shape[0]
uncertain_inputs = isinstance(X, VariationalPosterior)
uncertain_outputs = isinstance(Y, VariationalPosterior)
# from ..models.sslvm import Gaussian_Gamma
# if isinstance(likelihood, Gaussian_Gamma):
# beta = likelihood.expectation_beta()
# logL_R = -num_data*likelihood.expectation_logbeta()
# else:
beta = 1./np.fmax(likelihood.variance, 1e-6)
logL_R = -num_data*np.log(beta)
psi0, psi2, YRY, psi1, psi1Y, Shalf, psi1S = self.gatherPsiStat(kern, X, Z, Y, beta, uncertain_inputs)
#======================================================================
# Compute Common Components
#======================================================================
Kmm = kern.K(Z).copy()
if Kuu_sigma is not None:
diag.add(Kmm, Kuu_sigma)
else:
diag.add(Kmm, self.const_jitter)
Lm = jitchol(Kmm)
#LmInv = dtrtri(Lm)
if uncertain_inputs:
LmInvPsi2LmInvT = backsub_both_sides(Lm, psi2, 'right')
else:
LmInvPsi2LmInvT = tdot(dtrtrs(Lm, psi1.T)[0])/beta #tdot(psi1.dot(LmInv.T).T) /beta
Lambda = np.eye(Kmm.shape[0])+LmInvPsi2LmInvT
LL = jitchol(Lambda)
LmLL = Lm.dot(LL)
# LLInv = dtrtri(LL)
# LmLLInv = LLInv.dot(LmInv)
logdet_L = 2.*np.sum(np.log(np.diag(LL)))
b = dtrtrs(LmLL, psi1Y.T)[0].T #psi1Y.dot(LmLLInv.T)
bbt = np.square(b).sum()
v = dtrtrs(LmLL, b.T, trans=1)[0].T #b.dot(LmLLInv)
LLinvPsi1TYYTPsi1LLinvT = tdot(b.T)
if psi1S is not None:
psi1SLLinv = dtrtrs(LmLL, psi1S.T)[0].T #psi1S.dot(LmLLInv.T)
bbt += np.square(psi1SLLinv).sum()
LLinvPsi1TYYTPsi1LLinvT += tdot(psi1SLLinv.T)
psi1SP = dtrtrs(LmLL, psi1SLLinv.T, trans=1)[0].T #psi1SLLinv.dot(LmLLInv)
tmp = -backsub_both_sides(LL, LLinvPsi1TYYTPsi1LLinvT+output_dim*np.eye(input_dim))
dL_dpsi2R = backsub_both_sides(Lm, tmp+output_dim*np.eye(input_dim))/2
#tmp = -LLInv.T.dot(LLinvPsi1TYYTPsi1LLinvT+output_dim*np.eye(input_dim)).dot(LLInv)
#dL_dpsi2R = LmInv.T.dot(tmp+output_dim*np.eye(input_dim)).dot(LmInv)/2.
#======================================================================
# Compute log-likelihood
#======================================================================
logL = -(output_dim*(num_data*log_2_pi+logL_R+psi0-np.trace(LmInvPsi2LmInvT))+YRY- bbt)/2.-output_dim*logdet_L/2.
#======================================================================
# Compute dL_dKmm
#======================================================================
dL_dKmm = dL_dpsi2R - output_dim* backsub_both_sides(Lm, LmInvPsi2LmInvT)/2 #LmInv.T.dot(LmInvPsi2LmInvT).dot(LmInv)/2.
#======================================================================
# Compute the Posterior distribution of inducing points p(u|Y)
#======================================================================
LLInvLmT = dtrtrs(LL, Lm.T)[0]
cov = tdot(LLInvLmT.T)
wd_inv = backsub_both_sides(Lm, np.eye(input_dim)- backsub_both_sides(LL, np.identity(input_dim), transpose='left'), transpose='left')
post = Posterior(woodbury_inv=wd_inv, woodbury_vector=v.T, K=Kmm, mean=None, cov=cov, K_chol=Lm)
#======================================================================
# Compute dL_dthetaL for uncertian input and non-heter noise
#======================================================================
# if isinstance(likelihood, Gaussian_Gamma):
# from scipy.special import polygamma
# dL_dthetaL = ((YRY + output_dim*psi0)/2. - (dL_dpsi2R*psi2).sum() - np.trace(LLinvPsi1TYYTPsi1LLinvT))/-beta
# likelihood.q_a.gradient = num_data*output_dim/2.*polygamma(1, likelihood.q_a) + dL_dthetaL/likelihood.q_b
# likelihood.q_b.gradient = num_data*output_dim/(-2.*likelihood.q_b) +dL_dthetaL*(-likelihood.q_a/(likelihood.q_b*likelihood.q_b))
# else:
dL_dthetaL = (YRY*beta + beta*output_dim*psi0 - num_data*output_dim*beta)/2. - beta*(dL_dpsi2R*psi2).sum() - beta*np.trace(LLinvPsi1TYYTPsi1LLinvT)
#======================================================================
# Compute dL_dpsi
#======================================================================
dL_dpsi0 = -output_dim * (beta * np.ones((num_data,)))/2.
if uncertain_outputs:
m,s = Y.mean, Y.variance
dL_dpsi1 = beta*(np.dot(m,v)+Shalf[:,None]*psi1SP)
else:
dL_dpsi1 = beta*np.dot(Y,v)
if uncertain_inputs:
dL_dpsi2 = beta* dL_dpsi2R
else:
dL_dpsi1 += np.dot(psi1,dL_dpsi2R)*2.
dL_dpsi2 = None
if uncertain_inputs:
grad_dict = {'dL_dKmm': dL_dKmm,
'dL_dpsi0':dL_dpsi0,
'dL_dpsi1':dL_dpsi1,
'dL_dpsi2':dL_dpsi2,
'dL_dthetaL':dL_dthetaL}
else:
grad_dict = {'dL_dKmm': dL_dKmm,
'dL_dKdiag':dL_dpsi0,
'dL_dKnm':dL_dpsi1,
'dL_dthetaL':dL_dthetaL}
if uncertain_outputs:
m,s = Y.mean, Y.variance
psi1LmiLLi = dtrtrs(LmLL, psi1.T)[0].T
LLiLmipsi1Y = b.T
grad_dict['dL_dYmean'] = -m*beta+ psi1LmiLLi.dot(LLiLmipsi1Y)
grad_dict['dL_dYvar'] = beta/-2.+ np.square(psi1LmiLLi).sum(axis=1)/2
return post, logL, grad_dict
def inference_d(self, d, beta, Y, indexD, grad_dict, mid_res,
uncertain_inputs_r, uncertain_inputs_c, Mr, Mc):
idx_d = indexD == d
Y = Y[idx_d]
N, D = Y.shape[0], 1
beta = beta[d]
psi0_r, psi1_r, psi2_r = mid_res['psi0_r'], mid_res['psi1_r'], mid_res[
'psi2_r']
psi0_c, psi1_c, psi2_c = mid_res['psi0_c'], mid_res['psi1_c'], mid_res[
'psi2_c']
psi0_r, psi1_r, psi2_r = psi0_r[d], psi1_r[d:d + 1], psi2_r[d]
psi0_c, psi1_c, psi2_c = psi0_c[idx_d].sum(
), psi1_c[idx_d], psi2_c[idx_d].sum(0)
Lr = mid_res['Lr']
Lc = mid_res['Lc']
LcInvMLrInvT = mid_res['LcInvMLrInvT']
LcInvScLcInvT = mid_res['LcInvScLcInvT']
LrInvSrLrInvT = mid_res['LrInvSrLrInvT']
LcInvPsi2_cLcInvT = backsub_both_sides(Lc, psi2_c, 'right')
LrInvPsi2_rLrInvT = backsub_both_sides(Lr, psi2_r, 'right')
LcInvPsi1_cT = dtrtrs(Lc, psi1_c.T)[0]
LrInvPsi1_rT = dtrtrs(Lr, psi1_r.T)[0]
tr_LrInvPsi2_rLrInvT_LrInvSrLrInvT = (LrInvPsi2_rLrInvT *
LrInvSrLrInvT).sum()
tr_LcInvPsi2_cLcInvT_LcInvScLcInvT = (LcInvPsi2_cLcInvT *
LcInvScLcInvT).sum()
tr_LrInvPsi2_rLrInvT = np.trace(LrInvPsi2_rLrInvT)
tr_LcInvPsi2_cLcInvT = np.trace(LcInvPsi2_cLcInvT)
logL_A = - np.square(Y).sum() \
- (LcInvMLrInvT.T.dot(LcInvPsi2_cLcInvT).dot(LcInvMLrInvT)*LrInvPsi2_rLrInvT).sum() \
- tr_LrInvPsi2_rLrInvT_LrInvSrLrInvT* tr_LcInvPsi2_cLcInvT_LcInvScLcInvT \
+ 2 * (Y * LcInvPsi1_cT.T.dot(LcInvMLrInvT).dot(LrInvPsi1_rT)).sum() - psi0_c * psi0_r \
+ tr_LrInvPsi2_rLrInvT * tr_LcInvPsi2_cLcInvT
logL = -N * D / 2. * (np.log(2. * np.pi) -
np.log(beta)) + beta / 2. * logL_A
# ======= Gradients =====
tmp = beta* LcInvPsi2_cLcInvT.dot(LcInvMLrInvT).dot(LrInvPsi2_rLrInvT).dot(LcInvMLrInvT.T) \
+ beta* tr_LrInvPsi2_rLrInvT_LrInvSrLrInvT * LcInvPsi2_cLcInvT.dot(LcInvScLcInvT) \
- beta* LcInvMLrInvT.dot(LrInvPsi1_rT).dot(Y.T).dot(LcInvPsi1_cT.T) \
- beta/2. * tr_LrInvPsi2_rLrInvT* LcInvPsi2_cLcInvT
dL_dKuu_c = backsub_both_sides(Lc, tmp, 'left')
dL_dKuu_c += dL_dKuu_c.T
dL_dKuu_c *= 0.5
tmp = beta* LcInvMLrInvT.T.dot(LcInvPsi2_cLcInvT).dot(LcInvMLrInvT).dot(LrInvPsi2_rLrInvT) \
+ beta* tr_LcInvPsi2_cLcInvT_LcInvScLcInvT * LrInvPsi2_rLrInvT.dot(LrInvSrLrInvT) \
- beta* LrInvPsi1_rT.dot(Y.T).dot(LcInvPsi1_cT.T).dot(LcInvMLrInvT) \
- beta/2. * tr_LcInvPsi2_cLcInvT * LrInvPsi2_rLrInvT
dL_dKuu_r = backsub_both_sides(Lr, tmp, 'left')
dL_dKuu_r += dL_dKuu_r.T
dL_dKuu_r *= 0.5
#======================================================================
# Compute dL_dthetaL
#======================================================================
dL_dthetaL = -D * N * beta / 2. - logL_A * beta * beta / 2.
#======================================================================
# Compute dL_dqU
#======================================================================
tmp = -beta * LcInvPsi2_cLcInvT.dot(LcInvMLrInvT).dot(LrInvPsi2_rLrInvT)\
+ beta* LcInvPsi1_cT.dot(Y).dot(LrInvPsi1_rT.T)
dL_dqU_mean = dtrtrs(Lc, dtrtrs(Lr, tmp.T, trans=1)[0].T, trans=1)[0]
tmp = -beta / 2. * tr_LrInvPsi2_rLrInvT_LrInvSrLrInvT * LcInvPsi2_cLcInvT
dL_dqU_var_c = backsub_both_sides(Lc, tmp, 'left')
tmp = -beta / 2. * tr_LcInvPsi2_cLcInvT_LcInvScLcInvT * LrInvPsi2_rLrInvT
dL_dqU_var_r = backsub_both_sides(Lr, tmp, 'left')
#======================================================================
# Compute dL_dpsi
#======================================================================
dL_dpsi0_r = -psi0_c * beta / 2. * np.ones((D, ))
dL_dpsi0_c = -psi0_r * beta / 2. * np.ones((N, ))
dL_dpsi1_c = beta * dtrtrs(
Lc, (Y.dot(LrInvPsi1_rT.T).dot(LcInvMLrInvT.T)).T, trans=1)[0].T
dL_dpsi1_r = beta * dtrtrs(
Lr, (Y.T.dot(LcInvPsi1_cT.T).dot(LcInvMLrInvT)).T, trans=1)[0].T
tmp = beta / 2. * (
-LcInvMLrInvT.dot(LrInvPsi2_rLrInvT).dot(LcInvMLrInvT.T) -
tr_LrInvPsi2_rLrInvT_LrInvSrLrInvT * LcInvScLcInvT +
tr_LrInvPsi2_rLrInvT * np.eye(Mc))
dL_dpsi2_c = backsub_both_sides(Lc, tmp, 'left')
tmp = beta / 2. * (
-LcInvMLrInvT.T.dot(LcInvPsi2_cLcInvT).dot(LcInvMLrInvT) -
tr_LcInvPsi2_cLcInvT_LcInvScLcInvT * LrInvSrLrInvT +
tr_LcInvPsi2_cLcInvT * np.eye(Mr))
dL_dpsi2_r = backsub_both_sides(Lr, tmp, 'left')
grad_dict['dL_dthetaL'][d:d + 1] = dL_dthetaL
grad_dict['dL_dqU_mean'] += dL_dqU_mean
grad_dict['dL_dqU_var_c'] += dL_dqU_var_c
grad_dict['dL_dqU_var_r'] += dL_dqU_var_r
grad_dict['dL_dKuu_c'] += dL_dKuu_c
grad_dict['dL_dKuu_r'] += dL_dKuu_r
# if not uncertain_inputs_r:
# dL_dpsi1_r += (dL_dpsi2_r * psi1_r[:,:,None]).sum(1) + (dL_dpsi2_r * psi1_r[:,None,:]).sum(2)
# if not uncertain_inputs_c:
# dL_dpsi1_c += (dL_dpsi2_c * psi1_c[:,:,None]).sum(1) + (dL_dpsi2_c * psi1_c[:,None,:]).sum(2)
if not uncertain_inputs_r:
dL_dpsi1_r += psi1_r.dot(dL_dpsi2_r + dL_dpsi2_r.T)
if not uncertain_inputs_c:
dL_dpsi1_c += psi1_c.dot(dL_dpsi2_c + dL_dpsi2_c.T)
grad_dict['dL_dpsi0_c'][idx_d] += dL_dpsi0_c
grad_dict['dL_dpsi1_c'][idx_d] += dL_dpsi1_c
grad_dict['dL_dpsi2_c'][idx_d] += dL_dpsi2_c
grad_dict['dL_dpsi0_r'][d:d + 1] += dL_dpsi0_r
grad_dict['dL_dpsi1_r'][d:d + 1] += dL_dpsi1_r
grad_dict['dL_dpsi2_r'][d] += dL_dpsi2_r
return logL
def inference(self, kern_r, kern_c, Xr, Xc, Zr, Zc, likelihood, Y, qU_mean,
qU_var_r, qU_var_c, indexD, output_dim):
"""
The SVI-VarDTC inference
"""
N, D, Mr, Mc, Qr, Qc = Y.shape[0], output_dim, Zr.shape[0], Zc.shape[
0], Zr.shape[1], Zc.shape[1]
uncertain_inputs_r = isinstance(Xr, VariationalPosterior)
uncertain_inputs_c = isinstance(Xc, VariationalPosterior)
uncertain_outputs = isinstance(Y, VariationalPosterior)
grad_dict = self._init_grad_dict(N, D, Mr, Mc)
beta = 1. / likelihood.variance
if len(beta) == 1:
beta = np.zeros(D) + beta
psi0_r, psi1_r, psi2_r = self.gatherPsiStat(kern_r, Xr, Zr,
uncertain_inputs_r)
psi0_c, psi1_c, psi2_c = self.gatherPsiStat(kern_c, Xc, Zc,
uncertain_inputs_c)
#======================================================================
# Compute Common Components
#======================================================================
Kuu_r = kern_r.K(Zr).copy()
diag.add(Kuu_r, self.const_jitter)
Lr = jitchol(Kuu_r)
Kuu_c = kern_c.K(Zc).copy()
diag.add(Kuu_c, self.const_jitter)
Lc = jitchol(Kuu_c)
mu, Sr, Sc = qU_mean, qU_var_r, qU_var_c
LSr = jitchol(Sr)
LSc = jitchol(Sc)
LcInvMLrInvT = dtrtrs(Lc, dtrtrs(Lr, mu.T)[0].T)[0]
LcInvLSc = dtrtrs(Lc, LSc)[0]
LrInvLSr = dtrtrs(Lr, LSr)[0]
LcInvScLcInvT = tdot(LcInvLSc)
LrInvSrLrInvT = tdot(LrInvLSr)
tr_LrInvSrLrInvT = np.square(LrInvLSr).sum()
tr_LcInvScLcInvT = np.square(LcInvLSc).sum()
mid_res = {
'psi0_r': psi0_r,
'psi1_r': psi1_r,
'psi2_r': psi2_r,
'psi0_c': psi0_c,
'psi1_c': psi1_c,
'psi2_c': psi2_c,
'Lr': Lr,
'Lc': Lc,
'LcInvMLrInvT': LcInvMLrInvT,
'LcInvScLcInvT': LcInvScLcInvT,
'LrInvSrLrInvT': LrInvSrLrInvT,
}
#======================================================================
# Compute log-likelihood
#======================================================================
logL = 0.
for d in range(D):
logL += self.inference_d(d, beta, Y, indexD, grad_dict, mid_res,
uncertain_inputs_r, uncertain_inputs_c,
Mr, Mc)
logL += -Mc * (np.log(np.diag(Lr)).sum()-np.log(np.diag(LSr)).sum()) -Mr * (np.log(np.diag(Lc)).sum()-np.log(np.diag(LSc)).sum()) \
- np.square(LcInvMLrInvT).sum()/2. - tr_LrInvSrLrInvT * tr_LcInvScLcInvT/2. + Mr*Mc/2.
#======================================================================
# Compute dL_dKuu
#======================================================================
tmp = tdot(
LcInvMLrInvT
) / 2. + tr_LrInvSrLrInvT / 2. * LcInvScLcInvT - Mr / 2. * np.eye(Mc)
dL_dKuu_c = backsub_both_sides(Lc, tmp, 'left')
dL_dKuu_c += dL_dKuu_c.T
dL_dKuu_c *= 0.5
tmp = tdot(
LcInvMLrInvT.T
) / 2. + tr_LcInvScLcInvT / 2. * LrInvSrLrInvT - Mc / 2. * np.eye(Mr)
dL_dKuu_r = backsub_both_sides(Lr, tmp, 'left')
dL_dKuu_r += dL_dKuu_r.T
dL_dKuu_r *= 0.5
#======================================================================
# Compute dL_dqU
#======================================================================
tmp = -LcInvMLrInvT
dL_dqU_mean = dtrtrs(Lc, dtrtrs(Lr, tmp.T, trans=1)[0].T, trans=1)[0]
LScInv = dtrtri(LSc)
tmp = -tr_LrInvSrLrInvT / 2. * np.eye(Mc)
dL_dqU_var_c = backsub_both_sides(Lc, tmp,
'left') + tdot(LScInv.T) * Mr / 2.
LSrInv = dtrtri(LSr)
tmp = -tr_LcInvScLcInvT / 2. * np.eye(Mr)
dL_dqU_var_r = backsub_both_sides(Lr, tmp,
'left') + tdot(LSrInv.T) * Mc / 2.
#======================================================================
# Compute the Posterior distribution of inducing points p(u|Y)
#======================================================================
post = PosteriorMultioutput(LcInvMLrInvT=LcInvMLrInvT,
LcInvScLcInvT=LcInvScLcInvT,
LrInvSrLrInvT=LrInvSrLrInvT,
Lr=Lr,
Lc=Lc,
kern_r=kern_r,
Xr=Xr,
Zr=Zr)
#======================================================================
# Compute dL_dpsi
#======================================================================
grad_dict['dL_dqU_mean'] += dL_dqU_mean
grad_dict['dL_dqU_var_c'] += dL_dqU_var_c
grad_dict['dL_dqU_var_r'] += dL_dqU_var_r
grad_dict['dL_dKuu_c'] += dL_dKuu_c
grad_dict['dL_dKuu_r'] += dL_dKuu_r
if not uncertain_inputs_c:
grad_dict['dL_dKdiag_c'] = grad_dict['dL_dpsi0_c']
grad_dict['dL_dKfu_c'] = grad_dict['dL_dpsi1_c']
if not uncertain_inputs_r:
grad_dict['dL_dKdiag_r'] = grad_dict['dL_dpsi0_r']
grad_dict['dL_dKfu_r'] = grad_dict['dL_dpsi1_r']
return post, logL, grad_dict
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
sigma_n = likelihood.gaussian_variance(Y_metadata)
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
diag.add(Kmm, self.const_jitter)
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
dL_dthetaL = likelihood.exact_inference_gradients(dL_dR)
dL_dthetaL += 0.5 * alpha_const_term * num_data / sigma_n
grad_dict = {
'dL_dKmm': dL_dK,
'dL_dKdiag': dL_dR * self.alpha,
'dL_dKnm': dL_dU.T,
'dL_dthetaL': dL_dthetaL
}
#construct a posterior object
post = Posterior(woodbury_inv=Kmmi - P,
woodbury_vector=v,
K=Kmm,
mean=None,
cov=None,
K_chol=L)
return post, log_marginal, grad_dict
def inference(self,
kern,
X,
Z,
likelihood,
Y,
Y_metadata=None,
Lm=None,
dL_dKmm=None):
"""
The first phase of inference:
Compute: log-likelihood, dL_dKmm
Cached intermediate results: Kmm, KmmInv,
"""
num_data, output_dim = Y.shape
input_dim = Z.shape[0]
uncertain_inputs = isinstance(X, VariationalPosterior)
uncertain_outputs = isinstance(Y, VariationalPosterior)
beta = 1. / np.fmax(likelihood.variance, 1e-6)
psi0, psi2, YRY, psi1, psi1Y, Shalf, psi1S = self.gatherPsiStat(
kern, X, Z, Y, beta, uncertain_inputs)
#======================================================================
# Compute Common Components
#======================================================================
Kmm = kern.K(Z).copy()
diag.add(Kmm, self.const_jitter)
Lm = jitchol(Kmm)
#LmInv = dtrtri(Lm)
if uncertain_inputs:
LmInvPsi2LmInvT = backsub_both_sides(Lm, psi2, 'right')
else:
LmInvPsi2LmInvT = tdot(dtrtrs(
Lm, psi1.T)[0]) / beta #tdot(psi1.dot(LmInv.T).T) /beta
Lambda = np.eye(Kmm.shape[0]) + LmInvPsi2LmInvT
LL = jitchol(Lambda)
LmLL = Lm.dot(LL)
# LLInv = dtrtri(LL)
# LmLLInv = LLInv.dot(LmInv)
logdet_L = 2. * np.sum(np.log(np.diag(LL)))
b = dtrtrs(LmLL, psi1Y.T)[0].T #psi1Y.dot(LmLLInv.T)
bbt = np.square(b).sum()
v = dtrtrs(LmLL, b.T, trans=1)[0].T #b.dot(LmLLInv)
LLinvPsi1TYYTPsi1LLinvT = tdot(b.T)
if psi1S is not None:
psi1SLLinv = dtrtrs(LmLL, psi1S.T)[0].T #psi1S.dot(LmLLInv.T)
bbt += np.square(psi1SLLinv).sum()
LLinvPsi1TYYTPsi1LLinvT += tdot(psi1SLLinv.T)
psi1SP = dtrtrs(LmLL, psi1SLLinv.T,
trans=1)[0].T #psi1SLLinv.dot(LmLLInv)
tmp = -backsub_both_sides(
LL, LLinvPsi1TYYTPsi1LLinvT + output_dim * np.eye(input_dim))
dL_dpsi2R = backsub_both_sides(
Lm, tmp + output_dim * np.eye(input_dim)) / 2
#tmp = -LLInv.T.dot(LLinvPsi1TYYTPsi1LLinvT+output_dim*np.eye(input_dim)).dot(LLInv)
#dL_dpsi2R = LmInv.T.dot(tmp+output_dim*np.eye(input_dim)).dot(LmInv)/2.
#======================================================================
# Compute log-likelihood
#======================================================================
logL_R = -num_data * np.log(beta)
logL = -(
output_dim *
(num_data * log_2_pi + logL_R + psi0 - np.trace(LmInvPsi2LmInvT)) +
YRY - bbt) / 2. - output_dim * logdet_L / 2.
#======================================================================
# Compute dL_dKmm
#======================================================================
dL_dKmm = dL_dpsi2R - output_dim * backsub_both_sides(
Lm,
LmInvPsi2LmInvT) / 2 #LmInv.T.dot(LmInvPsi2LmInvT).dot(LmInv)/2.
#======================================================================
# Compute the Posterior distribution of inducing points p(u|Y)
#======================================================================
wd_inv = backsub_both_sides(
Lm,
np.eye(input_dim) -
backsub_both_sides(LL, np.identity(input_dim), transpose='left'),
transpose='left')
post = Posterior(woodbury_inv=wd_inv,
woodbury_vector=v.T,
K=Kmm,
mean=None,
cov=None,
K_chol=Lm)
#======================================================================
# Compute dL_dthetaL for uncertian input and non-heter noise
#======================================================================
dL_dthetaL = (YRY * beta + beta * output_dim * psi0 - num_data *
output_dim * beta) / 2. - beta * (dL_dpsi2R * psi2).sum(
) - beta * np.trace(LLinvPsi1TYYTPsi1LLinvT)
#======================================================================
# Compute dL_dpsi
#======================================================================
dL_dpsi0 = -output_dim * (beta * np.ones((num_data, ))) / 2.
if uncertain_outputs:
m, s = Y.mean, Y.variance
dL_dpsi1 = beta * (np.dot(m, v) + Shalf[:, None] * psi1SP)
else:
dL_dpsi1 = beta * np.dot(Y, v)
if uncertain_inputs:
dL_dpsi2 = beta * dL_dpsi2R
else:
dL_dpsi1 += np.dot(psi1, dL_dpsi2R) * 2.
dL_dpsi2 = None
if uncertain_inputs:
grad_dict = {
'dL_dKmm': dL_dKmm,
'dL_dpsi0': dL_dpsi0,
'dL_dpsi1': dL_dpsi1,
'dL_dpsi2': dL_dpsi2,
'dL_dthetaL': dL_dthetaL
}
else:
grad_dict = {
'dL_dKmm': dL_dKmm,
'dL_dKdiag': dL_dpsi0,
'dL_dKnm': dL_dpsi1,
'dL_dthetaL': dL_dthetaL
}
if uncertain_outputs:
m, s = Y.mean, Y.variance
psi1LmiLLi = dtrtrs(LmLL, psi1.T)[0].T #psi1.dot(LmLLInv.T)
LLiLmipsi1Y = b.T
grad_dict['dL_dYmean'] = -m * beta + psi1LmiLLi.dot(LLiLmipsi1Y)
grad_dict['dL_dYvar'] = beta / -2. + np.square(psi1LmiLLi).sum(
axis=1) / 2
return post, logL, grad_dict
def inference(self, kern, X, Z, likelihood, Y, qU_mean ,qU_var, Kuu_sigma=None):
"""
The SVI-VarDTC inference
"""
N, D, M, Q = Y.shape[0], Y.shape[1], Z.shape[0], Z.shape[1]
uncertain_inputs = isinstance(X, VariationalPosterior)
uncertain_outputs = isinstance(Y, VariationalPosterior)
beta = 1./likelihood.variance
psi0, psi2, YRY, psi1, psi1Y = self.gatherPsiStat(kern, X, Z, Y, beta, uncertain_inputs)
#======================================================================
# Compute Common Components
#======================================================================
Kuu = kern.K(Z).copy()
if Kuu_sigma is not None:
diag.add(Kuu, Kuu_sigma)
else:
diag.add(Kuu, self.const_jitter)
Lm = jitchol(Kuu)
mu, S = qU_mean, qU_var
Ls = jitchol(S)
LinvLs = dtrtrs(Lm, Ls)[0]
Linvmu = dtrtrs(Lm, mu)[0]
psi1YLinvT = dtrtrs(Lm,psi1Y.T)[0].T
self.mid = {
'qU_L': Ls,
'LinvLu': LinvLs,
'L':Lm,
'Linvmu': Linvmu}
if uncertain_inputs:
LmInvPsi2LmInvT = backsub_both_sides(Lm, psi2, 'right')
else:
LmInvPsi2LmInvT = tdot(dtrtrs(Lm, psi1.T)[0])/beta
LmInvSmuLmInvT = tdot(LinvLs)*D+tdot(Linvmu)
# logdet_L = np.sum(np.log(np.diag(Lm)))
# logdet_S = np.sum(np.log(np.diag(Ls)))
#======================================================================
# Compute log-likelihood
#======================================================================
logL_R = -N*np.log(beta)
logL = -N*D*log_2_pi/2. -D*logL_R/2. - D*psi0/2. - YRY/2. \
-(LmInvSmuLmInvT*LmInvPsi2LmInvT).sum()/2. + np.trace(LmInvPsi2LmInvT)*D/2.+(Linvmu*psi1YLinvT.T).sum()
#======================================================================
# Compute dL_dKmm
#======================================================================
tmp1 = backsub_both_sides(Lm,LmInvSmuLmInvT.dot(LmInvPsi2LmInvT), 'left')
tmp2 = Linvmu.dot(psi1YLinvT)
tmp3 = backsub_both_sides(Lm, - D*LmInvPsi2LmInvT -tmp2-tmp2.T, 'left')/2.
dL_dKmm = (tmp1+tmp1.T)/2. + tmp3
#======================================================================
# Compute dL_dthetaL for uncertian input and non-heter noise
#======================================================================
dL_dthetaL = -D*N*beta/2. -(- D*psi0/2. - YRY/2.-(LmInvSmuLmInvT*LmInvPsi2LmInvT).sum()/2. + np.trace(LmInvPsi2LmInvT)*D/2.+(Linvmu*psi1YLinvT.T).sum())*beta
#======================================================================
# Compute dL_dqU
#======================================================================
tmp1 = backsub_both_sides(Lm, - LmInvPsi2LmInvT, 'left')
dL_dqU_mean = tmp1.dot(mu) + dtrtrs(Lm, psi1YLinvT.T,trans=1)[0]
dL_dqU_var = D/2.*tmp1
#======================================================================
# Compute the Posterior distribution of inducing points p(u|Y)
#======================================================================
KuuInvmu = dtrtrs(Lm, Linvmu, trans=1)[0]
tmp = backsub_both_sides(Lm, np.eye(M) - tdot(LinvLs), 'left')
post = Posterior(woodbury_inv=tmp, woodbury_vector=KuuInvmu, K=Kuu, mean=mu, cov=S, K_chol=Lm)
#======================================================================
# Compute dL_dpsi
#======================================================================
dL_dpsi0 = -D * (beta * np.ones((N,)))/2.
if uncertain_outputs:
dL_dpsi1 = Y.mean.dot(dtrtrs(Lm,Linvmu,trans=1)[0].T)*beta
else:
dL_dpsi1 = Y.dot(dtrtrs(Lm,Linvmu,trans=1)[0].T)*beta
dL_dpsi2 = beta*backsub_both_sides(Lm, D*np.eye(M)-LmInvSmuLmInvT, 'left')/2.
if not uncertain_inputs:
dL_dpsi1 += psi1.dot(dL_dpsi2+dL_dpsi2.T)/beta
dL_dpsi2 = None
if uncertain_inputs:
grad_dict = {'dL_dKmm': dL_dKmm,
'dL_dpsi0':dL_dpsi0,
'dL_dpsi1':dL_dpsi1,
'dL_dpsi2':dL_dpsi2,
'dL_dthetaL':dL_dthetaL,
'dL_dqU_mean':dL_dqU_mean,
'dL_dqU_var':dL_dqU_var}
else:
grad_dict = {'dL_dKmm': dL_dKmm,
'dL_dKdiag':dL_dpsi0,
'dL_dKnm':dL_dpsi1,
'dL_dthetaL':dL_dthetaL,
'dL_dqU_mean':dL_dqU_mean,
'dL_dqU_var':dL_dqU_var}
if uncertain_outputs:
m,s = Y.mean, Y.variance
grad_dict['dL_dYmean'] = -m*beta+ dtrtrs(Lm,psi1.T)[0].T.dot(dtrtrs(Lm,mu)[0])
grad_dict['dL_dYvar'] = beta/-2.
return post, logL, grad_dict
def inference(self, kern, X, Z, likelihood, Y, qU):
"""
The SVI-VarDTC inference
"""
if isinstance(Y, np.ndarray) and np.any(np.isnan(Y)):
missing_data = True
N, M, Q = Y.shape[0], Z.shape[0], Z.shape[1]
Ds = Y.shape[1] - (np.isnan(Y)*1).sum(1)
Ymask = 1-np.isnan(Y)*1
Y_masked = np.zeros_like(Y)
Y_masked[Ymask==1] = Y[Ymask==1]
ND = Ymask.sum()
else:
missing_data = False
N, D, M, Q = Y.shape[0], Y.shape[1], Z.shape[0], Z.shape[1]
ND = N*D
uncertain_inputs = isinstance(X, VariationalPosterior)
uncertain_outputs = isinstance(Y, VariationalPosterior)
beta = 1./np.fmax(likelihood.variance, 1e-6)
psi0, psi2, YRY, psi1, psi1Y = self.gatherPsiStat(kern, X, Z, Y if not missing_data else Y_masked, beta, uncertain_inputs, D if not missing_data else Ds, missing_data)
#======================================================================
# Compute Common Components
#======================================================================
mu, S = qU.mean, qU.covariance
mupsi1Y = mu.dot(psi1Y)
Kmm = kern.K(Z).copy()
diag.add(Kmm, self.const_jitter)
Lm = jitchol(Kmm)
if missing_data:
S_mu = S[None,:,:]+mu.T[:,:,None]*mu.T[:,None,:]
NS_mu = S_mu.T.dot(Ymask.T).T
LmInv = dtrtri(Lm)
LmInvPsi2LmInvT = np.swapaxes(psi2.dot(LmInv.T),1,2).dot(LmInv.T)
LmInvSmuLmInvT = np.swapaxes(NS_mu.dot(LmInv.T),1,2).dot(LmInv.T)
B = mupsi1Y+ mupsi1Y.T +(Ds[:,None,None]*psi2).sum(0)
tmp = backsub_both_sides(Lm, B,'right')
logL = -ND*log_2_pi/2. +ND*np.log(beta)/2. - psi0/2. - YRY/2. \
-(LmInvSmuLmInvT*LmInvPsi2LmInvT).sum()/2. +np.trace(tmp)/2.
else:
S_mu = S*D+tdot(mu)
if uncertain_inputs:
LmInvPsi2LmInvT = backsub_both_sides(Lm, psi2, 'right')
else:
LmInvPsi2LmInvT = tdot(dtrtrs(Lm, psi1.T)[0])/beta #tdot(psi1.dot(LmInv.T).T) /beta
LmInvSmuLmInvT = backsub_both_sides(Lm, S_mu, 'right')
B = mupsi1Y+ mupsi1Y.T +D*psi2
tmp = backsub_both_sides(Lm, B,'right')
logL = -ND*log_2_pi/2. +ND*np.log(beta)/2. - psi0/2. - YRY/2. \
-(LmInvSmuLmInvT*LmInvPsi2LmInvT).sum()/2. +np.trace(tmp)/2.
#======================================================================
# Compute dL_dKmm
#======================================================================
dL_dKmm = np.eye(M)
#======================================================================
# Compute dL_dthetaL for uncertian input and non-heter noise
#======================================================================
dL_dthetaL = None #(YRY*beta + beta*output_dim*psi0 - num_data*output_dim*beta)/2. - beta*(dL_dpsi2R*psi2).sum() - beta*np.trace(LLinvPsi1TYYTPsi1LLinvT)
#======================================================================
# Compute dL_dpsi
#======================================================================
if missing_data:
dL_dpsi0 = -Ds * (beta * np.ones((N,)))/2.
else:
dL_dpsi0 = -D * (beta * np.ones((N,)))/2.
if uncertain_outputs:
Ym,Ys = Y.mean, Y.variance
dL_dpsi1 = dtrtrs(Lm, dtrtrs(Lm, Ym.dot(mu.T).T)[0], trans=1)[0].T*beta
else:
if missing_data:
dL_dpsi1 = dtrtrs(Lm, dtrtrs(Lm, (Y_masked).dot(mu.T).T)[0], trans=1)[0].T*beta
else:
dL_dpsi1 = dtrtrs(Lm, dtrtrs(Lm, Y.dot(mu.T).T)[0], trans=1)[0].T*beta
if uncertain_inputs:
if missing_data:
dL_dpsi2 = np.swapaxes((Ds[:,None,None]*np.eye(M)[None,:,:]-LmInvSmuLmInvT).dot(LmInv),1,2).dot(LmInv)*beta/2.
else:
dL_dpsi2 = beta*backsub_both_sides(Lm, D*np.eye(M)-LmInvSmuLmInvT, 'left')/2.
else:
dL_dpsi1 += beta*psi1.dot(dL_dpsi2+dL_dpsi2.T)
dL_dpsi2 = None
if uncertain_inputs:
grad_dict = {'dL_dKmm': dL_dKmm,
'dL_dpsi0':dL_dpsi0,
'dL_dpsi1':dL_dpsi1,
'dL_dpsi2':dL_dpsi2,
'dL_dthetaL':dL_dthetaL}
else:
grad_dict = {'dL_dKmm': dL_dKmm,
'dL_dKdiag':dL_dpsi0,
'dL_dKnm':dL_dpsi1,
'dL_dthetaL':dL_dthetaL}
if uncertain_outputs:
Ym = Y.mean
grad_dict['dL_dYmean'] = -Ym*beta+ dtrtrs(Lm,psi1.T)[0].T.dot(dtrtrs(Lm,mu)[0])
grad_dict['dL_dYvar'] = beta/-2.
return logL, grad_dict
def inference(self, kern, X, Z, likelihood, Y, qU):
"""
The SVI-VarDTC inference
"""
if isinstance(Y, np.ndarray) and np.any(np.isnan(Y)):
missing_data = True
N, M, Q = Y.shape[0], Z.shape[0], Z.shape[1]
Ds = Y.shape[1] - (np.isnan(Y) * 1).sum(1)
Ymask = 1 - np.isnan(Y) * 1
Y_masked = np.zeros_like(Y)
Y_masked[Ymask == 1] = Y[Ymask == 1]
ND = Ymask.sum()
else:
missing_data = False
N, D, M, Q = Y.shape[0], Y.shape[1], Z.shape[0], Z.shape[1]
ND = N * D
uncertain_inputs = isinstance(X, VariationalPosterior)
uncertain_outputs = isinstance(Y, VariationalPosterior)
beta = 1. / np.fmax(likelihood.variance, 1e-6)
psi0, psi2, YRY, psi1, psi1Y = self.gatherPsiStat(
kern, X, Z, Y if not missing_data else Y_masked, beta,
uncertain_inputs, D if not missing_data else Ds, missing_data)
#======================================================================
# Compute Common Components
#======================================================================
mu, S = qU.mean, qU.covariance
mupsi1Y = mu.dot(psi1Y)
Kmm = kern.K(Z).copy()
diag.add(Kmm, self.const_jitter)
Lm = jitchol(Kmm)
if missing_data:
S_mu = S[None, :, :] + mu.T[:, :, None] * mu.T[:, None, :]
NS_mu = S_mu.T.dot(Ymask.T).T
LmInv = dtrtri(Lm)
LmInvPsi2LmInvT = np.swapaxes(psi2.dot(LmInv.T), 1, 2).dot(LmInv.T)
LmInvSmuLmInvT = np.swapaxes(NS_mu.dot(LmInv.T), 1, 2).dot(LmInv.T)
B = mupsi1Y + mupsi1Y.T + (Ds[:, None, None] * psi2).sum(0)
tmp = backsub_both_sides(Lm, B, 'right')
logL = -ND*log_2_pi/2. +ND*np.log(beta)/2. - psi0/2. - YRY/2. \
-(LmInvSmuLmInvT*LmInvPsi2LmInvT).sum()/2. +np.trace(tmp)/2.
else:
S_mu = S * D + tdot(mu)
if uncertain_inputs:
LmInvPsi2LmInvT = backsub_both_sides(Lm, psi2, 'right')
else:
LmInvPsi2LmInvT = tdot(dtrtrs(
Lm, psi1.T)[0]) / beta #tdot(psi1.dot(LmInv.T).T) /beta
LmInvSmuLmInvT = backsub_both_sides(Lm, S_mu, 'right')
B = mupsi1Y + mupsi1Y.T + D * psi2
tmp = backsub_both_sides(Lm, B, 'right')
logL = -ND*log_2_pi/2. +ND*np.log(beta)/2. - psi0/2. - YRY/2. \
-(LmInvSmuLmInvT*LmInvPsi2LmInvT).sum()/2. +np.trace(tmp)/2.
#======================================================================
# Compute dL_dKmm
#======================================================================
dL_dKmm = np.eye(M)
#======================================================================
# Compute dL_dthetaL for uncertian input and non-heter noise
#======================================================================
dL_dthetaL = None #(YRY*beta + beta*output_dim*psi0 - num_data*output_dim*beta)/2. - beta*(dL_dpsi2R*psi2).sum() - beta*np.trace(LLinvPsi1TYYTPsi1LLinvT)
#======================================================================
# Compute dL_dpsi
#======================================================================
if missing_data:
dL_dpsi0 = -Ds * (beta * np.ones((N, ))) / 2.
else:
dL_dpsi0 = -D * (beta * np.ones((N, ))) / 2.
if uncertain_outputs:
Ym, Ys = Y.mean, Y.variance
dL_dpsi1 = dtrtrs(Lm, dtrtrs(Lm,
Ym.dot(mu.T).T)[0],
trans=1)[0].T * beta
else:
if missing_data:
dL_dpsi1 = dtrtrs(
Lm, dtrtrs(Lm,
(Y_masked).dot(mu.T).T)[0], trans=1)[0].T * beta
else:
dL_dpsi1 = dtrtrs(Lm, dtrtrs(Lm,
Y.dot(mu.T).T)[0],
trans=1)[0].T * beta
if uncertain_inputs:
if missing_data:
dL_dpsi2 = np.swapaxes(
(Ds[:, None, None] * np.eye(M)[None, :, :] -
LmInvSmuLmInvT).dot(LmInv), 1, 2).dot(LmInv) * beta / 2.
else:
dL_dpsi2 = beta * backsub_both_sides(
Lm,
D * np.eye(M) - LmInvSmuLmInvT, 'left') / 2.
else:
dL_dpsi1 += beta * psi1.dot(dL_dpsi2 + dL_dpsi2.T)
dL_dpsi2 = None
if uncertain_inputs:
grad_dict = {
'dL_dKmm': dL_dKmm,
'dL_dpsi0': dL_dpsi0,
'dL_dpsi1': dL_dpsi1,
'dL_dpsi2': dL_dpsi2,
'dL_dthetaL': dL_dthetaL
}
else:
grad_dict = {
'dL_dKmm': dL_dKmm,
'dL_dKdiag': dL_dpsi0,
'dL_dKnm': dL_dpsi1,
'dL_dthetaL': dL_dthetaL
}
if uncertain_outputs:
Ym = Y.mean
grad_dict['dL_dYmean'] = -Ym * beta + dtrtrs(Lm, psi1.T)[0].T.dot(
dtrtrs(Lm, mu)[0])
grad_dict['dL_dYvar'] = beta / -2.
return logL, grad_dict
def inference(self,
kern,
X,
Z,
likelihood,
Y,
indexD,
output_dim,
Y_metadata=None,
Lm=None,
dL_dKmm=None,
Kuu_sigma=None):
"""
The first phase of inference:
Compute: log-likelihood, dL_dKmm
Cached intermediate results: Kmm, KmmInv,
"""
input_dim = Z.shape[0]
uncertain_inputs = isinstance(X, VariationalPosterior)
beta = 1. / likelihood.variance
if len(beta) == 1:
beta = np.zeros(output_dim) + beta
beta_exp = np.zeros(indexD.shape[0])
for d in range(output_dim):
beta_exp[indexD == d] = beta[d]
psi0, psi1, psi2 = self.gatherPsiStat(kern, X, Z, Y, beta,
uncertain_inputs)
psi2_sum = (beta_exp[:, None, None] * psi2).sum(0) / output_dim
#======================================================================
# Compute Common Components
#======================================================================
Kmm = kern.K(Z).copy()
if Kuu_sigma is not None:
diag.add(Kmm, Kuu_sigma)
else:
diag.add(Kmm, self.const_jitter)
Lm = jitchol(Kmm)
logL = 0.
dL_dthetaL = np.zeros(output_dim)
dL_dKmm = np.zeros_like(Kmm)
dL_dpsi0 = np.zeros_like(psi0)
dL_dpsi1 = np.zeros_like(psi1)
dL_dpsi2 = np.zeros_like(psi2)
wv = np.empty((Kmm.shape[0], output_dim))
for d in range(output_dim):
idx_d = indexD == d
Y_d = Y[idx_d]
N_d = Y_d.shape[0]
beta_d = beta[d]
psi2_d = psi2[idx_d].sum(0) * beta_d
psi1Y = Y_d.T.dot(psi1[idx_d]) * beta_d
psi0_d = psi0[idx_d].sum() * beta_d
YRY_d = np.square(Y_d).sum() * beta_d
LmInvPsi2LmInvT = backsub_both_sides(Lm, psi2_d, 'right')
Lambda = np.eye(Kmm.shape[0]) + LmInvPsi2LmInvT
LL = jitchol(Lambda)
LmLL = Lm.dot(LL)
b = dtrtrs(LmLL, psi1Y.T)[0].T
bbt = np.square(b).sum()
v = dtrtrs(LmLL, b.T, trans=1)[0].T
LLinvPsi1TYYTPsi1LLinvT = tdot(b.T)
tmp = -backsub_both_sides(LL, LLinvPsi1TYYTPsi1LLinvT)
dL_dpsi2R = backsub_both_sides(Lm, tmp + np.eye(input_dim)) / 2
logL_R = -N_d * np.log(beta_d)
logL += -((N_d * log_2_pi + logL_R + psi0_d -
np.trace(LmInvPsi2LmInvT)) + YRY_d - bbt) / 2.
dL_dKmm += dL_dpsi2R - backsub_both_sides(Lm, LmInvPsi2LmInvT) / 2
dL_dthetaL[d:d +
1] = (YRY_d * beta_d + beta_d * psi0_d - N_d *
beta_d) / 2. - beta_d * (dL_dpsi2R * psi2_d).sum(
) - beta_d * np.trace(LLinvPsi1TYYTPsi1LLinvT)
dL_dpsi0[idx_d] = -beta_d / 2.
dL_dpsi1[idx_d] = beta_d * np.dot(Y_d, v)
dL_dpsi2[idx_d] = beta_d * dL_dpsi2R
wv[:, d] = v
LmInvPsi2LmInvT = backsub_both_sides(Lm, psi2_sum, 'right')
Lambda = np.eye(Kmm.shape[0]) + LmInvPsi2LmInvT
LL = jitchol(Lambda)
LmLL = Lm.dot(LL)
logdet_L = 2. * np.sum(np.log(np.diag(LL)))
dL_dpsi2R_common = dpotri(LmLL)[0] / -2.
dL_dpsi2 += dL_dpsi2R_common[None, :, :] * beta_exp[:, None, None]
for d in range(output_dim):
dL_dthetaL[d] += (dL_dpsi2R_common * psi2[indexD == d].sum(0)
).sum() * -beta[d] * beta[d]
dL_dKmm += dL_dpsi2R_common * output_dim
logL += -output_dim * logdet_L / 2.
#======================================================================
# Compute dL_dKmm
#======================================================================
# dL_dKmm = dL_dpsi2R - output_dim* backsub_both_sides(Lm, LmInvPsi2LmInvT)/2 #LmInv.T.dot(LmInvPsi2LmInvT).dot(LmInv)/2.
#======================================================================
# Compute the Posterior distribution of inducing points p(u|Y)
#======================================================================
LLInvLmT = dtrtrs(LL, Lm.T)[0]
cov = tdot(LLInvLmT.T)
wd_inv = backsub_both_sides(
Lm,
np.eye(input_dim) -
backsub_both_sides(LL, np.identity(input_dim), transpose='left'),
transpose='left')
post = Posterior(woodbury_inv=wd_inv,
woodbury_vector=wv,
K=Kmm,
mean=None,
cov=cov,
K_chol=Lm)
#======================================================================
# Compute dL_dthetaL for uncertian input and non-heter noise
#======================================================================
# for d in range(output_dim):
# dL_dthetaL[d:d+1] += - beta[d]*beta[d]*(dL_dpsi2R[None,:,:] * psi2[indexD==d]/output_dim).sum()
# dL_dthetaL += - (dL_dpsi2R[None,:,:] * psi2_sum*D beta*(dL_dpsi2R*psi2).sum()
#======================================================================
# Compute dL_dpsi
#======================================================================
if not uncertain_inputs:
dL_dpsi1 += (psi1[:, None, :] * dL_dpsi2).sum(2) * 2.
if uncertain_inputs:
grad_dict = {
'dL_dKmm': dL_dKmm,
'dL_dpsi0': dL_dpsi0,
'dL_dpsi1': dL_dpsi1,
'dL_dpsi2': dL_dpsi2,
'dL_dthetaL': dL_dthetaL
}
else:
grad_dict = {
'dL_dKmm': dL_dKmm,
'dL_dKdiag': dL_dpsi0,
'dL_dKnm': dL_dpsi1,
'dL_dthetaL': dL_dthetaL
}
return post, logL, grad_dict
def inference(self, kern_r, kern_c, Xr, Xc, Zr, Zc, likelihood, Y, qU_mean,
qU_var_r, qU_var_c):
"""
The SVI-VarDTC inference
"""
N, D, Mr, Mc, Qr, Qc = Y.shape[0], Y.shape[1], Zr.shape[0], Zc.shape[
0], Zr.shape[1], Zc.shape[1]
uncertain_inputs_r = isinstance(Xr, VariationalPosterior)
uncertain_inputs_c = isinstance(Xc, VariationalPosterior)
uncertain_outputs = isinstance(Y, VariationalPosterior)
beta = 1. / likelihood.variance
psi0_r, psi1_r, psi2_r = self.gatherPsiStat(kern_r, Xr, Zr,
uncertain_inputs_r)
psi0_c, psi1_c, psi2_c = self.gatherPsiStat(kern_c, Xc, Zc,
uncertain_inputs_c)
#======================================================================
# Compute Common Components
#======================================================================
Kuu_r = kern_r.K(Zr).copy()
diag.add(Kuu_r, self.const_jitter)
Lr = jitchol(Kuu_r)
Kuu_c = kern_c.K(Zc).copy()
diag.add(Kuu_c, self.const_jitter)
Lc = jitchol(Kuu_c)
mu, Sr, Sc = qU_mean, qU_var_r, qU_var_c
LSr = jitchol(Sr)
LSc = jitchol(Sc)
LcInvMLrInvT = dtrtrs(Lc, dtrtrs(Lr, mu.T)[0].T)[0]
LcInvPsi2_cLcInvT = backsub_both_sides(Lc, psi2_c, 'right')
LrInvPsi2_rLrInvT = backsub_both_sides(Lr, psi2_r, 'right')
LcInvLSc = dtrtrs(Lc, LSc)[0]
LrInvLSr = dtrtrs(Lr, LSr)[0]
LcInvScLcInvT = tdot(LcInvLSc)
LrInvSrLrInvT = tdot(LrInvLSr)
LcInvPsi1_cT = dtrtrs(Lc, psi1_c.T)[0]
LrInvPsi1_rT = dtrtrs(Lr, psi1_r.T)[0]
tr_LrInvPsi2_rLrInvT_LrInvSrLrInvT = (LrInvPsi2_rLrInvT *
LrInvSrLrInvT).sum()
tr_LcInvPsi2_cLcInvT_LcInvScLcInvT = (LcInvPsi2_cLcInvT *
LcInvScLcInvT).sum()
tr_LrInvSrLrInvT = np.square(LrInvLSr).sum()
tr_LcInvScLcInvT = np.square(LcInvLSc).sum()
tr_LrInvPsi2_rLrInvT = np.trace(LrInvPsi2_rLrInvT)
tr_LcInvPsi2_cLcInvT = np.trace(LcInvPsi2_cLcInvT)
#======================================================================
# Compute log-likelihood
#======================================================================
logL_A = - np.square(Y).sum() \
- (LcInvMLrInvT.T.dot(LcInvPsi2_cLcInvT).dot(LcInvMLrInvT)*LrInvPsi2_rLrInvT).sum() \
- tr_LrInvPsi2_rLrInvT_LrInvSrLrInvT* tr_LcInvPsi2_cLcInvT_LcInvScLcInvT \
+ 2 * (Y * LcInvPsi1_cT.T.dot(LcInvMLrInvT).dot(LrInvPsi1_rT)).sum() - psi0_c * psi0_r \
+ tr_LrInvPsi2_rLrInvT * tr_LcInvPsi2_cLcInvT
logL = -N*D/2.*(np.log(2.*np.pi)-np.log(beta)) + beta/2.* logL_A \
-Mc * (np.log(np.diag(Lr)).sum()-np.log(np.diag(LSr)).sum()) -Mr * (np.log(np.diag(Lc)).sum()-np.log(np.diag(LSc)).sum()) \
- np.square(LcInvMLrInvT).sum()/2. - tr_LrInvSrLrInvT * tr_LcInvScLcInvT/2. + Mr*Mc/2.
#======================================================================
# Compute dL_dKuu
#======================================================================
tmp = beta* LcInvPsi2_cLcInvT.dot(LcInvMLrInvT).dot(LrInvPsi2_rLrInvT).dot(LcInvMLrInvT.T) \
+ beta* tr_LrInvPsi2_rLrInvT_LrInvSrLrInvT * LcInvPsi2_cLcInvT.dot(LcInvScLcInvT) \
- beta* LcInvMLrInvT.dot(LrInvPsi1_rT).dot(Y.T).dot(LcInvPsi1_cT.T) \
- beta/2. * tr_LrInvPsi2_rLrInvT* LcInvPsi2_cLcInvT - Mr/2.*np.eye(Mc) \
+ tdot(LcInvMLrInvT)/2. + tr_LrInvSrLrInvT/2. * LcInvScLcInvT
dL_dKuu_c = backsub_both_sides(Lc, tmp, 'left')
dL_dKuu_c += dL_dKuu_c.T
dL_dKuu_c *= 0.5
tmp = beta* LcInvMLrInvT.T.dot(LcInvPsi2_cLcInvT).dot(LcInvMLrInvT).dot(LrInvPsi2_rLrInvT) \
+ beta* tr_LcInvPsi2_cLcInvT_LcInvScLcInvT * LrInvPsi2_rLrInvT.dot(LrInvSrLrInvT) \
- beta* LrInvPsi1_rT.dot(Y.T).dot(LcInvPsi1_cT.T).dot(LcInvMLrInvT) \
- beta/2. * tr_LcInvPsi2_cLcInvT * LrInvPsi2_rLrInvT - Mc/2.*np.eye(Mr) \
+ tdot(LcInvMLrInvT.T)/2. + tr_LcInvScLcInvT/2. * LrInvSrLrInvT
dL_dKuu_r = backsub_both_sides(Lr, tmp, 'left')
dL_dKuu_r += dL_dKuu_r.T
dL_dKuu_r *= 0.5
#======================================================================
# Compute dL_dthetaL
#======================================================================
dL_dthetaL = -D * N * beta / 2. - logL_A * beta * beta / 2.
#======================================================================
# Compute dL_dqU
#======================================================================
tmp = -beta * LcInvPsi2_cLcInvT.dot(LcInvMLrInvT).dot(LrInvPsi2_rLrInvT)\
+ beta* LcInvPsi1_cT.dot(Y).dot(LrInvPsi1_rT.T) - LcInvMLrInvT
dL_dqU_mean = dtrtrs(Lc, dtrtrs(Lr, tmp.T, trans=1)[0].T, trans=1)[0]
LScInv = dtrtri(LSc)
tmp = -beta / 2. * tr_LrInvPsi2_rLrInvT_LrInvSrLrInvT * LcInvPsi2_cLcInvT - tr_LrInvSrLrInvT / 2. * np.eye(
Mc)
dL_dqU_var_c = backsub_both_sides(Lc, tmp,
'left') + tdot(LScInv.T) * Mr / 2.
LSrInv = dtrtri(LSr)
tmp = -beta / 2. * tr_LcInvPsi2_cLcInvT_LcInvScLcInvT * LrInvPsi2_rLrInvT - tr_LcInvScLcInvT / 2. * np.eye(
Mr)
dL_dqU_var_r = backsub_both_sides(Lr, tmp,
'left') + tdot(LSrInv.T) * Mc / 2.
#======================================================================
# Compute the Posterior distribution of inducing points p(u|Y)
#======================================================================
post = PosteriorMultioutput(LcInvMLrInvT=LcInvMLrInvT,
LcInvScLcInvT=LcInvScLcInvT,
LrInvSrLrInvT=LrInvSrLrInvT,
Lr=Lr,
Lc=Lc,
kern_r=kern_r,
Xr=Xr,
Zr=Zr)
#======================================================================
# Compute dL_dpsi
#======================================================================
dL_dpsi0_r = -psi0_c * beta / 2. * np.ones((D, ))
dL_dpsi0_c = -psi0_r * beta / 2. * np.ones((N, ))
dL_dpsi1_c = beta * dtrtrs(
Lc, (Y.dot(LrInvPsi1_rT.T).dot(LcInvMLrInvT.T)).T, trans=1)[0].T
dL_dpsi1_r = beta * dtrtrs(
Lr, (Y.T.dot(LcInvPsi1_cT.T).dot(LcInvMLrInvT)).T, trans=1)[0].T
tmp = beta / 2. * (
-LcInvMLrInvT.dot(LrInvPsi2_rLrInvT).dot(LcInvMLrInvT.T) -
tr_LrInvPsi2_rLrInvT_LrInvSrLrInvT * LcInvScLcInvT +
tr_LrInvPsi2_rLrInvT * np.eye(Mc))
dL_dpsi2_c = backsub_both_sides(Lc, tmp, 'left')
tmp = beta / 2. * (
-LcInvMLrInvT.T.dot(LcInvPsi2_cLcInvT).dot(LcInvMLrInvT) -
tr_LcInvPsi2_cLcInvT_LcInvScLcInvT * LrInvSrLrInvT +
tr_LcInvPsi2_cLcInvT * np.eye(Mr))
dL_dpsi2_r = backsub_both_sides(Lr, tmp, 'left')
if not uncertain_inputs_r:
dL_dpsi1_r += psi1_r.dot(dL_dpsi2_r + dL_dpsi2_r.T)
if not uncertain_inputs_c:
dL_dpsi1_c += psi1_c.dot(dL_dpsi2_c + dL_dpsi2_c.T)
grad_dict = {
'dL_dthetaL': dL_dthetaL,
'dL_dqU_mean': dL_dqU_mean,
'dL_dqU_var_c': dL_dqU_var_c,
'dL_dqU_var_r': dL_dqU_var_r,
'dL_dKuu_c': dL_dKuu_c,
'dL_dKuu_r': dL_dKuu_r,
}
if uncertain_inputs_c:
grad_dict['dL_dpsi0_c'] = dL_dpsi0_c
grad_dict['dL_dpsi1_c'] = dL_dpsi1_c
grad_dict['dL_dpsi2_c'] = dL_dpsi2_c
else:
grad_dict['dL_dKdiag_c'] = dL_dpsi0_c
grad_dict['dL_dKfu_c'] = dL_dpsi1_c
if uncertain_inputs_r:
grad_dict['dL_dpsi0_r'] = dL_dpsi0_r
grad_dict['dL_dpsi1_r'] = dL_dpsi1_r
grad_dict['dL_dpsi2_r'] = dL_dpsi2_r
else:
grad_dict['dL_dKdiag_r'] = dL_dpsi0_r
grad_dict['dL_dKfu_r'] = dL_dpsi1_r
return post, logL, grad_dict
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
sigma_n = likelihood.gaussian_variance(Y_metadata)
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
diag.add(Kmm, self.const_jitter)
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
dL_dthetaL = likelihood.exact_inference_gradients(dL_dR)
dL_dthetaL += 0.5*alpha_const_term*num_data / sigma_n
grad_dict = {'dL_dKmm': dL_dK, 'dL_dKdiag':dL_dR * self.alpha, 'dL_dKnm':dL_dU.T, 'dL_dthetaL':dL_dthetaL}
#construct a posterior object
post = Posterior(woodbury_inv=Kmmi-P, woodbury_vector=v, K=Kmm, mean=None, cov=None, K_chol=L)
return post, log_marginal, grad_dict
def inference(self,
kern,
X,
Z,
likelihood,
Y,
qU_mean,
qU_var,
Kuu_sigma=None):
"""
The SVI-VarDTC inference
"""
N, D, M, Q = Y.shape[0], Y.shape[1], Z.shape[0], Z.shape[1]
uncertain_inputs = isinstance(X, VariationalPosterior)
uncertain_outputs = isinstance(Y, VariationalPosterior)
beta = 1. / likelihood.variance
psi0, psi2, YRY, psi1, psi1Y = self.gatherPsiStat(
kern, X, Z, Y, beta, uncertain_inputs)
#======================================================================
# Compute Common Components
#======================================================================
Kuu = kern.K(Z).copy()
if Kuu_sigma is not None:
diag.add(Kuu, Kuu_sigma)
else:
diag.add(Kuu, self.const_jitter)
Lm = jitchol(Kuu)
mu, S = qU_mean, qU_var
Ls = jitchol(S)
LinvLs = dtrtrs(Lm, Ls)[0]
Linvmu = dtrtrs(Lm, mu)[0]
psi1YLinvT = dtrtrs(Lm, psi1Y.T)[0].T
self.mid = {'qU_L': Ls, 'LinvLu': LinvLs, 'L': Lm, 'Linvmu': Linvmu}
if uncertain_inputs:
LmInvPsi2LmInvT = backsub_both_sides(Lm, psi2, 'right')
else:
LmInvPsi2LmInvT = tdot(dtrtrs(Lm, psi1.T)[0]) / beta
LmInvSmuLmInvT = tdot(LinvLs) * D + tdot(Linvmu)
# logdet_L = np.sum(np.log(np.diag(Lm)))
# logdet_S = np.sum(np.log(np.diag(Ls)))
#======================================================================
# Compute log-likelihood
#======================================================================
logL_R = -N * np.log(beta)
logL = -N*D*log_2_pi/2. -D*logL_R/2. - D*psi0/2. - YRY/2. \
-(LmInvSmuLmInvT*LmInvPsi2LmInvT).sum()/2. + np.trace(LmInvPsi2LmInvT)*D/2.+(Linvmu*psi1YLinvT.T).sum()
#======================================================================
# Compute dL_dKmm
#======================================================================
tmp1 = backsub_both_sides(Lm, LmInvSmuLmInvT.dot(LmInvPsi2LmInvT),
'left')
tmp2 = Linvmu.dot(psi1YLinvT)
tmp3 = backsub_both_sides(Lm, -D * LmInvPsi2LmInvT - tmp2 - tmp2.T,
'left') / 2.
dL_dKmm = (tmp1 + tmp1.T) / 2. + tmp3
#======================================================================
# Compute dL_dthetaL for uncertian input and non-heter noise
#======================================================================
dL_dthetaL = -D * N * beta / 2. - (
-D * psi0 / 2. - YRY / 2. -
(LmInvSmuLmInvT * LmInvPsi2LmInvT).sum() / 2. +
np.trace(LmInvPsi2LmInvT) * D / 2. +
(Linvmu * psi1YLinvT.T).sum()) * beta
#======================================================================
# Compute dL_dqU
#======================================================================
tmp1 = backsub_both_sides(Lm, -LmInvPsi2LmInvT, 'left')
dL_dqU_mean = tmp1.dot(mu) + dtrtrs(Lm, psi1YLinvT.T, trans=1)[0]
dL_dqU_var = D / 2. * tmp1
#======================================================================
# Compute the Posterior distribution of inducing points p(u|Y)
#======================================================================
KuuInvmu = dtrtrs(Lm, Linvmu, trans=1)[0]
tmp = backsub_both_sides(Lm, np.eye(M) - tdot(LinvLs), 'left')
post = Posterior(woodbury_inv=tmp,
woodbury_vector=KuuInvmu,
K=Kuu,
mean=mu,
cov=S,
K_chol=Lm)
#======================================================================
# Compute dL_dpsi
#======================================================================
dL_dpsi0 = -D * (beta * np.ones((N, ))) / 2.
if uncertain_outputs:
dL_dpsi1 = Y.mean.dot(dtrtrs(Lm, Linvmu, trans=1)[0].T) * beta
else:
dL_dpsi1 = Y.dot(dtrtrs(Lm, Linvmu, trans=1)[0].T) * beta
dL_dpsi2 = beta * backsub_both_sides(Lm,
D * np.eye(M) - LmInvSmuLmInvT,
'left') / 2.
if not uncertain_inputs:
dL_dpsi1 += psi1.dot(dL_dpsi2 + dL_dpsi2.T) / beta
dL_dpsi2 = None
if uncertain_inputs:
grad_dict = {
'dL_dKmm': dL_dKmm,
'dL_dpsi0': dL_dpsi0,
'dL_dpsi1': dL_dpsi1,
'dL_dpsi2': dL_dpsi2,
'dL_dthetaL': dL_dthetaL,
'dL_dqU_mean': dL_dqU_mean,
'dL_dqU_var': dL_dqU_var
}
else:
grad_dict = {
'dL_dKmm': dL_dKmm,
'dL_dKdiag': dL_dpsi0,
'dL_dKnm': dL_dpsi1,
'dL_dthetaL': dL_dthetaL,
'dL_dqU_mean': dL_dqU_mean,
'dL_dqU_var': dL_dqU_var
}
if uncertain_outputs:
m, s = Y.mean, Y.variance
grad_dict['dL_dYmean'] = -m * beta + dtrtrs(Lm, psi1.T)[0].T.dot(
dtrtrs(Lm, mu)[0])
grad_dict['dL_dYvar'] = beta / -2.
return post, logL, grad_dict
def inference(self, kern, X, Z, likelihood, Y, Y_metadata=None, Lm=None, dL_dKmm=None, fixed_covs_kerns=None, **kw):
_, output_dim = Y.shape
uncertain_inputs = isinstance(X, VariationalPosterior)
#see whether we've got a different noise variance for each datum
beta = 1./np.fmax(likelihood.gaussian_variance(Y_metadata), 1e-6)
# VVT_factor is a matrix such that tdot(VVT_factor) = VVT...this is for efficiency!
#self.YYTfactor = self.get_YYTfactor(Y)
#VVT_factor = self.get_VVTfactor(self.YYTfactor, beta)
het_noise = beta.size > 1
if het_noise:
raise(NotImplementedError("Heteroscedastic noise not implemented, should be possible though, feel free to try implementing it :)"))
if beta.ndim == 1:
beta = beta[:, None]
# do the inference:
num_inducing = Z.shape[0]
num_data = Y.shape[0]
# kernel computations, using BGPLVM notation
Kmm = kern.K(Z).copy()
diag.add(Kmm, self.const_jitter)
if Lm is None:
Lm = jitchol(Kmm)
# The rather complex computations of A, and the psi stats
if uncertain_inputs:
psi0 = kern.psi0(Z, X)
psi1 = kern.psi1(Z, X)
if het_noise:
psi2_beta = np.sum([kern.psi2(Z,X[i:i+1,:]) * beta_i for i,beta_i in enumerate(beta)],0)
else:
psi2_beta = kern.psi2(Z,X) * beta
LmInv = dtrtri(Lm)
A = LmInv.dot(psi2_beta.dot(LmInv.T))
else:
psi0 = kern.Kdiag(X)
psi1 = kern.K(X, Z)
if het_noise:
tmp = psi1 * (np.sqrt(beta))
else:
tmp = psi1 * (np.sqrt(beta))
tmp, _ = dtrtrs(Lm, tmp.T, lower=1)
A = tdot(tmp)
# factor B
B = np.eye(num_inducing) + A
LB = jitchol(B)
# back substutue C into psi1Vf
#tmp, _ = dtrtrs(Lm, psi1.T.dot(VVT_factor), lower=1, trans=0)
#_LBi_Lmi_psi1Vf, _ = dtrtrs(LB, tmp, lower=1, trans=0)
#tmp, _ = dtrtrs(LB, _LBi_Lmi_psi1Vf, lower=1, trans=1)
#Cpsi1Vf, _ = dtrtrs(Lm, tmp, lower=1, trans=1)
# data fit and derivative of L w.r.t. Kmm
#delit = tdot(_LBi_Lmi_psi1Vf)
# Expose YYT to get additional covariates in (YYT + Kgg):
tmp, _ = dtrtrs(Lm, psi1.T, lower=1, trans=0)
_LBi_Lmi_psi1, _ = dtrtrs(LB, tmp, lower=1, trans=0)
tmp, _ = dtrtrs(LB, _LBi_Lmi_psi1, lower=1, trans=1)
Cpsi1, _ = dtrtrs(Lm, tmp, lower=1, trans=1)
# TODO: cache this:
# Compute fixed covariates covariance:
if fixed_covs_kerns is not None:
K_fixed = 0
for name, [cov, k] in fixed_covs_kerns.iteritems():
K_fixed += k.K(cov)
#trYYT = self.get_trYYT(Y)
YYT_covs = (tdot(Y) + K_fixed)
data_term = beta**2 * YYT_covs
trYYT_covs = np.trace(YYT_covs)
else:
data_term = beta**2 * tdot(Y)
trYYT_covs = self.get_trYYT(Y)
#trYYT = self.get_trYYT(Y)
delit = mdot(_LBi_Lmi_psi1, data_term, _LBi_Lmi_psi1.T)
data_fit = np.trace(delit)
DBi_plus_BiPBi = backsub_both_sides(LB, output_dim * np.eye(num_inducing) + delit)
if dL_dKmm is None:
delit = -0.5 * DBi_plus_BiPBi
delit += -0.5 * B * output_dim
delit += output_dim * np.eye(num_inducing)
# Compute dL_dKmm
dL_dKmm = backsub_both_sides(Lm, delit)
# derivatives of L w.r.t. psi
dL_dpsi0, dL_dpsi1, dL_dpsi2 = _compute_dL_dpsi(num_inducing, num_data, output_dim, beta, Lm,
data_term, Cpsi1, DBi_plus_BiPBi,
psi1, het_noise, uncertain_inputs)
# log marginal likelihood
log_marginal = _compute_log_marginal_likelihood(likelihood, num_data, output_dim, beta, het_noise,
psi0, A, LB, trYYT_covs, data_fit, Y)
if self.save_per_dim:
self.saved_vals = [psi0, A, LB, _LBi_Lmi_psi1, beta]
# No heteroscedastics, so no _LBi_Lmi_psi1Vf:
# For the interested reader, try implementing the heteroscedastic version, it should be possible
_LBi_Lmi_psi1Vf = None # Is just here for documentation, so you can see, what it was.
#noise derivatives
dL_dR = _compute_dL_dR(likelihood,
het_noise, uncertain_inputs, LB,
_LBi_Lmi_psi1Vf, DBi_plus_BiPBi, Lm, A,
psi0, psi1, beta,
data_fit, num_data, output_dim, trYYT_covs, Y, None)
dL_dthetaL = likelihood.exact_inference_gradients(dL_dR,Y_metadata)
#put the gradients in the right places
if uncertain_inputs:
grad_dict = {'dL_dKmm': dL_dKmm,
'dL_dpsi0':dL_dpsi0,
'dL_dpsi1':dL_dpsi1,
'dL_dpsi2':dL_dpsi2,
'dL_dthetaL':dL_dthetaL}
else:
grad_dict = {'dL_dKmm': dL_dKmm,
'dL_dKdiag':dL_dpsi0,
'dL_dKnm':dL_dpsi1,
'dL_dthetaL':dL_dthetaL}
if fixed_covs_kerns is not None:
# For now, we do not take the gradients, we can compute them,
# but the maximum likelihood solution is to switch off the additional covariates....
dL_dcovs = beta * np.eye(K_fixed.shape[0]) - beta**2*tdot(_LBi_Lmi_psi1.T)
grad_dict['dL_dcovs'] = -.5 * dL_dcovs
#get sufficient things for posterior prediction
#TODO: do we really want to do this in the loop?
if 1:
woodbury_vector = (beta*Cpsi1).dot(Y)
else:
import ipdb; ipdb.set_trace()
psi1V = np.dot(Y.T*beta, psi1).T
tmp, _ = dtrtrs(Lm, psi1V, lower=1, trans=0)
tmp, _ = dpotrs(LB, tmp, lower=1)
woodbury_vector, _ = dtrtrs(Lm, tmp, lower=1, trans=1)
Bi, _ = dpotri(LB, lower=1)
symmetrify(Bi)
Bi = -dpotri(LB, lower=1)[0]
diag.add(Bi, 1)
woodbury_inv = backsub_both_sides(Lm, Bi)
#construct a posterior object
post = Posterior(woodbury_inv=woodbury_inv, woodbury_vector=woodbury_vector, K=Kmm, mean=None, cov=None, K_chol=Lm)
return post, log_marginal, grad_dict