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 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 woodbury_inv(self):
"""
The inverse of the woodbury matrix, in the gaussian likelihood case it is defined as
$$
(K_{xx} + \Sigma_{xx})^{-1}
\Sigma_{xx} := \texttt{Likelihood.variance / Approximate likelihood covariance}
$$
"""
if self._woodbury_inv is None:
if self._woodbury_chol is not None:
self._woodbury_inv, _ = dpotri(self._woodbury_chol, lower=1)
symmetrify(self._woodbury_inv)
elif self._covariance is not None:
B = np.atleast_3d(self._K) - np.atleast_3d(self._covariance)
self._woodbury_inv = np.empty_like(B)
for i in range(B.shape[-1]):
tmp, _ = dpotrs(self.K_chol, B[:, :, i])
self._woodbury_inv[:, :, i], _ = dpotrs(self.K_chol, tmp.T)
return self._woodbury_inv
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 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
