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EP is back.
This commit is contained in:
Ricardo
parent
a03d037736
commit
3d76664af0
4 changed files with 238 additions and 27 deletions
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@ -96,15 +96,11 @@ def toy_linear_1d_classification_laplace(seed=default_seed, optimize=True, plot=
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# Optimize
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if optimize:
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#m.update_likelihood_approximation()
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# Parameters optimization:
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try:
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m.optimize('scg', messages=1)
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except Exception as e:
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return m
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#m.pseudo_EM()
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# Plot
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if plot:
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fig, axes = pb.subplots(2, 1)
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@ -133,10 +129,7 @@ def sparse_toy_linear_1d_classification(num_inducing=10, seed=default_seed, opti
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# Optimize
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if optimize:
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#m.update_likelihood_approximation()
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# Parameters optimization:
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#m.optimize()
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m.pseudo_EM()
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m.optimize()
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# Plot
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if plot:
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@ -24,6 +24,13 @@ class EP(LatentFunctionInference):
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self.old_mutilde, self.old_vtilde = None, None
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self._ep_approximation = None
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def on_optimization_start(self):
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self._ep_approximation = None
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def on_optimization_end(self):
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# TODO: update approximation in the end as well? Maybe even with a switch?
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pass
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def inference(self, kern, X, likelihood, Y, Y_metadata=None, Z=None):
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num_data, output_dim = X.shape
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assert output_dim ==1, "ep in 1D only (for now!)"
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@ -47,8 +54,6 @@ class EP(LatentFunctionInference):
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return Posterior(woodbury_inv=Wi, woodbury_vector=alpha, K=K), log_marginal, {'dL_dK':dL_dK, 'dL_dthetaL':dL_dthetaL}
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def expectation_propagation(self, K, Y, likelihood, Y_metadata):
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num_data, data_dim = Y.shape
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@ -113,4 +118,3 @@ class EP(LatentFunctionInference):
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mu_tilde = v_tilde/tau_tilde
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return mu, Sigma, mu_tilde, tau_tilde, Z_hat
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@ -1,14 +1,56 @@
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import numpy as np
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from ...util.linalg import pdinv,jitchol,DSYR,tdot,dtrtrs, dpotrs
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from expectation_propagation import EP
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from ...util import diag
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from ...util.linalg import mdot, jitchol, backsub_both_sides, tdot, dtrtrs, dtrtri, dpotri, dpotrs, symmetrify, DSYR
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from ...util.misc import param_to_array
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from ...core.parameterization.variational import VariationalPosterior
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from . import LatentFunctionInference
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from posterior import Posterior
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log_2_pi = np.log(2*np.pi)
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class EPDTC(EP):
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def __init__(self, epsilon=1e-6, eta=1., delta=1.):
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class EPDTC(LatentFunctionInference):
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const_jitter = 1e-6
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def __init__(self, epsilon=1e-6, eta=1., delta=1., limit=1):
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from ...util.caching import Cacher
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self.limit = limit
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self.get_trYYT = Cacher(self._get_trYYT, limit)
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self.get_YYTfactor = Cacher(self._get_YYTfactor, limit)
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self.epsilon, self.eta, self.delta = epsilon, eta, delta
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self.reset()
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def set_limit(self, limit):
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self.get_trYYT.limit = limit
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self.get_YYTfactor.limit = limit
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def _get_trYYT(self, Y):
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return param_to_array(np.sum(np.square(Y)))
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def __getstate__(self):
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# has to be overridden, as Cacher objects cannot be pickled.
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return self.limit
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def __setstate__(self, state):
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# has to be overridden, as Cacher objects cannot be pickled.
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self.limit = state
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from ...util.caching import Cacher
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self.get_trYYT = Cacher(self._get_trYYT, self.limit)
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self.get_YYTfactor = Cacher(self._get_YYTfactor, self.limit)
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def _get_YYTfactor(self, Y):
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"""
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find a matrix L which satisfies LLT = YYT.
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Note that L may have fewer columns than Y.
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"""
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N, D = Y.shape
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if (N>=D):
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return param_to_array(Y)
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else:
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return jitchol(tdot(Y))
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def get_VVTfactor(self, Y, prec):
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return Y * prec # TODO chache this, and make it effective
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def reset(self):
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self.old_mutilde, self.old_vtilde = None, None
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self._ep_approximation = None
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@ -20,28 +62,131 @@ class EPDTC(EP):
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Kmm = kern.K(Z)
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Kmn = kern.K(Z,X)
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Lm = jitchol(Kmm)
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Lmi = dtrtrs(Lm,np.eye(Lm.shape[0]))[0]
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Kmmi = np.dot(Lmi.T,Lmi)
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KmmiKmn = np.dot(Kmmi,Kmn)
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K = np.dot(Kmn.T,KmmiKmn)
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if self._ep_approximation is None:
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mu, Sigma, mu_tilde, tau_tilde, Z_hat = self._ep_approximation = self.expectation_propagation(Kmm, Kmn, Y, likelihood, Y_metadata)
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else:
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mu, Sigma, mu_tilde, tau_tilde, Z_hat = self._ep_approximation
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Wi, LW, LWi, W_logdet = pdinv(K + np.diag(1./tau_tilde))
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alpha, _ = dpotrs(LW, mu_tilde, lower=1)
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if isinstance(X, VariationalPosterior):
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uncertain_inputs = True
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psi0 = kern.psi0(Z, X)
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psi1 = Kmn.T#kern.psi1(Z, X)
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psi2 = kern.psi2(Z, X)
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else:
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uncertain_inputs = False
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psi0 = kern.Kdiag(X)
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psi1 = Kmn.T#kern.K(X, Z)
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psi2 = None
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log_marginal = 0.5*(-num_data * log_2_pi - W_logdet - np.sum(alpha * mu_tilde)) # TODO: add log Z_hat??
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#see whether we're using variational uncertain inputs
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dL_dK = 0.5 * (tdot(alpha[:,None]) - Wi)
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_, output_dim = Y.shape
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#see whether we've got a different noise variance for each datum
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#beta = 1./np.fmax(likelihood.gaussian_variance(Y_metadata), 1e-6)
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beta = tau_tilde
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VVT_factor = beta[:,None]*mu_tilde[:,None]
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trYYT = self.get_trYYT(mu_tilde[:,None])
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# do the inference:
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het_noise = beta.size > 1
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num_inducing = Z.shape[0]
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num_data = Y.shape[0]
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# kernel computations, using BGPLVM notation
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Kmm = kern.K(Z).copy()
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diag.add(Kmm, self.const_jitter)
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Lm = jitchol(Kmm)
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# The rather complex computations of A
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if uncertain_inputs:
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if het_noise:
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psi2_beta = psi2 * (beta.flatten().reshape(num_data, 1, 1)).sum(0)
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else:
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psi2_beta = psi2.sum(0) * beta
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LmInv = dtrtri(Lm)
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A = LmInv.dot(psi2_beta.dot(LmInv.T))
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else:
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if het_noise:
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tmp = psi1 * (np.sqrt(beta.reshape(num_data, 1)))
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else:
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tmp = psi1 * (np.sqrt(beta))
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tmp, _ = dtrtrs(Lm, tmp.T, lower=1)
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A = tdot(tmp) #print A.sum()
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# factor B
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B = np.eye(num_inducing) + A
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LB = jitchol(B)
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psi1Vf = np.dot(psi1.T, VVT_factor)
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# back substutue C into psi1Vf
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tmp, _ = dtrtrs(Lm, psi1Vf, lower=1, trans=0)
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_LBi_Lmi_psi1Vf, _ = dtrtrs(LB, tmp, lower=1, trans=0)
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tmp, _ = dtrtrs(LB, _LBi_Lmi_psi1Vf, lower=1, trans=1)
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Cpsi1Vf, _ = dtrtrs(Lm, tmp, lower=1, trans=1)
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# data fit and derivative of L w.r.t. Kmm
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delit = tdot(_LBi_Lmi_psi1Vf)
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data_fit = np.trace(delit)
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DBi_plus_BiPBi = backsub_both_sides(LB, output_dim * np.eye(num_inducing) + delit)
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delit = -0.5 * DBi_plus_BiPBi
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delit += -0.5 * B * output_dim
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delit += output_dim * np.eye(num_inducing)
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# Compute dL_dKmm
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dL_dKmm = backsub_both_sides(Lm, delit)
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# derivatives of L w.r.t. psi
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dL_dpsi0, dL_dpsi1, dL_dpsi2 = _compute_dL_dpsi(num_inducing, num_data, output_dim, beta, Lm,
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VVT_factor, Cpsi1Vf, DBi_plus_BiPBi,
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psi1, het_noise, uncertain_inputs)
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# log marginal likelihood
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log_marginal = _compute_log_marginal_likelihood(likelihood, num_data, output_dim, beta, het_noise,
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psi0, A, LB, trYYT, data_fit, VVT_factor)
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#put the gradients in the right places
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dL_dR = _compute_dL_dR(likelihood,
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het_noise, uncertain_inputs, LB,
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_LBi_Lmi_psi1Vf, DBi_plus_BiPBi, Lm, A,
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psi0, psi1, beta,
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data_fit, num_data, output_dim, trYYT, mu_tilde[:,None])
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dL_dthetaL = 0#likelihood.exact_inference_gradients(dL_dR,Y_metadata)
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if uncertain_inputs:
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grad_dict = {'dL_dKmm': dL_dKmm,
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'dL_dpsi0':dL_dpsi0,
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'dL_dpsi1':dL_dpsi1,
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'dL_dpsi2':dL_dpsi2,
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'dL_dthetaL':dL_dthetaL}
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else:
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grad_dict = {'dL_dKmm': dL_dKmm,
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'dL_dKdiag':dL_dpsi0,
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'dL_dKnm':dL_dpsi1,
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'dL_dthetaL':dL_dthetaL}
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#get sufficient things for posterior prediction
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#TODO: do we really want to do this in the loop?
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if VVT_factor.shape[1] == Y.shape[1]:
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woodbury_vector = Cpsi1Vf # == Cpsi1V
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else:
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print 'foobar'
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psi1V = np.dot(mu_tilde[:,None].T*beta, psi1).T
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tmp, _ = dtrtrs(Lm, psi1V, lower=1, trans=0)
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tmp, _ = dpotrs(LB, tmp, lower=1)
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woodbury_vector, _ = dtrtrs(Lm, tmp, lower=1, trans=1)
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Bi, _ = dpotri(LB, lower=1)
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symmetrify(Bi)
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Bi = -dpotri(LB, lower=1)[0]
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diag.add(Bi, 1)
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woodbury_inv = backsub_both_sides(Lm, Bi)
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#construct a posterior object
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post = Posterior(woodbury_inv=woodbury_inv, woodbury_vector=woodbury_vector, K=Kmm, mean=None, cov=None, K_chol=Lm)
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return post, log_marginal, grad_dict
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dL_dthetaL = np.zeros(likelihood.size)#TODO: derivatives of the likelihood parameters
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return Posterior(woodbury_inv=Wi, woodbury_vector=alpha, K=K), log_marginal, {'dL_dK':dL_dK, 'dL_dthetaL':dL_dthetaL}
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@ -129,3 +274,69 @@ class EPDTC(EP):
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mu_tilde = v_tilde/tau_tilde
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return mu, Sigma, mu_tilde, tau_tilde, Z_hat
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def _compute_dL_dpsi(num_inducing, num_data, output_dim, beta, Lm, VVT_factor, Cpsi1Vf, DBi_plus_BiPBi, psi1, het_noise, uncertain_inputs):
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dL_dpsi0 = -0.5 * output_dim * (beta[:,None] * np.ones([num_data, 1])).flatten()
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dL_dpsi1 = np.dot(VVT_factor, Cpsi1Vf.T)
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dL_dpsi2_beta = 0.5 * backsub_both_sides(Lm, output_dim * np.eye(num_inducing) - DBi_plus_BiPBi)
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if het_noise:
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if uncertain_inputs:
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dL_dpsi2 = beta[:, None, None] * dL_dpsi2_beta[None, :, :]
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else:
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dL_dpsi1 += 2.*np.dot(dL_dpsi2_beta, (psi1 * beta.reshape(num_data, 1)).T).T
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dL_dpsi2 = None
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else:
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dL_dpsi2 = beta * dL_dpsi2_beta
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if uncertain_inputs:
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# repeat for each of the N psi_2 matrices
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dL_dpsi2 = np.repeat(dL_dpsi2[None, :, :], num_data, axis=0)
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else:
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# subsume back into psi1 (==Kmn)
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dL_dpsi1 += 2.*np.dot(psi1, dL_dpsi2)
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dL_dpsi2 = None
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return dL_dpsi0, dL_dpsi1, dL_dpsi2
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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):
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# the partial derivative vector for the likelihood
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if likelihood.size == 0:
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# save computation here.
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dL_dR = None
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elif het_noise:
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if uncertain_inputs:
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raise NotImplementedError, "heteroscedatic derivates with uncertain inputs not implemented"
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else:
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#from ...util.linalg import chol_inv
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#LBi = chol_inv(LB)
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LBi, _ = dtrtrs(LB,np.eye(LB.shape[0]))
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Lmi_psi1, nil = dtrtrs(Lm, psi1.T, lower=1, trans=0)
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_LBi_Lmi_psi1, _ = dtrtrs(LB, Lmi_psi1, lower=1, trans=0)
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dL_dR = -0.5 * beta + 0.5 * (beta*Y)**2
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dL_dR += 0.5 * output_dim * (psi0 - np.sum(Lmi_psi1**2,0))[:,None] * beta**2
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dL_dR += 0.5*np.sum(mdot(LBi.T,LBi,Lmi_psi1)*Lmi_psi1,0)[:,None]*beta**2
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dL_dR += -np.dot(_LBi_Lmi_psi1Vf.T,_LBi_Lmi_psi1).T * Y * beta**2
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dL_dR += 0.5*np.dot(_LBi_Lmi_psi1Vf.T,_LBi_Lmi_psi1).T**2 * beta**2
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else:
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# likelihood is not heteroscedatic
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dL_dR = -0.5 * num_data * output_dim * beta + 0.5 * trYYT * beta ** 2
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dL_dR += 0.5 * output_dim * (psi0.sum() * beta ** 2 - np.trace(A) * beta)
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dL_dR += beta * (0.5 * np.sum(A * DBi_plus_BiPBi) - data_fit)
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return dL_dR
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def _compute_log_marginal_likelihood(likelihood, num_data, output_dim, beta, het_noise, psi0, A, LB, trYYT, data_fit,Y):
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#compute log marginal likelihood
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if het_noise:
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lik_1 = -0.5 * num_data * output_dim * np.log(2. * np.pi) + 0.5 * np.sum(np.log(beta)) - 0.5 * np.sum(beta * np.square(Y).sum(axis=-1))
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lik_2 = -0.5 * output_dim * (np.sum(beta.flatten() * psi0) - np.trace(A))
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else:
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lik_1 = -0.5 * num_data * output_dim * (np.log(2. * np.pi) - np.log(beta)) - 0.5 * beta * trYYT
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lik_2 = -0.5 * output_dim * (np.sum(beta * psi0) - np.trace(A))
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lik_3 = -output_dim * (np.sum(np.log(np.diag(LB))))
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lik_4 = 0.5 * data_fit
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log_marginal = lik_1 + lik_2 + lik_3 + lik_4
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return log_marginal
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@ -227,3 +227,6 @@ class Bernoulli(Likelihood):
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ns = np.ones_like(gp, dtype=int)
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Ysim = np.random.binomial(ns, self.gp_link.transf(gp))
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return Ysim.reshape(orig_shape)
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def exact_inference_gradients(self, dL_dKdiag,Y_metadata=None):
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pass
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