diff --git a/GPy/inference/latent_function_inference/expectation_propagation.py b/GPy/inference/latent_function_inference/expectation_propagation.py index 02beedb3..81c020df 100644 --- a/GPy/inference/latent_function_inference/expectation_propagation.py +++ b/GPy/inference/latent_function_inference/expectation_propagation.py @@ -6,9 +6,111 @@ from paramz import ObsAr from . import ExactGaussianInference, VarDTC from ...util import diag from .posterior import PosteriorEP as Posterior +from ...likelihoods import Gaussian log_2_pi = np.log(2*np.pi) + +#Four wrapper classes to help modularisation of different EP versions +class marginalMoments(object): + def __init__(self, num_data): + self.Z_hat = np.empty(num_data,dtype=np.float64) + self.mu_hat = np.empty(num_data,dtype=np.float64) + self.sigma2_hat = np.empty(num_data,dtype=np.float64) + + +class cavityParams(object): + def __init__(self, num_data): + self.tau = np.empty(num_data,dtype=np.float64) + self.v = np.empty(num_data,dtype=np.float64) + def _update_i(self, eta, ga_approx, post_params, i): + self.tau[i] = 1./post_params.Sigma_diag[i] - eta*ga_approx.tau[i] + self.v[i] = post_params.mu[i]/post_params.Sigma_diag[i] - eta*ga_approx.v[i] + + +class gaussianApproximation(object): + def __init__(self, v, tau): + self.tau = tau + self.v = v + def _update_i(self, eta, delta, post_params, marg_moments, i): + #Site parameters update + delta_tau = delta/eta*(1./marg_moments.sigma2_hat[i] - 1./post_params.Sigma_diag[i]) + delta_v = delta/eta*(marg_moments.mu_hat[i]/marg_moments.sigma2_hat[i] - post_params.mu[i]/post_params.Sigma_diag[i]) + tau_tilde_prev = self.tau[i] + self.tau[i] += delta_tau + + # Enforce positivity of tau_tilde. Even though this is guaranteed for logconcave sites, it is still possible + # to get negative values due to numerical errors. Moreover, the value of tau_tilde should be positive in order to + # update the marginal likelihood without runnint into instabilities issues. + if self.tau[i] < np.finfo(float).eps: + self.tau[i] = np.finfo(float).eps + delta_tau = self.tau[i] - tau_tilde_prev + self.v[i] += delta_v + + return (delta_tau, delta_v) + + +class posteriorParamsBase(object): + def __init__(self, mu, Sigma_diag): + self.mu = mu + self.Sigma_diag = Sigma_diag + def _update_rank1(self, *arg): + pass + + def _recompute(self, *arg): + pass + +class posteriorParams(posteriorParamsBase): + def __init__(self, mu, Sigma, L=None): + self.Sigma = Sigma + self.L = L + Sigma_diag = np.diag(self.Sigma) + super(posteriorParams, self).__init__(mu, Sigma_diag) + + def _update_rank1(self, delta_tau, ga_approx, i): + ci = delta_tau/(1.+ delta_tau*self.Sigma_diag[i]) + DSYR(self.Sigma, self.Sigma[:,i].copy(), -ci) + self.mu = np.dot(self.Sigma, ga_approx.v) + + @staticmethod + def _recompute(K, ga_approx): + num_data = len(ga_approx.tau) + tau_tilde_root = np.sqrt(ga_approx.tau) + Sroot_tilde_K = tau_tilde_root[:,None] * K + B = np.eye(num_data) + Sroot_tilde_K * tau_tilde_root[None,:] + L = jitchol(B) + V, _ = dtrtrs(L, Sroot_tilde_K, lower=1) + Sigma = K - np.dot(V.T,V) #K - KS^(1/2)BS^(1/2)K = (K^(-1) + \Sigma^(-1))^(-1) + mu = np.dot(Sigma,ga_approx.v) + return posteriorParams(mu=mu, Sigma=Sigma, L=L) + +class posteriorParamsDTC(posteriorParamsBase): + def __init__(self, mu, Sigma_diag): + super(posteriorParamsDTC, self).__init__(mu, Sigma_diag) + + def _update_rank1(self, LLT, Kmn, delta_v, delta_tau, i): + #DSYR(Sigma, Sigma[:,i].copy(), -delta_tau/(1.+ delta_tau*Sigma[i,i])) + DSYR(LLT,Kmn[:,i].copy(),delta_tau) + L = jitchol(LLT) + V,info = dtrtrs(L,Kmn,lower=1) + self.Sigma_diag = np.maximum(np.sum(V*V,-2), np.finfo(float).eps) #diag(K_nm (L L^\top)^(-1)) K_mn + si = np.sum(V.T*V[:,i],-1) #(V V^\top)[:,i] + self.mu += (delta_v-delta_tau*self.mu[i])*si + #mu = np.dot(Sigma, v_tilde) + + @staticmethod + def _recompute(LLT0, Kmn, ga_approx): + LLT = LLT0 + np.dot(Kmn*ga_approx.tau[None,:],Kmn.T) + L = jitchol(LLT) + V, _ = dtrtrs(L,Kmn,lower=1) + #Sigma_diag = np.sum(V*V,-2) + #Knmv_tilde = np.dot(Kmn,v_tilde) + #mu = np.dot(V2.T,Knmv_tilde) + Sigma = np.dot(V.T,V) + mu = np.dot(Sigma, ga_approx.v) + Sigma_diag = np.diag(Sigma).copy() + return posteriorParamsDTC(mu, Sigma_diag), LLT + class EPBase(object): def __init__(self, epsilon=1e-6, eta=1., delta=1., always_reset=False, max_iters=np.inf, ep_mode="alternated", parallel_updates=False): """ @@ -28,6 +130,7 @@ class EPBase(object): :parallel_updates: boolean. If true, updates of the parameters of the sites in parallel """ super(EPBase, self).__init__() + self.always_reset = always_reset self.epsilon, self.eta, self.delta, self.max_iters = epsilon, eta, delta, max_iters self.ep_mode = ep_mode @@ -35,7 +138,7 @@ class EPBase(object): self.reset() def reset(self): - self.old_mutilde, self.old_vtilde = None, None + self.ga_approx_old = None self._ep_approximation = None def on_optimization_start(self): @@ -45,6 +148,11 @@ class EPBase(object): # TODO: update approximation in the end as well? Maybe even with a switch? pass + def _stop_criteria(self, ga_approx): + tau_diff = np.mean(np.square(ga_approx.tau-self.ga_approx_old.tau)) + v_diff = np.mean(np.square(ga_approx.v-self.ga_approx_old.v)) + return ((tau_diff < self.epsilon) and (v_diff < self.epsilon)) + def __setstate__(self, state): super(EPBase, self).__setstate__(state[0]) self.epsilon, self.eta, self.delta = state[1] @@ -67,19 +175,18 @@ class EP(EPBase, ExactGaussianInference): if self.ep_mode=="nested": #Force EP at each step of the optimization self._ep_approximation = None - mu, Sigma, mu_tilde, tau_tilde, log_Z_tilde = self._ep_approximation = self.expectation_propagation(K, Y, likelihood, Y_metadata) + post_params, ga_approx, cav_params, log_Z_tilde = self._ep_approximation = self.expectation_propagation(K, Y, likelihood, Y_metadata) elif self.ep_mode=="alternated": if getattr(self, '_ep_approximation', None) is None: #if we don't yet have the results of runnign EP, run EP and store the computed factors in self._ep_approximation - mu, Sigma, mu_tilde, tau_tilde, log_Z_tilde = self._ep_approximation = self.expectation_propagation(K, Y, likelihood, Y_metadata) + post_params, ga_approx, cav_params, log_Z_tilde = self._ep_approximation = self.expectation_propagation(K, Y, likelihood, Y_metadata) else: #if we've already run EP, just use the existing approximation stored in self._ep_approximation - mu, Sigma, mu_tilde, tau_tilde, log_Z_tilde = self._ep_approximation + post_params, ga_approx, cav_params, log_Z_tilde = self._ep_approximation else: raise ValueError("ep_mode value not valid") - v_tilde = mu_tilde * tau_tilde - return self._inference(K, tau_tilde, v_tilde, likelihood, Y_metadata=Y_metadata, Z_tilde=log_Z_tilde.sum()) + return self._inference(Y, K, ga_approx, cav_params, likelihood, Y_metadata=Y_metadata, Z_tilde=log_Z_tilde) def expectation_propagation(self, K, Y, likelihood, Y_metadata): @@ -90,41 +197,57 @@ class EP(EPBase, ExactGaussianInference): # than ObsArrays Y = Y.values.copy() - #Initial values - Marginal moments - Z_hat = np.empty(num_data,dtype=np.float64) - mu_hat = np.empty(num_data,dtype=np.float64) - sigma2_hat = np.empty(num_data,dtype=np.float64) + #Initial values - Marginal moments, cavity params, gaussian approximation params and posterior params + marg_moments = marginalMoments(num_data) + cav_params = cavityParams(num_data) + ga_approx, post_params = self._init_approximations(K, num_data) - tau_cav = np.empty(num_data,dtype=np.float64) - v_cav = np.empty(num_data,dtype=np.float64) + #Approximation + stop = False + iterations = 0 + while not stop and (iterations < self.max_iters): + self._local_updates(num_data, cav_params, post_params, marg_moments, ga_approx, likelihood, Y, Y_metadata) + #(re) compute Sigma and mu using full Cholesky decompy + post_params = posteriorParams._recompute(K, ga_approx) + + #monitor convergence + if iterations > 0: + stop = self._stop_criteria(ga_approx) + self.ga_approx_old = gaussianApproximation(ga_approx.v.copy(), ga_approx.tau.copy()) + iterations += 1 + + # Z_tilde after removing the terms that can lead to infinite terms due to tau_tilde close to zero. + # This terms cancel with the coreresponding terms in the marginal loglikelihood + log_Z_tilde = self._log_Z_tilde(marg_moments, ga_approx, cav_params) + # - 0.5*np.log(tau_tilde) + 0.5*(v_tilde*v_tilde*1./tau_tilde) + return (post_params, ga_approx, cav_params, log_Z_tilde) + + def _init_approximations(self, K, num_data): #initial values - Gaussian factors #Initial values - Posterior distribution parameters: q(f|X,Y) = N(f|mu,Sigma) - if self.old_mutilde is None: - tau_tilde, mu_tilde, v_tilde = np.zeros((3, num_data)) + if self.ga_approx_old is None: + v_tilde, tau_tilde = np.zeros((2, num_data)) + ga_approx = gaussianApproximation(v_tilde, tau_tilde) Sigma = K.copy() diag.add(Sigma, 1e-7) mu = np.zeros(num_data) + post_params = posteriorParams(mu, Sigma) else: - assert self.old_mutilde.size == num_data, "data size mis-match: did you change the data? try resetting!" - mu_tilde, v_tilde = self.old_mutilde, self.old_vtilde - tau_tilde = v_tilde/mu_tilde - mu, Sigma, _ = self._ep_compute_posterior(K, tau_tilde, v_tilde) - diag.add(Sigma, 1e-7) + assert self.ga_approx_old.v.size == num_data, "data size mis-match: did you change the data? try resetting!" + ga_approx = gaussianApproximation(self.ga_approx_old.v, self.ga_approx_old.tau) + post_params = posteriorParams._recompute(K, ga_approx) + diag.add(post_params.Sigma, 1e-7) # TODO: Check the log-marginal under both conditions and choose the best one + return (ga_approx, post_params) - #Approximation - tau_diff = self.epsilon + 1. - v_diff = self.epsilon + 1. - tau_tilde_old = np.nan - v_tilde_old = np.nan - iterations = 0 - while ((tau_diff > self.epsilon) or (v_diff > self.epsilon)) and (iterations < self.max_iters): - update_order = np.random.permutation(num_data) + def _local_updates(self, num_data, cav_params, post_params, marg_moments, ga_approx, likelihood, Y, Y_metadata, update_order=None): + if update_order is None: + update_order = np.random.permutation(num_data) for i in update_order: #Cavity distribution parameters - tau_cav[i] = 1./Sigma[i,i] - self.eta*tau_tilde[i] - v_cav[i] = mu[i]/Sigma[i,i] - self.eta*v_tilde[i] + cav_params._update_i(self.eta, ga_approx, post_params, i) + if Y_metadata is not None: # Pick out the relavent metadata for Yi Y_metadata_i = {} @@ -133,91 +256,45 @@ class EP(EPBase, ExactGaussianInference): else: Y_metadata_i = None #Marginal moments - Z_hat[i], mu_hat[i], sigma2_hat[i] = likelihood.moments_match_ep(Y[i], tau_cav[i], v_cav[i], Y_metadata_i=Y_metadata_i) - #Site parameters update - delta_tau = self.delta/self.eta*(1./sigma2_hat[i] - 1./Sigma[i,i]) - delta_v = self.delta/self.eta*(mu_hat[i]/sigma2_hat[i] - mu[i]/Sigma[i,i]) - tau_tilde_prev = tau_tilde[i] - tau_tilde[i] += delta_tau + marg_moments.Z_hat[i], marg_moments.mu_hat[i], marg_moments.sigma2_hat[i] = likelihood.moments_match_ep(Y[i], cav_params.tau[i], cav_params.v[i], Y_metadata_i=Y_metadata_i) - # Enforce positivity of tau_tilde. Even though this is guaranteed for logconcave sites, it is still possible - # to get negative values due to numerical errors. Moreover, the value of tau_tilde should be positive in order to - # update the marginal likelihood without inestability issues. - if tau_tilde[i] < np.finfo(float).eps: - tau_tilde[i] = np.finfo(float).eps - delta_tau = tau_tilde[i] - tau_tilde_prev - v_tilde[i] += delta_v + #Site parameters update + delta_tau, delta_v = ga_approx._update_i(self.eta, self.delta, post_params, marg_moments, i) if self.parallel_updates == False: - #Posterior distribution parameters update - ci = delta_tau/(1.+ delta_tau*Sigma[i,i]) - DSYR(Sigma, Sigma[:,i].copy(), -ci) - mu = np.dot(Sigma, v_tilde) + post_params._update_rank1(delta_tau, ga_approx, i) - #(re) compute Sigma and mu using full Cholesky decompy - mu, Sigma, _ = self._ep_compute_posterior(K, tau_tilde, v_tilde) + def _log_Z_tilde(self, marg_moments, ga_approx, cav_params): + return np.sum((np.log(marg_moments.Z_hat) + 0.5*np.log(2*np.pi) + 0.5*np.log(1+ga_approx.tau/cav_params.tau) - 0.5 * ((ga_approx.v)**2 * 1./(cav_params.tau + ga_approx.tau)) + + 0.5*(cav_params.v * ( ( (ga_approx.tau/cav_params.tau) * cav_params.v - 2.0 * ga_approx.v ) * 1./(cav_params.tau + ga_approx.tau))))) - #monitor convergence - if iterations > 0: - tau_diff = np.mean(np.square(tau_tilde-tau_tilde_old)) - v_diff = np.mean(np.square(v_tilde-v_tilde_old)) - tau_tilde_old = tau_tilde.copy() - v_tilde_old = v_tilde.copy() - iterations += 1 - mu_tilde = v_tilde/tau_tilde - mu_cav = v_cav/tau_cav - sigma2_sigma2tilde = 1./tau_cav + 1./tau_tilde - - # Z_tilde after removing the terms that can lead to infinite terms due to tau_tilde close to zero. - # This terms cancel with the coreresponding terms in the marginal loglikelihood - log_Z_tilde = (np.log(Z_hat) + 0.5*np.log(2*np.pi) + 0.5*np.log(1+tau_tilde/tau_cav) - - 0.5 * ((v_tilde)**2 * 1./(tau_cav + tau_tilde)) + 0.5*(v_cav * ( ( (tau_tilde/tau_cav) * v_cav - 2.0 * v_tilde ) * 1./(tau_cav + tau_tilde)))) - # - 0.5*np.log(tau_tilde) + 0.5*(v_tilde*v_tilde*1./tau_tilde) - - self.old_mutilde = mu_tilde - self.old_vtilde = v_tilde - - return mu, Sigma, mu_tilde, tau_tilde, log_Z_tilde - - def _ep_compute_posterior(self, K, tau_tilde, v_tilde): - num_data = len(tau_tilde) - tau_tilde_root = np.sqrt(tau_tilde) - Sroot_tilde_K = tau_tilde_root[:,None] * K - B = np.eye(num_data) + Sroot_tilde_K * tau_tilde_root[None,:] - L = jitchol(B) - V, _ = dtrtrs(L, Sroot_tilde_K, lower=1) - Sigma = K - np.dot(V.T,V) #K - KS^(1/2)BS^(1/2)K = (K^(-1) + \Sigma^(-1))^(-1) - mu = np.dot(Sigma,v_tilde) - return (mu, Sigma, L) - - def _ep_marginal(self, K, tau_tilde, v_tilde, Z_tilde): - mu, Sigma, L = self._ep_compute_posterior(K, tau_tilde, v_tilde) + def _ep_marginal(self, K, ga_approx, Z_tilde): + post_params = posteriorParams._recompute(K, ga_approx) # Gaussian log marginal excluding terms that can go to infinity due to arbitrarily small tau_tilde. # These terms cancel out with the terms excluded from Z_tilde - B_logdet = np.sum(2.0*np.log(np.diag(L))) - log_marginal = 0.5*(-len(tau_tilde) * log_2_pi - B_logdet + np.sum(v_tilde * np.dot(Sigma,v_tilde))) + B_logdet = np.sum(2.0*np.log(np.diag(post_params.L))) + log_marginal = 0.5*(-len(ga_approx.tau) * log_2_pi - B_logdet + np.sum(ga_approx.v * np.dot(post_params.Sigma,ga_approx.v))) log_marginal += Z_tilde - return log_marginal, mu, Sigma, L + return log_marginal, post_params - def _inference(self, K, tau_tilde, v_tilde, likelihood, Z_tilde, Y_metadata=None): - log_marginal, mu, Sigma, L = self._ep_marginal(K, tau_tilde, v_tilde, Z_tilde) + def _inference(self, Y, K, ga_approx, cav_params, likelihood, Z_tilde, Y_metadata=None): + log_marginal, post_params = self._ep_marginal(K, ga_approx, Z_tilde) - tau_tilde_root = np.sqrt(tau_tilde) + tau_tilde_root = np.sqrt(ga_approx.tau) Sroot_tilde_K = tau_tilde_root[:,None] * K - aux_alpha , _ = dpotrs(L, np.dot(Sroot_tilde_K, v_tilde), lower=1) - alpha = (v_tilde - tau_tilde_root * aux_alpha)[:,None] #(K + Sigma^(\tilde))^(-1) /mu^(/tilde) - LWi, _ = dtrtrs(L, np.diag(tau_tilde_root), lower=1) + 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 = likelihood.exact_inference_gradients(np.diag(dL_dK), Y_metadata) - + dL_dthetaL = likelihood.ep_gradients(Y, cav_params.tau, cav_params.v, np.diag(dL_dK), Y_metadata=Y_metadata, quad_mode='gh') return Posterior(woodbury_inv=Wi, woodbury_vector=alpha, K=K), log_marginal, {'dL_dK':dL_dK, 'dL_dthetaL':dL_dthetaL, 'dL_dm':alpha} @@ -244,24 +321,25 @@ class EPDTC(EPBase, VarDTC): if self.ep_mode=="nested": #Force EP at each step of the optimization self._ep_approximation = None - mu, Sigma_diag, mu_tilde, tau_tilde, log_Z_tilde = self._ep_approximation = self.expectation_propagation(Kmm, Kmn, Y, likelihood, Y_metadata) + post_params, ga_approx, log_Z_tilde = self._ep_approximation = self.expectation_propagation(Kmm, Kmn, Y, likelihood, Y_metadata) elif self.ep_mode=="alternated": if getattr(self, '_ep_approximation', None) is None: #if we don't yet have the results of runnign EP, run EP and store the computed factors in self._ep_approximation - mu, Sigma_diag, mu_tilde, tau_tilde, log_Z_tilde = self._ep_approximation = self.expectation_propagation(Kmm, Kmn, Y, likelihood, Y_metadata) + post_params, ga_approx, log_Z_tilde = self._ep_approximation = self.expectation_propagation(Kmm, Kmn, Y, likelihood, Y_metadata) else: #if we've already run EP, just use the existing approximation stored in self._ep_approximation - mu, Sigma_diag, mu_tilde, tau_tilde, log_Z_tilde = self._ep_approximation + post_params, ga_approx, log_Z_tilde = self._ep_approximation else: raise ValueError("ep_mode value not valid") - return super(EPDTC, self).inference(kern, X, Z, likelihood, mu_tilde, + mu_tilde = ga_approx.v / ga_approx.tau.astype(float) + + return super(EPDTC, self).inference(kern, X, Z, likelihood, ObsAr(mu_tilde[:,None]), mean_function=mean_function, Y_metadata=Y_metadata, - precision=tau_tilde, + precision=ga_approx.tau, Lm=Lm, dL_dKmm=dL_dKmm, - psi0=psi0, psi1=psi1, psi2=psi2, Z_tilde=log_Z_tilde.sum()) - + psi0=psi0, psi1=psi1, psi2=psi2, Z_tilde=log_Z_tilde) def expectation_propagation(self, Kmm, Kmn, Y, likelihood, Y_metadata): @@ -272,119 +350,91 @@ class EPDTC(EPBase, VarDTC): # than ObsArrays Y = Y.values.copy() - #Initial values - Marginal moments - Z_hat = np.zeros(num_data,dtype=np.float64) - mu_hat = np.zeros(num_data,dtype=np.float64) - sigma2_hat = np.zeros(num_data,dtype=np.float64) + #Initial values - Marginal moments, cavity params, gaussian approximation params and posterior params + marg_moments = marginalMoments(num_data) + cav_params = cavityParams(num_data) + ga_approx, post_params, LLT0, LLT = self._init_approximations(Kmm, Kmn, num_data) - tau_cav = np.empty(num_data,dtype=np.float64) - v_cav = np.empty(num_data,dtype=np.float64) + #Approximation + stop = False + iterations = 0 + while not stop and (iterations < self.max_iters): + self._local_updates(num_data, LLT0, LLT, Kmn, cav_params, post_params, marg_moments, ga_approx, likelihood, Y, Y_metadata) + #(re) compute Sigma, Sigma_diag and mu using full Cholesky decompy + post_params, LLT = posteriorParamsDTC._recompute(LLT0, Kmn, ga_approx) + post_params.Sigma_diag = np.maximum(post_params.Sigma_diag, np.finfo(float).eps) + #monitor convergence + if iterations > 0: + stop = self._stop_criteria(ga_approx) + self.ga_approx_old = gaussianApproximation(ga_approx.v.copy(), ga_approx.tau.copy()) + iterations += 1 + + log_Z_tilde = self._log_Z_tilde(marg_moments, ga_approx, cav_params) + + return post_params, ga_approx, log_Z_tilde + + def _log_Z_tilde(self, marg_moments, ga_approx, cav_params): + mu_tilde = ga_approx.v/ga_approx.tau + mu_cav = cav_params.v/cav_params.tau + sigma2_sigma2tilde = 1./cav_params.tau + 1./ga_approx.tau + + return np.sum((np.log(marg_moments.Z_hat) + 0.5*np.log(2*np.pi) + 0.5*np.log(sigma2_sigma2tilde) + + 0.5*((mu_cav - mu_tilde)**2) / (sigma2_sigma2tilde))) + + def _init_approximations(self, Kmm, Kmn, num_data): #initial values - Gaussian factors #Initial values - Posterior distribution parameters: q(f|X,Y) = N(f|mu,Sigma) LLT0 = Kmm.copy() Lm = jitchol(LLT0) #K_m = L_m L_m^\top - Vm,info = dtrtrs(Lm,Kmn,lower=1) + Vm,info = dtrtrs(Lm, Kmn,lower=1) # Lmi = dtrtri(Lm) # Kmmi = np.dot(Lmi.T,Lmi) # KmmiKmn = np.dot(Kmmi,Kmn) # Qnn_diag = np.sum(Kmn*KmmiKmn,-2) Qnn_diag = np.sum(Vm*Vm,-2) #diag(Knm Kmm^(-1) Kmn) #diag.add(LLT0, 1e-8) - if self.old_mutilde is None: + if self.ga_approx_old is None: #Initial values - Posterior distribution parameters: q(f|X,Y) = N(f|mu,Sigma) LLT = LLT0.copy() #Sigma = K.copy() mu = np.zeros(num_data) Sigma_diag = Qnn_diag.copy() + 1e-8 - tau_tilde, mu_tilde, v_tilde = np.zeros((3, num_data)) + v_tilde, tau_tilde = np.zeros((2, num_data)) + ga_approx = gaussianApproximation(v_tilde, tau_tilde) + post_params = posteriorParamsDTC(mu, Sigma_diag) + else: - assert self.old_mutilde.size == num_data, "data size mis-match: did you change the data? try resetting!" - mu_tilde, v_tilde = self.old_mutilde, self.old_vtilde - tau_tilde = v_tilde/mu_tilde - mu, Sigma_diag, LLT = self._ep_compute_posterior(LLT0, Kmn, tau_tilde, v_tilde) - Sigma_diag += 1e-8 + assert self.ga_approx_old.v.size == num_data, "data size mis-match: did you change the data? try resetting!" + ga_approx = gaussianApproximation(self.ga_approx_old.v, self.ga_approx_old.tau) + post_params, LLT = posteriorParamsDTC._recompute(LLT0, Kmn, ga_approx) + post_params.Sigma_diag += 1e-8 + # TODO: Check the log-marginal under both conditions and choose the best one - #Approximation - tau_diff = self.epsilon + 1. - v_diff = self.epsilon + 1. - tau_tilde_old = np.nan - v_tilde_old = np.nan - iterations = 0 - while ((tau_diff > self.epsilon) or (v_diff > self.epsilon)) and (iterations < self.max_iters): + return (ga_approx, post_params, LLT0, LLT) + + def _local_updates(self, num_data, LLT0, LLT, Kmn, cav_params, post_params, marg_moments, ga_approx, likelihood, Y, Y_metadata, update_order=None): + if update_order is None: update_order = np.random.permutation(num_data) - for i in update_order: - #Cavity distribution parameters - tau_cav[i] = 1./Sigma_diag[i] - self.eta*tau_tilde[i] - v_cav[i] = mu[i]/Sigma_diag[i] - self.eta*v_tilde[i] - if Y_metadata is not None: - # Pick out the relavent metadata for Yi - Y_metadata_i = {} - for key in Y_metadata.keys(): - Y_metadata_i[key] = Y_metadata[key][i, :] - else: - Y_metadata_i = None + for i in update_order: - #Marginal moments - Z_hat[i], mu_hat[i], sigma2_hat[i] = likelihood.moments_match_ep(Y[i], tau_cav[i], v_cav[i], Y_metadata_i=Y_metadata_i) - #Site parameters update - delta_tau = self.delta/self.eta*(1./sigma2_hat[i] - 1./Sigma_diag[i]) - delta_v = self.delta/self.eta*(mu_hat[i]/sigma2_hat[i] - mu[i]/Sigma_diag[i]) - tau_tilde_prev = tau_tilde[i] - tau_tilde[i] += delta_tau + #Cavity distribution parameters + cav_params._update_i(self.eta, ga_approx, post_params, i) - # Enforce positivity of tau_tilde. Even though this is guaranteed for logconcave sites, it is still possible - # to get negative values due to numerical errors. Moreover, the value of tau_tilde should be positive in order to - # update the marginal likelihood without inestability issues. - if tau_tilde[i] < np.finfo(float).eps: - tau_tilde[i] = np.finfo(float).eps - delta_tau = tau_tilde[i] - tau_tilde_prev - v_tilde[i] += delta_v - #Posterior distribution parameters update - if self.parallel_updates == False: - #DSYR(Sigma, Sigma[:,i].copy(), -delta_tau/(1.+ delta_tau*Sigma[i,i])) - DSYR(LLT,Kmn[:,i].copy(),delta_tau) - L = jitchol(LLT) - V,info = dtrtrs(L,Kmn,lower=1) - Sigma_diag = np.maximum(np.sum(V*V,-2), np.finfo(float).eps) #diag(K_nm (L L^\top)^(-1)) K_mn - si = np.sum(V.T*V[:,i],-1) #(V V^\top)[:,i] - mu += (delta_v-delta_tau*mu[i])*si - #mu = np.dot(Sigma, v_tilde) + if Y_metadata is not None: + # Pick out the relavent metadata for Yi + Y_metadata_i = {} + for key in Y_metadata.keys(): + Y_metadata_i[key] = Y_metadata[key][i, :] + else: + Y_metadata_i = None - #(re) compute Sigma, Sigma_diag and mu using full Cholesky decompy - mu, Sigma_diag, LLT = self._ep_compute_posterior(LLT0, Kmn, tau_tilde, v_tilde) - Sigma_diag = np.maximum(Sigma_diag, np.finfo(float).eps) + #Marginal moments + marg_moments.Z_hat[i], marg_moments.mu_hat[i], marg_moments.sigma2_hat[i] = likelihood.moments_match_ep(Y[i], cav_params.tau[i], cav_params.v[i], Y_metadata_i=Y_metadata_i) + #Site parameters update + delta_tau, delta_v = ga_approx._update_i(self.eta, self.delta, post_params, marg_moments, i) - #monitor convergence - if iterations>0: - tau_diff = np.mean(np.square(tau_tilde-tau_tilde_old)) - v_diff = np.mean(np.square(v_tilde-v_tilde_old)) - tau_tilde_old = tau_tilde.copy() - v_tilde_old = v_tilde.copy() - iterations += 1 - - mu_tilde = v_tilde/tau_tilde - mu_cav = v_cav/tau_cav - sigma2_sigma2tilde = 1./tau_cav + 1./tau_tilde - - log_Z_tilde = (np.log(Z_hat) + 0.5*np.log(2*np.pi) + 0.5*np.log(sigma2_sigma2tilde) - + 0.5*((mu_cav - mu_tilde)**2) / (sigma2_sigma2tilde)) - - self.old_mutilde = mu_tilde - self.old_vtilde = v_tilde - - return mu, Sigma_diag, ObsAr(mu_tilde[:,None]), tau_tilde, log_Z_tilde - - def _ep_compute_posterior(self, LLT0, Kmn, tau_tilde, v_tilde): - LLT = LLT0 + np.dot(Kmn*tau_tilde[None,:],Kmn.T) - L = jitchol(LLT) - V, _ = dtrtrs(L,Kmn,lower=1) - #Sigma_diag = np.sum(V*V,-2) - #Knmv_tilde = np.dot(Kmn,v_tilde) - #mu = np.dot(V2.T,Knmv_tilde) - Sigma = np.dot(V.T,V) - mu = np.dot(Sigma,v_tilde) - Sigma_diag = np.diag(Sigma).copy() - - return (mu, Sigma_diag, LLT) + #Posterior distribution parameters update + if self.parallel_updates == False: + post_params._update_rank1(LLT, Kmn, delta_v, delta_tau, i) diff --git a/GPy/likelihoods/binomial.py b/GPy/likelihoods/binomial.py index e63c009a..61440ec9 100644 --- a/GPy/likelihoods/binomial.py +++ b/GPy/likelihoods/binomial.py @@ -66,7 +66,14 @@ class Binomial(Likelihood): np.testing.assert_array_equal(N.shape, y.shape) nchoosey = special.gammaln(N+1) - special.gammaln(y+1) - special.gammaln(N-y+1) - return nchoosey + y*np.log(inv_link_f) + (N-y)*np.log(1.-inv_link_f) + + Ny = N-y + t1 = np.zeros(y.shape) + t2 = np.zeros(y.shape) + t1[y>0] = y[y>0]*np.log(inv_link_f[y>0]) + t2[Ny>0] = Ny[Ny>0]*np.log(1.-inv_link_f[Ny>0]) + + return nchoosey + t1 + t2 def dlogpdf_dlink(self, inv_link_f, y, Y_metadata=None): """ @@ -86,7 +93,13 @@ class Binomial(Likelihood): N = Y_metadata['trials'] np.testing.assert_array_equal(N.shape, y.shape) - return y/inv_link_f - (N-y)/(1.-inv_link_f) + Ny = N-y + t1 = np.zeros(y.shape) + t2 = np.zeros(y.shape) + t1[y>0] = y[y>0]/inv_link_f[y>0] + t2[Ny>0] = (Ny[Ny>0])/(1.-inv_link_f[Ny>0]) + + return t1 - t2 def d2logpdf_dlink2(self, inv_link_f, y, Y_metadata=None): """ @@ -111,7 +124,13 @@ class Binomial(Likelihood): """ N = Y_metadata['trials'] np.testing.assert_array_equal(N.shape, y.shape) - return -y/np.square(inv_link_f) - (N-y)/np.square(1.-inv_link_f) + Ny = N-y + t1 = np.zeros(y.shape) + t2 = np.zeros(y.shape) + t1[y>0] = -y[y>0]/np.square(inv_link_f[y>0]) + t2[Ny>0] = -(Ny[Ny>0])/np.square(1.-inv_link_f[Ny>0]) + return t1+t2 + def d3logpdf_dlink3(self, inv_link_f, y, Y_metadata=None): """ @@ -135,8 +154,14 @@ class Binomial(Likelihood): N = Y_metadata['trials'] np.testing.assert_array_equal(N.shape, y.shape) - inv_link_f2 = np.square(inv_link_f) - return 2*y/inv_link_f**3 - 2*(N-y)/(1.-inv_link_f)**3 + #inv_link_f2 = np.square(inv_link_f) #TODO Remove. Why is this here? + + Ny = N-y + t1 = np.zeros(y.shape) + t2 = np.zeros(y.shape) + t1[y>0] = 2*y[y>0]/inv_link_f[y>0]**3 + t2[Ny>0] = - 2*(Ny[Ny>0])/(1.-inv_link_f[Ny>0])**3 + return t1 + t2 def samples(self, gp, Y_metadata=None, **kw): """ diff --git a/GPy/likelihoods/gaussian.py b/GPy/likelihoods/gaussian.py index 533c6558..04fd6a33 100644 --- a/GPy/likelihoods/gaussian.py +++ b/GPy/likelihoods/gaussian.py @@ -57,7 +57,10 @@ class Gaussian(Likelihood): def update_gradients(self, grad): self.variance.gradient = grad - def exact_inference_gradients(self, dL_dKdiag,Y_metadata=None): + def ep_gradients(self, Y, cav_tau, cav_v, dL_dKdiag, Y_metadata=None, quad_mode='gk', boost_grad=1.): + return self.exact_inference_gradients(dL_dKdiag) + + def exact_inference_gradients(self, dL_dKdiag, Y_metadata=None): return dL_dKdiag.sum() def _preprocess_values(self, Y): diff --git a/GPy/likelihoods/likelihood.py b/GPy/likelihoods/likelihood.py index c5b2094f..308c6a76 100644 --- a/GPy/likelihoods/likelihood.py +++ b/GPy/likelihoods/likelihood.py @@ -6,8 +6,12 @@ from scipy import stats,special import scipy as sp from . import link_functions from ..util.misc import chain_1, chain_2, chain_3, blockify_dhess_dtheta, blockify_third, blockify_hessian, safe_exp +from ..util.quad_integrate import quadgk_int from scipy.integrate import quad +from functools import partial + import warnings + from ..core.parameterization import Parameterized class Likelihood(Parameterized): @@ -223,6 +227,91 @@ class Likelihood(Parameterized): self.__gh_points = np.polynomial.hermite.hermgauss(T) return self.__gh_points + def ep_gradients(self, Y, cav_tau, cav_v, dL_dKdiag, Y_metadata=None, quad_mode='gk', boost_grad=1.): + if self.size > 0: + shape = Y.shape + tau,v,Y = cav_tau.flatten(), cav_v.flatten(),Y.flatten() + mu = v/tau + sigma2 = 1./tau + + # assert Y.shape == v.shape + dlik_dtheta = np.empty((self.size, Y.shape[0])) + # for j in range(self.size): + Y_metadata_list = [] + for index in range(len(Y)): + Y_metadata_i = {} + if Y_metadata is not None: + for key in Y_metadata.keys(): + Y_metadata_i[key] = Y_metadata[key][index,:] + Y_metadata_list.append(Y_metadata_i) + + if quad_mode == 'gk': + f = partial(self.integrate_gk) + quads = zip(*map(f, Y.flatten(), mu.flatten(), np.sqrt(sigma2.flatten()), Y_metadata_list)) + quads = np.vstack(quads) + quads.reshape(self.size, shape[0], shape[1]) + elif quad_mode == 'gh': + f = partial(self.integrate_gh) + quads = zip(*map(f, Y.flatten(), mu.flatten(), np.sqrt(sigma2.flatten()))) + quads = np.hstack(quads) + quads = quads.T + else: + raise Exception("no other quadrature mode available") + # do a gaussian-hermite integration + dL_dtheta_avg = boost_grad * np.nanmean(quads, axis=1) + dL_dtheta = boost_grad * np.nansum(quads, axis=1) + # dL_dtheta = boost_grad * np.nansum(dlik_dtheta, axis=1) + else: + dL_dtheta = np.zeros(self.num_params) + return dL_dtheta + + + def integrate_gk(self, Y, mu, sigma, Y_metadata_i=None): + # gaussian-kronrod integration. + fmin = -np.inf + fmax = np.inf + SQRT_2PI = np.sqrt(2.*np.pi) + def generate_integral(f): + a = np.exp(self.logpdf_link(f, Y, Y_metadata_i)) * np.exp(-0.5 * np.square((f - mu) / sigma)) / ( + SQRT_2PI * sigma) + fn1 = a * self.dlogpdf_dtheta(f, Y, Y_metadata_i) + fn = fn1 + return fn + + dF_dtheta_i = quadgk_int(generate_integral, fmin=fmin, fmax=fmax) + return dF_dtheta_i + + def integrate_gh(self, Y, mu, sigma, Y_metadata_i=None, gh_points=None): + # gaussian-hermite quadrature. + # "calculate site derivatives E_f{d logp(y_i|f_i)/da} where a is a likelihood parameter + # and the expectation is over the exact marginal posterior, which is not gaussian- and is + # unnormalised product of the cavity distribution(a Gaussian) and the exact likelihood term. + # + # calculate the expectation wrt the approximate marginal posterior, which should be approximately the same. + # . This term is needed for evaluating the + # gradients of the marginal likelihood estimate Z_EP wrt likelihood parameters." + # "writing it explicitly " + # use them for gaussian-hermite quadrature + + SQRT_2PI = np.sqrt(2.*np.pi) + if gh_points is None: + gh_x, gh_w = self._gh_points(32) + else: + gh_x, gh_w = gh_points + + X = gh_x[None,:]*np.sqrt(2.)*sigma + mu + + # Here X is a grid vector of possible fi values, while Y is just a single value which will be broadcasted. + a = np.exp(self.logpdf_link(X, Y, Y_metadata_i)) + a = a.repeat(self.num_params,0) + b = self.dlogpdf_dtheta(X, Y, Y_metadata_i) + old_shape = b.shape + fn = np.array([i*j for i,j in zip(a.flatten(), b.flatten())]) + fn = fn.reshape(old_shape) + + dF_dtheta_i = np.dot(fn, gh_w)/np.sqrt(np.pi) + return dF_dtheta_i + def variational_expectations(self, Y, m, v, gh_points=None, Y_metadata=None): """ Use Gauss-Hermite Quadrature to compute diff --git a/GPy/testing/ep_likelihood_tests.py b/GPy/testing/ep_likelihood_tests.py new file mode 100644 index 00000000..70efe210 --- /dev/null +++ b/GPy/testing/ep_likelihood_tests.py @@ -0,0 +1,147 @@ + +import numpy as np +import unittest +import GPy +from GPy.models import GradientChecker + +fixed_seed = 10 +from nose.tools import with_setup, nottest + + +# this file will contain some high level tests, this is not unit testing, but will give us a higher level estimate +# if things are going well under the hood. +class TestObservationModels(unittest.TestCase): + def setUp(self): + np.random.seed(fixed_seed) + self.N = 100 + self.D = 2 + self.X = np.random.rand(self.N, self.D) + + self.real_noise_std = 0.05 + noise = np.random.randn(*self.X[:, 0].shape) * self.real_noise_std + self.Y = (np.sin(self.X[:, 0] * 2 * np.pi) + noise)[:, None] + self.num_points = self.X.shape[0] + self.f = np.random.rand(self.N, 1) + self.binary_Y = np.asarray(np.random.rand(self.N) > 0.5, dtype=np.int)[:, None] + # self.binary_Y[self.binary_Y == 0.0] = -1.0 + self.positive_Y = np.exp(self.Y.copy()) + + self.Y_noisy = self.Y.copy() + self.Y_verynoisy = self.Y.copy() + self.Y_noisy[75] += 1.3 + + self.init_var = 0.15 + self.deg_free = 4. + censored = np.zeros_like(self.Y) + random_inds = np.random.choice(self.N, int(self.N / 2), replace=True) + censored[random_inds] = 1 + self.Y_metadata = dict() + self.Y_metadata['censored'] = censored + self.kernel1 = GPy.kern.RBF(self.X.shape[1]) + GPy.kern.White(self.X.shape[1]) + + def tearDown(self): + self.Y = None + self.X = None + self.binary_Y =None + self.positive_Y = None + self.kernel1 = None + + @with_setup(setUp, tearDown) + def testEPClassification(self): + bernoulli = GPy.likelihoods.Bernoulli() + laplace_inf = GPy.inference.latent_function_inference.Laplace() + + ep_inf_alt = GPy.inference.latent_function_inference.EP(ep_mode='alternated') + ep_inf_nested = GPy.inference.latent_function_inference.EP(ep_mode='nested') + ep_inf_fractional = GPy.inference.latent_function_inference.EP(ep_mode='nested', eta=0.9) + + m1 = GPy.core.GP(self.X, self.binary_Y.copy(), kernel=self.kernel1.copy(), likelihood=bernoulli.copy(), inference_method=laplace_inf) + m1.randomize() + + m2 = GPy.core.GP(self.X, self.binary_Y.copy(), kernel=self.kernel1.copy(), likelihood=bernoulli.copy(), inference_method=ep_inf_alt) + m2.randomize() + + m3 = GPy.core.GP(self.X, self.binary_Y.copy(), kernel=self.kernel1.copy(), likelihood=bernoulli.copy(), inference_method=ep_inf_nested) + m3.randomize() + # + m4 = GPy.core.GP(self.X, self.binary_Y.copy(), kernel=self.kernel1.copy(), likelihood=bernoulli.copy(), inference_method=ep_inf_fractional) + m4.randomize() + + optimizer = 'bfgs' + + #do gradcheck here ... + # self.assertTrue(m1.checkgrad()) + # self.assertTrue(m2.checkgrad()) + # self.assertTrue(m3.checkgrad()) + # self.assertTrue(m4.checkgrad()) + + m1.optimize(optimizer=optimizer, max_iters=300) + m2.optimize(optimizer=optimizer, max_iters=300) + m3.optimize(optimizer=optimizer, max_iters=300) + m4.optimize(optimizer=optimizer, max_iters=300) + + # taking laplace predictions as the ground truth + probs_mean_lap, probs_var_lap = m1.predict(self.X) + probs_mean_ep_alt, probs_var_ep_alt = m2.predict(self.X) + probs_mean_ep_nested, probs_var_ep_nested = m3.predict(self.X) + + # for simple single dimension data , marginal likelihood for laplace and EP approximations should not be so far apart. + self.assertAlmostEqual(m1.log_likelihood(), m2.log_likelihood(),delta=1) + self.assertAlmostEqual(m1.log_likelihood(), m3.log_likelihood(), delta=1) + self.assertAlmostEqual(m1.log_likelihood(), m4.log_likelihood(), delta=5) + + GPy.util.classification.conf_matrix(probs_mean_lap, self.binary_Y) + GPy.util.classification.conf_matrix(probs_mean_ep_alt, self.binary_Y) + GPy.util.classification.conf_matrix(probs_mean_ep_nested, self.binary_Y) + + @nottest + def rmse(self, Y, Ystar): + return np.sqrt(np.mean((Y - Ystar) ** 2)) + + @with_setup(setUp, tearDown) + def test_EP_with_StudentT(self): + studentT = GPy.likelihoods.StudentT(deg_free=self.deg_free, sigma2=self.init_var) + laplace_inf = GPy.inference.latent_function_inference.Laplace() + + ep_inf_alt = GPy.inference.latent_function_inference.EP(ep_mode='alternated') + ep_inf_nested = GPy.inference.latent_function_inference.EP(ep_mode='nested') + ep_inf_frac = GPy.inference.latent_function_inference.EP(ep_mode='nested', eta=0.7) + + m1 = GPy.core.GP(self.X.copy(), self.Y_noisy.copy(), kernel=self.kernel1.copy(), likelihood=studentT.copy(), inference_method=laplace_inf) + # optimize + m1['.*white'].constrain_fixed(1e-5) + m1.randomize() + + m2 = GPy.core.GP(self.X.copy(), self.Y_noisy.copy(), kernel=self.kernel1.copy(), likelihood=studentT.copy(), inference_method=ep_inf_alt) + m2['.*white'].constrain_fixed(1e-5) + # m2.constrain_bounded('.*t_scale2', 0.001, 10) + m2.randomize() + + # m3 = GPy.core.GP(self.X, self.Y_noisy.copy(), kernel=self.kernel1, likelihood=studentT.copy(), inference_method=ep_inf_nested) + # m3['.*white'].constrain_fixed(1e-5) + # # m3.constrain_bounded('.*t_scale2', 0.001, 10) + # m3.randomize() + + optimizer='bfgs' + m1.optimize(optimizer=optimizer,max_iters=400) + m2.optimize(optimizer=optimizer, max_iters=400) + # m3.optimize(optimizer=optimizer, max_iters=500) + + self.assertAlmostEqual(m1.log_likelihood(), m2.log_likelihood(),delta=200) + + # self.assertAlmostEqual(m1.log_likelihood(), m3.log_likelihood(), 3) + + preds_mean_lap, preds_var_lap = m1.predict(self.X) + preds_mean_alt, preds_var_alt = m2.predict(self.X) + # preds_mean_nested, preds_var_nested = m3.predict(self.X) + rmse_lap = self.rmse(preds_mean_lap, self.Y) + rmse_alt = self.rmse(preds_mean_alt, self.Y) + # rmse_nested = self.rmse(preds_mean_nested, self.Y_noisy) + + if rmse_alt > rmse_lap: + self.assertAlmostEqual(rmse_lap, rmse_alt, delta=1.5) + # m3.optimize(optimizer=optimizer, max_iters=500) + + +if __name__ == "__main__": + unittest.main() \ No newline at end of file diff --git a/GPy/testing/inference_tests.py b/GPy/testing/inference_tests.py index dbde1329..816d5488 100644 --- a/GPy/testing/inference_tests.py +++ b/GPy/testing/inference_tests.py @@ -61,6 +61,18 @@ class InferenceGPEP(unittest.TestCase): Y = lik.samples(f).reshape(-1,1) return X, Y + def genNoisyData(self): + np.random.seed(1) + X = np.random.rand(100,1) + self.real_std = 0.1 + noise = np.random.randn(*X[:, 0].shape)*self.real_std + Y = (np.sin(X[:, 0]*2*np.pi) + noise)[:, None] + self.f = np.random.rand(X.shape[0],1) + Y_extra_noisy = Y.copy() + Y_extra_noisy[50] += 4. + # Y_extra_noisy[80:83] -= 2. + return X, Y, Y_extra_noisy + def test_inference_EP(self): from paramz import ObsAr X, Y = self.genData() @@ -73,11 +85,45 @@ class InferenceGPEP(unittest.TestCase): inference_method=inf, likelihood=lik) K = self.model.kern.K(X) - mu, Sigma, mu_tilde, tau_tilde, log_Z_tilde = self.model.inference_method.expectation_propagation(K, ObsAr(Y), lik, None) - v_tilde = mu_tilde * tau_tilde - p, m, d = self.model.inference_method._inference(K, tau_tilde, v_tilde, lik, Y_metadata=None, Z_tilde=log_Z_tilde.sum()) - p0, m0, d0 = super(GPy.inference.latent_function_inference.expectation_propagation.EP, inf).inference(k, X,lik ,mu_tilde[:,None], mean_function=None, variance=1./tau_tilde, K=K, Z_tilde=log_Z_tilde.sum() + np.sum(- 0.5*np.log(tau_tilde) + 0.5*(v_tilde*v_tilde*1./tau_tilde))) + post_params, ga_approx, cav_params, log_Z_tilde = self.model.inference_method.expectation_propagation(K, ObsAr(Y), lik, None) + + mu_tilde = ga_approx.v / ga_approx.tau.astype(float) + p, m, d = self.model.inference_method._inference(Y, K, ga_approx, cav_params, lik, Y_metadata=None, Z_tilde=log_Z_tilde) + p0, m0, d0 = super(GPy.inference.latent_function_inference.expectation_propagation.EP, inf).inference(k, X,lik ,mu_tilde[:,None], mean_function=None, variance=1./ga_approx.tau, K=K, Z_tilde=log_Z_tilde + np.sum(- 0.5*np.log(ga_approx.tau) + 0.5*(ga_approx.v*ga_approx.v*1./ga_approx.tau))) + + assert (np.sum(np.array([m - m0, + np.sum(d['dL_dK'] - d0['dL_dK']), + np.sum(d['dL_dthetaL'] - d0['dL_dthetaL']), + np.sum(d['dL_dm'] - d0['dL_dm']), + np.sum(p._woodbury_vector - p0._woodbury_vector), + np.sum(p.woodbury_inv - p0.woodbury_inv)])) < 1e6) + + # NOTE: adding a test like above for parameterized likelihood- the above test is + # only for probit likelihood which does not have any tunable hyperparameter which is why + # the term in dictionary of gradients: dL_dthetaL will always be zero. So here we repeat tests for + # student-t likelihood and heterodescastic gaussian noise case. This test simply checks if the posterior + # and gradients of log marginal are roughly the same for inference through EP and exact gaussian inference using + # the gaussian approximation for the individual likelihood site terms. For probit likelihood, it is possible to + # calculate moments analytically, but for other likelihoods, we will need to use numerical quadrature techniques, + # and it is possible that any error might creep up because of quadrature implementation. + def test_inference_EP_non_classification(self): + from paramz import ObsAr + X, Y, Y_extra_noisy = self.genNoisyData() + deg_freedom = 5. + init_noise_var = 0.08 + lik_studentT = GPy.likelihoods.StudentT(deg_free=deg_freedom, sigma2=init_noise_var) + # like_gaussian_noise = GPy.likelihoods.MixedNoise() + k = GPy.kern.RBF(1, variance=2., lengthscale=1.1) + ep_inf_alt = GPy.inference.latent_function_inference.expectation_propagation.EP(max_iters=4, delta=0.5) + # ep_inf_nested = GPy.inference.latent_function_inference.expectation_propagation.EP(ep_mode='nested', max_iters=100, delta=0.5) + m = GPy.core.GP(X=X,Y=Y_extra_noisy,kernel=k,likelihood=lik_studentT,inference_method=ep_inf_alt) + K = m.kern.K(X) + post_params, ga_approx, cav_params, log_Z_tilde = m.inference_method.expectation_propagation(K, ObsAr(Y_extra_noisy), lik_studentT, None) + + mu_tilde = ga_approx.v / ga_approx.tau.astype(float) + p, m, d = m.inference_method._inference(Y_extra_noisy, K, ga_approx, cav_params, lik_studentT, Y_metadata=None, Z_tilde=log_Z_tilde) + p0, m0, d0 = super(GPy.inference.latent_function_inference.expectation_propagation.EP, ep_inf_alt).inference(k, X,lik_studentT ,mu_tilde[:,None], mean_function=None, variance=1./ga_approx.tau, K=K, Z_tilde=log_Z_tilde + np.sum(- 0.5*np.log(ga_approx.tau) + 0.5*(ga_approx.v*ga_approx.v*1./ga_approx.tau))) assert (np.sum(np.array([m - m0, np.sum(d['dL_dK'] - d0['dL_dK']), diff --git a/GPy/testing/quadrature_tests.py b/GPy/testing/quadrature_tests.py new file mode 100644 index 00000000..e519d87e --- /dev/null +++ b/GPy/testing/quadrature_tests.py @@ -0,0 +1,39 @@ +from __future__ import print_function, division +import numpy as np +import GPy +import warnings +from ..util.quad_integrate import quadgk_int, quadvgk + + + +class QuadTests(np.testing.TestCase): + """ + test file for checking implementation of gaussian-kronrod quadrature. + we will take a function which can be integrated analytically and check if quadgk result is similar or not! + through this file we can test how numerically accurate quadrature implementation in native numpy or manual code is. + """ + def setUp(self): + pass + + def test_infinite_quad(self): + def f(x): + return np.exp(-0.5*x**2)*np.power(x,np.arange(3)[:,None]) + quad_int_val = quadgk_int(f) + real_val = np.sqrt(np.pi * 2) + np.testing.assert_almost_equal(real_val, quad_int_val[0], decimal=7) + + def test_finite_quad(self): + def f2(x): + return x**2 + quad_int_val = quadvgk(f2, 1.,2.) + real_val = 7/3. + np.testing.assert_almost_equal(real_val, quad_int_val, decimal=5) + +if __name__ == '__main__': + def f(x): + return np.exp(-0.5 * x ** 2) * np.power(x, np.arange(3)[:, None]) + + quad_int_val = quadgk_int(f) + real_val = np.sqrt(np.pi*2) + np.testing.assert_almost_equal(real_val, quad_int_val[0], decimal=7) + print(quadgk_int(f)) diff --git a/GPy/util/__init__.py b/GPy/util/__init__.py index 685551fd..4994ddcb 100644 --- a/GPy/util/__init__.py +++ b/GPy/util/__init__.py @@ -17,3 +17,4 @@ from . import multioutput from . import parallel from . import functions from . import cluster_with_offset +from . import quad_integrate diff --git a/GPy/util/quad_integrate.py b/GPy/util/quad_integrate.py new file mode 100644 index 00000000..f3711ada --- /dev/null +++ b/GPy/util/quad_integrate.py @@ -0,0 +1,119 @@ +""" +The file for utilities related to integration by quadrature methods +- will contain implementation for gaussian-kronrod integration. + +""" +import numpy as np + +def getSubs(Subs, XK, NK=1): + M = (Subs[1, :] - Subs[0, :]) / 2 + C = (Subs[1, :] + Subs[0, :]) / 2 + I = XK[:, None] * M + np.ones((NK, 1)) * C + # A = [Subs(1,:); I] + A = np.vstack((Subs[0, :], I)) + # B = [I;Subs(2,:)] + B = np.vstack((I, Subs[1, :])) + # Subs = [reshape(A, 1, []); + A = A.flatten() + # reshape(B, 1, [])]; + B = B.flatten() + Subs = np.vstack((A,B)) + # Subs = np.concatenate((A, B), axis=0) + return Subs + +def quadvgk(feval, fmin, fmax, tol1=1e-5, tol2=1e-5): + """ + numpy implementation makes use of the code here: http://se.mathworks.com/matlabcentral/fileexchange/18801-quadvgk + We here use gaussian kronrod integration already used in gpstuff for evaluating one dimensional integrals. + This is vectorised quadrature which means that several functions can be evaluated at the same time over a grid of + points. + :param f: + :param fmin: + :param fmax: + :param difftol: + :return: + """ + + XK = np.array([-0.991455371120813, -0.949107912342759, -0.864864423359769, -0.741531185599394, + -0.586087235467691, -0.405845151377397, -0.207784955007898, 0., + 0.207784955007898, 0.405845151377397, 0.586087235467691, + 0.741531185599394, 0.864864423359769, 0.949107912342759, 0.991455371120813]) + WK = np.array([0.022935322010529, 0.063092092629979, 0.104790010322250, 0.140653259715525, + 0.169004726639267, 0.190350578064785, 0.204432940075298, 0.209482141084728, + 0.204432940075298, 0.190350578064785, 0.169004726639267, + 0.140653259715525, 0.104790010322250, 0.063092092629979, 0.022935322010529]) + # 7-point Gaussian weightings + WG = np.array([0.129484966168870, 0.279705391489277, 0.381830050505119, 0.417959183673469, + 0.381830050505119, 0.279705391489277, 0.129484966168870]) + + NK = WK.size + G = np.arange(2,NK,2) + tol1 = 1e-4 + tol2 = 1e-4 + Subs = np.array([[fmin],[fmax]]) + # number of functions to evaluate in the feval vector of functions. + NF = feval(np.zeros(1)).size + Q = np.zeros(NF) + neval = 0 + while Subs.size > 0: + Subs = getSubs(Subs,XK) + M = (Subs[1,:] - Subs[0,:]) / 2 + C = (Subs[1,:] + Subs[0,:]) / 2 + # NM = length(M); + NM = M.size + # x = reshape(XK * M + ones(NK, 1) * C, 1, []); + x = XK[:,None]*M + C + x = x.flatten() + FV = feval(x) + # FV = FV[:,None] + Q1 = np.zeros((NF, NM)) + Q2 = np.zeros((NF, NM)) + + # for n=1:NF + # F = reshape(FV(n,:), NK, []); + # Q1(n,:) = M. * sum((WK * ones(1, NM)). * F); + # Q2(n,:) = M. * sum((WG * ones(1, NM)). * F(G,:)); + # end + # for i in range(NF): + # F = FV + # F = F.reshape((NK,-1)) + # temp_mat = np.sum(np.multiply(WK[:,None]*np.ones((1,NM)), F),axis=0) + # Q1[i,:] = np.multiply(M, temp_mat) + # temp_mat = np.sum(np.multiply(WG[:,None]*np.ones((1, NM)), F[G-1,:]), axis=0) + # Q2[i,:] = np.multiply(M, temp_mat) + # ind = np.where(np.logical_or(np.max(np.abs(Q1 -Q2) / Q1) < tol1, (Subs[1,:] - Subs[0,:]) <= tol2) > 0)[0] + # Q = Q + np.sum(Q1[:,ind], axis=1) + # np.delete(Subs, ind,axis=1) + + Q1 = np.dot(FV.reshape(NF, NK, NM).swapaxes(2,1),WK)*M + Q2 = np.dot(FV.reshape(NF, NK, NM).swapaxes(2,1)[:,:,1::2],WG)*M + #ind = np.nonzero(np.logical_or(np.max(np.abs((Q1-Q2)/Q1), 0) < difftol , M < xtol))[0] + ind = np.nonzero(np.logical_or(np.max(np.abs((Q1-Q2)), 0) < tol1 , (Subs[1,:] - Subs[0,:]) < tol2))[0] + Q = Q + np.sum(Q1[:,ind], axis=1) + Subs = np.delete(Subs, ind, axis=1) + return Q + +def quadgk_int(f, fmin=-np.inf, fmax=np.inf, difftol=0.1): + """ + Integrate f from fmin to fmax, + do integration by substitution + x = r / (1-r**2) + when r goes from -1 to 1 , x goes from -inf to inf. + the interval for quadgk function is from -1 to +1, so we transform the space from (-inf,inf) to (-1,1) + :param f: + :param fmin: + :param fmax: + :param difftol: + :return: + """ + difftol = 1e-4 + def trans_func(r): + r2 = np.square(r) + x = r / (1-r2) + dx_dr = (1 + r2)/(1-r2)**2 + return f(x)*dx_dr + + integrand = quadvgk(trans_func, -1., 1., difftol, difftol) + return integrand + +