import numpy as np from scipy import stats from ..util.linalg import pdinv,mdot,jitchol,chol_inv,DSYR,tdot,dtrtrs from likelihood import likelihood class EP(likelihood): def __init__(self,data,noise_model): """ Expectation Propagation :param data: data to model :type data: numpy array :param noise_model: noise distribution :type noise_model: A GPy noise model """ self.noise_model = noise_model self.data = data self.num_data, self.output_dim = self.data.shape self.is_heteroscedastic = True self.num_params = 0 #Initial values - Likelihood approximation parameters: #p(y|f) = t(f|tau_tilde,v_tilde) self.tau_tilde = np.zeros(self.num_data) self.v_tilde = np.zeros(self.num_data) #initial values for the GP variables self.Y = np.zeros((self.num_data,1)) self.covariance_matrix = np.eye(self.num_data) self.precision = np.ones(self.num_data)[:,None] self.Z = 0 self.YYT = None self.V = self.precision * self.Y self.VVT_factor = self.V self.trYYT = 0. super(EP, self).__init__() def restart(self): self.tau_tilde = np.zeros(self.num_data) self.v_tilde = np.zeros(self.num_data) self.Y = np.zeros((self.num_data,1)) self.covariance_matrix = np.eye(self.num_data) self.precision = np.ones(self.num_data)[:,None] self.Z = 0 self.YYT = None self.V = self.precision * self.Y self.VVT_factor = self.V self.trYYT = 0. def predictive_values(self,mu,var,full_cov,**noise_args): if full_cov: raise NotImplementedError, "Cannot make correlated predictions with an EP likelihood" return self.noise_model.predictive_values(mu,var,**noise_args) def log_predictive_density(self, y_test, mu_star, var_star): """ Calculation of the log predictive density .. math: p(y_{*}|D) = p(y_{*}|f_{*})p(f_{*}|\mu_{*}\\sigma^{2}_{*}) :param y_test: test observations (y_{*}) :type y_test: (Nx1) array :param mu_star: predictive mean of gaussian p(f_{*}|mu_{*}, var_{*}) :type mu_star: (Nx1) array :param var_star: predictive variance of gaussian p(f_{*}|mu_{*}, var_{*}) :type var_star: (Nx1) array """ return self.noise_model.log_predictive_density(y_test, mu_star, var_star) def _get_params(self): #return np.zeros(0) return self.noise_model._get_params() def _get_param_names(self): #return [] return self.noise_model._get_param_names() def _set_params(self,p): #pass # TODO: the EP likelihood might want to take some parameters... self.noise_model._set_params(p) def _gradients(self,partial): #return np.zeros(0) # TODO: the EP likelihood might want to take some parameters... return self.noise_model._gradients(partial) def _compute_GP_variables(self): #Variables to be called from GP mu_tilde = self.v_tilde/self.tau_tilde #When calling EP, this variable is used instead of Y in the GP model sigma_sum = 1./self.tau_ + 1./self.tau_tilde mu_diff_2 = (self.v_/self.tau_ - mu_tilde)**2 self.Z = np.sum(np.log(self.Z_hat)) + 0.5*np.sum(np.log(sigma_sum)) + 0.5*np.sum(mu_diff_2/sigma_sum) #Normalization constant, aka Z_ep self.Z += 0.5*self.num_data*np.log(2*np.pi) self.Y = mu_tilde[:,None] self.YYT = np.dot(self.Y,self.Y.T) self.covariance_matrix = np.diag(1./self.tau_tilde) self.precision = self.tau_tilde[:,None] self.V = self.precision * self.Y self.VVT_factor = self.V self.trYYT = np.trace(self.YYT) def fit_full(self, K, epsilon=1e-3,power_ep=[1.,1.]): """ The expectation-propagation algorithm. For nomenclature see Rasmussen & Williams 2006. :param epsilon: Convergence criterion, maximum squared difference allowed between mean updates to stop iterations (float) :type epsilon: float :param power_ep: Power EP parameters :type power_ep: list of floats """ self.epsilon = epsilon self.eta, self.delta = power_ep #Initial values - Posterior distribution parameters: q(f|X,Y) = N(f|mu,Sigma) mu = np.zeros(self.num_data) Sigma = K.copy() """ Initial values - Cavity distribution parameters: q_(f|mu_,sigma2_) = Product{q_i(f|mu_i,sigma2_i)} sigma_ = 1./tau_ mu_ = v_/tau_ """ self.tau_ = np.empty(self.num_data,dtype=float) self.v_ = np.empty(self.num_data,dtype=float) #Initial values - Marginal moments z = np.empty(self.num_data,dtype=float) self.Z_hat = np.empty(self.num_data,dtype=float) phi = np.empty(self.num_data,dtype=float) mu_hat = np.empty(self.num_data,dtype=float) sigma2_hat = np.empty(self.num_data,dtype=float) #Approximation epsilon_np1 = self.epsilon + 1. epsilon_np2 = self.epsilon + 1. self.iterations = 0 self.np1 = [self.tau_tilde.copy()] self.np2 = [self.v_tilde.copy()] while epsilon_np1 > self.epsilon or epsilon_np2 > self.epsilon: update_order = np.random.permutation(self.num_data) for i in update_order: #Cavity distribution parameters self.tau_[i] = 1./Sigma[i,i] - self.eta*self.tau_tilde[i] self.v_[i] = mu[i]/Sigma[i,i] - self.eta*self.v_tilde[i] #Marginal moments self.Z_hat[i], mu_hat[i], sigma2_hat[i] = self.noise_model.moments_match(self.data[i],self.tau_[i],self.v_[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]) self.tau_tilde[i] += Delta_tau self.v_tilde[i] += Delta_v #Posterior distribution parameters update DSYR(Sigma,Sigma[:,i].copy(), -float(Delta_tau/(1.+ Delta_tau*Sigma[i,i]))) mu = np.dot(Sigma,self.v_tilde) self.iterations += 1 #Sigma recomptutation with Cholesky decompositon Sroot_tilde_K = np.sqrt(self.tau_tilde)[:,None]*K B = np.eye(self.num_data) + np.sqrt(self.tau_tilde)[None,:]*Sroot_tilde_K L = jitchol(B) V,info = dtrtrs(L,Sroot_tilde_K,lower=1) Sigma = K - np.dot(V.T,V) mu = np.dot(Sigma,self.v_tilde) epsilon_np1 = sum((self.tau_tilde-self.np1[-1])**2)/self.num_data epsilon_np2 = sum((self.v_tilde-self.np2[-1])**2)/self.num_data self.np1.append(self.tau_tilde.copy()) self.np2.append(self.v_tilde.copy()) return self._compute_GP_variables() def fit_DTC(self, Kmm, Kmn, epsilon=1e-3,power_ep=[1.,1.]): """ The expectation-propagation algorithm with sparse pseudo-input. For nomenclature see ... 2013. :param epsilon: Convergence criterion, maximum squared difference allowed between mean updates to stop iterations (float) :type epsilon: float :param power_ep: Power EP parameters :type power_ep: list of floats """ self.epsilon = epsilon self.eta, self.delta = power_ep num_inducing = Kmm.shape[0] #TODO: this doesn't work with uncertain inputs! """ Prior approximation parameters: q(f|X) = int_{df}{N(f|KfuKuu_invu,diag(Kff-Qff)*N(u|0,Kuu)} = N(f|0,Sigma0) Sigma0 = Qnn = Knm*Kmmi*Kmn """ KmnKnm = np.dot(Kmn,Kmn.T) Lm = jitchol(Kmm) Lmi = chol_inv(Lm) Kmmi = np.dot(Lmi.T,Lmi) KmmiKmn = np.dot(Kmmi,Kmn) Qnn_diag = np.sum(Kmn*KmmiKmn,-2) LLT0 = Kmm.copy() #Kmmi, Lm, Lmi, Kmm_logdet = pdinv(Kmm) #KmnKnm = np.dot(Kmn, Kmn.T) #KmmiKmn = np.dot(Kmmi,Kmn) #Qnn_diag = np.sum(Kmn*KmmiKmn,-2) #LLT0 = Kmm.copy() """ Posterior approximation: q(f|y) = N(f| mu, Sigma) Sigma = Diag + P*R.T*R*P.T + K mu = w + P*Gamma """ mu = np.zeros(self.num_data) LLT = Kmm.copy() Sigma_diag = Qnn_diag.copy() """ Initial values - Cavity distribution parameters: q_(g|mu_,sigma2_) = Product{q_i(g|mu_i,sigma2_i)} sigma_ = 1./tau_ mu_ = v_/tau_ """ self.tau_ = np.empty(self.num_data,dtype=float) self.v_ = np.empty(self.num_data,dtype=float) #Initial values - Marginal moments z = np.empty(self.num_data,dtype=float) self.Z_hat = np.empty(self.num_data,dtype=float) phi = np.empty(self.num_data,dtype=float) mu_hat = np.empty(self.num_data,dtype=float) sigma2_hat = np.empty(self.num_data,dtype=float) #Approximation epsilon_np1 = 1 epsilon_np2 = 1 self.iterations = 0 np1 = [self.tau_tilde.copy()] np2 = [self.v_tilde.copy()] while epsilon_np1 > self.epsilon or epsilon_np2 > self.epsilon: update_order = np.random.permutation(self.num_data) for i in update_order: #Cavity distribution parameters self.tau_[i] = 1./Sigma_diag[i] - self.eta*self.tau_tilde[i] self.v_[i] = mu[i]/Sigma_diag[i] - self.eta*self.v_tilde[i] #Marginal moments self.Z_hat[i], mu_hat[i], sigma2_hat[i] = self.noise_model.moments_match(self.data[i],self.tau_[i],self.v_[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]) self.tau_tilde[i] += Delta_tau self.v_tilde[i] += Delta_v #Posterior distribution parameters update DSYR(LLT,Kmn[:,i].copy(),Delta_tau) #LLT = LLT + np.outer(Kmn[:,i],Kmn[:,i])*Delta_tau L = jitchol(LLT) #cholUpdate(L,Kmn[:,i]*np.sqrt(Delta_tau)) V,info = dtrtrs(L,Kmn,lower=1) Sigma_diag = np.sum(V*V,-2) si = np.sum(V.T*V[:,i],-1) mu += (Delta_v-Delta_tau*mu[i])*si self.iterations += 1 #Sigma recomputation with Cholesky decompositon LLT = LLT0 + np.dot(Kmn*self.tau_tilde[None,:],Kmn.T) L = jitchol(LLT) V,info = dtrtrs(L,Kmn,lower=1) V2,info = dtrtrs(L.T,V,lower=0) Sigma_diag = np.sum(V*V,-2) Knmv_tilde = np.dot(Kmn,self.v_tilde) mu = np.dot(V2.T,Knmv_tilde) epsilon_np1 = sum((self.tau_tilde-np1[-1])**2)/self.num_data epsilon_np2 = sum((self.v_tilde-np2[-1])**2)/self.num_data np1.append(self.tau_tilde.copy()) np2.append(self.v_tilde.copy()) self._compute_GP_variables() def fit_FITC(self, Kmm, Kmn, Knn_diag, epsilon=1e-3,power_ep=[1.,1.]): """ The expectation-propagation algorithm with sparse pseudo-input. For nomenclature see Naish-Guzman and Holden, 2008. :param epsilon: Convergence criterion, maximum squared difference allowed between mean updates to stop iterations (float) :type epsilon: float :param power_ep: Power EP parameters :type power_ep: list of floats """ self.epsilon = epsilon self.eta, self.delta = power_ep num_inducing = Kmm.shape[0] """ Prior approximation parameters: q(f|X) = int_{df}{N(f|KfuKuu_invu,diag(Kff-Qff)*N(u|0,Kuu)} = N(f|0,Sigma0) Sigma0 = diag(Knn-Qnn) + Qnn, Qnn = Knm*Kmmi*Kmn """ Lm = jitchol(Kmm) Lmi = chol_inv(Lm) Kmmi = np.dot(Lmi.T,Lmi) P0 = Kmn.T KmnKnm = np.dot(P0.T, P0) KmmiKmn = np.dot(Kmmi,P0.T) Qnn_diag = np.sum(P0.T*KmmiKmn,-2) Diag0 = Knn_diag - Qnn_diag R0 = jitchol(Kmmi).T """ Posterior approximation: q(f|y) = N(f| mu, Sigma) Sigma = Diag + P*R.T*R*P.T + K mu = w + P*Gamma """ self.w = np.zeros(self.num_data) self.Gamma = np.zeros(num_inducing) mu = np.zeros(self.num_data) P = P0.copy() R = R0.copy() Diag = Diag0.copy() Sigma_diag = Knn_diag RPT0 = np.dot(R0,P0.T) """ Initial values - Cavity distribution parameters: q_(g|mu_,sigma2_) = Product{q_i(g|mu_i,sigma2_i)} sigma_ = 1./tau_ mu_ = v_/tau_ """ self.tau_ = np.empty(self.num_data,dtype=float) self.v_ = np.empty(self.num_data,dtype=float) #Initial values - Marginal moments z = np.empty(self.num_data,dtype=float) self.Z_hat = np.empty(self.num_data,dtype=float) phi = np.empty(self.num_data,dtype=float) mu_hat = np.empty(self.num_data,dtype=float) sigma2_hat = np.empty(self.num_data,dtype=float) #Approximation epsilon_np1 = 1 epsilon_np2 = 1 self.iterations = 0 self.np1 = [self.tau_tilde.copy()] self.np2 = [self.v_tilde.copy()] while epsilon_np1 > self.epsilon or epsilon_np2 > self.epsilon: update_order = np.random.permutation(self.num_data) for i in update_order: #Cavity distribution parameters self.tau_[i] = 1./Sigma_diag[i] - self.eta*self.tau_tilde[i] self.v_[i] = mu[i]/Sigma_diag[i] - self.eta*self.v_tilde[i] #Marginal moments self.Z_hat[i], mu_hat[i], sigma2_hat[i] = self.noise_model.moments_match(self.data[i],self.tau_[i],self.v_[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]) self.tau_tilde[i] += Delta_tau self.v_tilde[i] += Delta_v #Posterior distribution parameters update dtd1 = Delta_tau*Diag[i] + 1. dii = Diag[i] Diag[i] = dii - (Delta_tau * dii**2.)/dtd1 pi_ = P[i,:].reshape(1,num_inducing) P[i,:] = pi_ - (Delta_tau*dii)/dtd1 * pi_ Rp_i = np.dot(R,pi_.T) RTR = np.dot(R.T,np.dot(np.eye(num_inducing) - Delta_tau/(1.+Delta_tau*Sigma_diag[i]) * np.dot(Rp_i,Rp_i.T),R)) R = jitchol(RTR).T self.w[i] += (Delta_v - Delta_tau*self.w[i])*dii/dtd1 self.Gamma += (Delta_v - Delta_tau*mu[i])*np.dot(RTR,P[i,:].T) RPT = np.dot(R,P.T) Sigma_diag = Diag + np.sum(RPT.T*RPT.T,-1) mu = self.w + np.dot(P,self.Gamma) self.iterations += 1 #Sigma recomptutation with Cholesky decompositon Iplus_Dprod_i = 1./(1.+ Diag0 * self.tau_tilde) Diag = Diag0 * Iplus_Dprod_i P = Iplus_Dprod_i[:,None] * P0 safe_diag = np.where(Diag0 < self.tau_tilde, self.tau_tilde/(1.+Diag0*self.tau_tilde), (1. - Iplus_Dprod_i)/Diag0) L = jitchol(np.eye(num_inducing) + np.dot(RPT0,safe_diag[:,None]*RPT0.T)) R,info = dtrtrs(L,R0,lower=1) RPT = np.dot(R,P.T) Sigma_diag = Diag + np.sum(RPT.T*RPT.T,-1) self.w = Diag * self.v_tilde self.Gamma = np.dot(R.T, np.dot(RPT,self.v_tilde)) mu = self.w + np.dot(P,self.Gamma) epsilon_np1 = sum((self.tau_tilde-self.np1[-1])**2)/self.num_data epsilon_np2 = sum((self.v_tilde-self.np2[-1])**2)/self.num_data self.np1.append(self.tau_tilde.copy()) self.np2.append(self.v_tilde.copy()) return self._compute_GP_variables()