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Merge pull request #542 from SheffieldML/EP_refactor
Merge in the EP refactoring
This commit is contained in:
Zhenwen Dai
GitHub
commit
0fb273a41c
9 changed files with 721 additions and 202 deletions
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@ -6,9 +6,111 @@ from paramz import ObsAr
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from . import ExactGaussianInference, VarDTC
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from ...util import diag
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from .posterior import PosteriorEP as Posterior
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from ...likelihoods import Gaussian
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log_2_pi = np.log(2*np.pi)
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#Four wrapper classes to help modularisation of different EP versions
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class marginalMoments(object):
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def __init__(self, num_data):
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self.Z_hat = np.empty(num_data,dtype=np.float64)
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self.mu_hat = np.empty(num_data,dtype=np.float64)
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self.sigma2_hat = np.empty(num_data,dtype=np.float64)
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class cavityParams(object):
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def __init__(self, num_data):
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self.tau = np.empty(num_data,dtype=np.float64)
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self.v = np.empty(num_data,dtype=np.float64)
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def _update_i(self, eta, ga_approx, post_params, i):
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self.tau[i] = 1./post_params.Sigma_diag[i] - eta*ga_approx.tau[i]
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self.v[i] = post_params.mu[i]/post_params.Sigma_diag[i] - eta*ga_approx.v[i]
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class gaussianApproximation(object):
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def __init__(self, v, tau):
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self.tau = tau
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self.v = v
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def _update_i(self, eta, delta, post_params, marg_moments, i):
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#Site parameters update
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delta_tau = delta/eta*(1./marg_moments.sigma2_hat[i] - 1./post_params.Sigma_diag[i])
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delta_v = delta/eta*(marg_moments.mu_hat[i]/marg_moments.sigma2_hat[i] - post_params.mu[i]/post_params.Sigma_diag[i])
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tau_tilde_prev = self.tau[i]
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self.tau[i] += delta_tau
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# Enforce positivity of tau_tilde. Even though this is guaranteed for logconcave sites, it is still possible
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# to get negative values due to numerical errors. Moreover, the value of tau_tilde should be positive in order to
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# update the marginal likelihood without runnint into instabilities issues.
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if self.tau[i] < np.finfo(float).eps:
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self.tau[i] = np.finfo(float).eps
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delta_tau = self.tau[i] - tau_tilde_prev
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self.v[i] += delta_v
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return (delta_tau, delta_v)
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class posteriorParamsBase(object):
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def __init__(self, mu, Sigma_diag):
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self.mu = mu
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self.Sigma_diag = Sigma_diag
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def _update_rank1(self, *arg):
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pass
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def _recompute(self, *arg):
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pass
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class posteriorParams(posteriorParamsBase):
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def __init__(self, mu, Sigma, L=None):
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self.Sigma = Sigma
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self.L = L
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Sigma_diag = np.diag(self.Sigma)
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super(posteriorParams, self).__init__(mu, Sigma_diag)
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def _update_rank1(self, delta_tau, ga_approx, i):
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ci = delta_tau/(1.+ delta_tau*self.Sigma_diag[i])
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DSYR(self.Sigma, self.Sigma[:,i].copy(), -ci)
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self.mu = np.dot(self.Sigma, ga_approx.v)
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@staticmethod
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def _recompute(K, ga_approx):
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num_data = len(ga_approx.tau)
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tau_tilde_root = np.sqrt(ga_approx.tau)
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Sroot_tilde_K = tau_tilde_root[:,None] * K
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B = np.eye(num_data) + Sroot_tilde_K * tau_tilde_root[None,:]
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L = jitchol(B)
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V, _ = dtrtrs(L, Sroot_tilde_K, lower=1)
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Sigma = K - np.dot(V.T,V) #K - KS^(1/2)BS^(1/2)K = (K^(-1) + \Sigma^(-1))^(-1)
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mu = np.dot(Sigma,ga_approx.v)
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return posteriorParams(mu=mu, Sigma=Sigma, L=L)
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class posteriorParamsDTC(posteriorParamsBase):
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def __init__(self, mu, Sigma_diag):
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super(posteriorParamsDTC, self).__init__(mu, Sigma_diag)
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def _update_rank1(self, LLT, Kmn, delta_v, delta_tau, i):
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#DSYR(Sigma, Sigma[:,i].copy(), -delta_tau/(1.+ delta_tau*Sigma[i,i]))
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DSYR(LLT,Kmn[:,i].copy(),delta_tau)
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L = jitchol(LLT)
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V,info = dtrtrs(L,Kmn,lower=1)
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self.Sigma_diag = np.maximum(np.sum(V*V,-2), np.finfo(float).eps) #diag(K_nm (L L^\top)^(-1)) K_mn
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si = np.sum(V.T*V[:,i],-1) #(V V^\top)[:,i]
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self.mu += (delta_v-delta_tau*self.mu[i])*si
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#mu = np.dot(Sigma, v_tilde)
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@staticmethod
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def _recompute(LLT0, Kmn, ga_approx):
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LLT = LLT0 + np.dot(Kmn*ga_approx.tau[None,:],Kmn.T)
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L = jitchol(LLT)
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V, _ = dtrtrs(L,Kmn,lower=1)
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#Sigma_diag = np.sum(V*V,-2)
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#Knmv_tilde = np.dot(Kmn,v_tilde)
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#mu = np.dot(V2.T,Knmv_tilde)
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Sigma = np.dot(V.T,V)
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mu = np.dot(Sigma, ga_approx.v)
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Sigma_diag = np.diag(Sigma).copy()
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return posteriorParamsDTC(mu, Sigma_diag), LLT
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class EPBase(object):
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def __init__(self, epsilon=1e-6, eta=1., delta=1., always_reset=False, max_iters=np.inf, ep_mode="alternated", parallel_updates=False):
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"""
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@ -28,6 +130,7 @@ class EPBase(object):
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:parallel_updates: boolean. If true, updates of the parameters of the sites in parallel
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"""
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super(EPBase, self).__init__()
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self.always_reset = always_reset
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self.epsilon, self.eta, self.delta, self.max_iters = epsilon, eta, delta, max_iters
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self.ep_mode = ep_mode
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@ -35,7 +138,7 @@ class EPBase(object):
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self.reset()
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def reset(self):
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self.old_mutilde, self.old_vtilde = None, None
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self.ga_approx_old = None
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self._ep_approximation = None
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def on_optimization_start(self):
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@ -45,6 +148,11 @@ class EPBase(object):
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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 _stop_criteria(self, ga_approx):
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tau_diff = np.mean(np.square(ga_approx.tau-self.ga_approx_old.tau))
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v_diff = np.mean(np.square(ga_approx.v-self.ga_approx_old.v))
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return ((tau_diff < self.epsilon) and (v_diff < self.epsilon))
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def __setstate__(self, state):
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super(EPBase, self).__setstate__(state[0])
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self.epsilon, self.eta, self.delta = state[1]
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@ -67,19 +175,18 @@ class EP(EPBase, ExactGaussianInference):
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if self.ep_mode=="nested":
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#Force EP at each step of the optimization
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self._ep_approximation = None
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mu, Sigma, mu_tilde, tau_tilde, log_Z_tilde = self._ep_approximation = self.expectation_propagation(K, Y, likelihood, Y_metadata)
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post_params, ga_approx, cav_params, log_Z_tilde = self._ep_approximation = self.expectation_propagation(K, Y, likelihood, Y_metadata)
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elif self.ep_mode=="alternated":
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if getattr(self, '_ep_approximation', None) is None:
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#if we don't yet have the results of runnign EP, run EP and store the computed factors in self._ep_approximation
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mu, Sigma, mu_tilde, tau_tilde, log_Z_tilde = self._ep_approximation = self.expectation_propagation(K, Y, likelihood, Y_metadata)
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post_params, ga_approx, cav_params, log_Z_tilde = self._ep_approximation = self.expectation_propagation(K, Y, likelihood, Y_metadata)
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else:
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#if we've already run EP, just use the existing approximation stored in self._ep_approximation
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mu, Sigma, mu_tilde, tau_tilde, log_Z_tilde = self._ep_approximation
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post_params, ga_approx, cav_params, log_Z_tilde = self._ep_approximation
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else:
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raise ValueError("ep_mode value not valid")
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v_tilde = mu_tilde * tau_tilde
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return self._inference(K, tau_tilde, v_tilde, likelihood, Y_metadata=Y_metadata, Z_tilde=log_Z_tilde.sum())
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return self._inference(Y, K, ga_approx, cav_params, likelihood, Y_metadata=Y_metadata, Z_tilde=log_Z_tilde)
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def expectation_propagation(self, K, Y, likelihood, Y_metadata):
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@ -90,41 +197,57 @@ class EP(EPBase, ExactGaussianInference):
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# than ObsArrays
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Y = Y.values.copy()
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#Initial values - Marginal moments
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Z_hat = np.empty(num_data,dtype=np.float64)
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mu_hat = np.empty(num_data,dtype=np.float64)
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sigma2_hat = np.empty(num_data,dtype=np.float64)
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#Initial values - Marginal moments, cavity params, gaussian approximation params and posterior params
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marg_moments = marginalMoments(num_data)
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cav_params = cavityParams(num_data)
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ga_approx, post_params = self._init_approximations(K, num_data)
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tau_cav = np.empty(num_data,dtype=np.float64)
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v_cav = np.empty(num_data,dtype=np.float64)
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#Approximation
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stop = False
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iterations = 0
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while not stop and (iterations < self.max_iters):
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self._local_updates(num_data, cav_params, post_params, marg_moments, ga_approx, likelihood, Y, Y_metadata)
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#(re) compute Sigma and mu using full Cholesky decompy
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post_params = posteriorParams._recompute(K, ga_approx)
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#monitor convergence
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if iterations > 0:
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stop = self._stop_criteria(ga_approx)
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self.ga_approx_old = gaussianApproximation(ga_approx.v.copy(), ga_approx.tau.copy())
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iterations += 1
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# Z_tilde after removing the terms that can lead to infinite terms due to tau_tilde close to zero.
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# This terms cancel with the coreresponding terms in the marginal loglikelihood
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log_Z_tilde = self._log_Z_tilde(marg_moments, ga_approx, cav_params)
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# - 0.5*np.log(tau_tilde) + 0.5*(v_tilde*v_tilde*1./tau_tilde)
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return (post_params, ga_approx, cav_params, log_Z_tilde)
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def _init_approximations(self, K, num_data):
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#initial values - Gaussian factors
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#Initial values - Posterior distribution parameters: q(f|X,Y) = N(f|mu,Sigma)
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if self.old_mutilde is None:
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tau_tilde, mu_tilde, v_tilde = np.zeros((3, num_data))
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if self.ga_approx_old is None:
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v_tilde, tau_tilde = np.zeros((2, num_data))
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ga_approx = gaussianApproximation(v_tilde, tau_tilde)
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Sigma = K.copy()
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diag.add(Sigma, 1e-7)
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mu = np.zeros(num_data)
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post_params = posteriorParams(mu, Sigma)
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else:
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assert self.old_mutilde.size == num_data, "data size mis-match: did you change the data? try resetting!"
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mu_tilde, v_tilde = self.old_mutilde, self.old_vtilde
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tau_tilde = v_tilde/mu_tilde
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mu, Sigma, _ = self._ep_compute_posterior(K, tau_tilde, v_tilde)
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diag.add(Sigma, 1e-7)
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assert self.ga_approx_old.v.size == num_data, "data size mis-match: did you change the data? try resetting!"
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ga_approx = gaussianApproximation(self.ga_approx_old.v, self.ga_approx_old.tau)
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post_params = posteriorParams._recompute(K, ga_approx)
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diag.add(post_params.Sigma, 1e-7)
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# TODO: Check the log-marginal under both conditions and choose the best one
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return (ga_approx, post_params)
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#Approximation
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tau_diff = self.epsilon + 1.
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v_diff = self.epsilon + 1.
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tau_tilde_old = np.nan
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v_tilde_old = np.nan
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iterations = 0
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while ((tau_diff > self.epsilon) or (v_diff > self.epsilon)) and (iterations < self.max_iters):
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update_order = np.random.permutation(num_data)
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def _local_updates(self, num_data, cav_params, post_params, marg_moments, ga_approx, likelihood, Y, Y_metadata, update_order=None):
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if update_order is None:
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update_order = np.random.permutation(num_data)
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for i in update_order:
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#Cavity distribution parameters
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tau_cav[i] = 1./Sigma[i,i] - self.eta*tau_tilde[i]
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v_cav[i] = mu[i]/Sigma[i,i] - self.eta*v_tilde[i]
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cav_params._update_i(self.eta, ga_approx, post_params, i)
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if Y_metadata is not None:
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# Pick out the relavent metadata for Yi
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Y_metadata_i = {}
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@ -133,91 +256,45 @@ class EP(EPBase, ExactGaussianInference):
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else:
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Y_metadata_i = None
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#Marginal moments
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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)
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#Site parameters update
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delta_tau = self.delta/self.eta*(1./sigma2_hat[i] - 1./Sigma[i,i])
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delta_v = self.delta/self.eta*(mu_hat[i]/sigma2_hat[i] - mu[i]/Sigma[i,i])
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tau_tilde_prev = tau_tilde[i]
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tau_tilde[i] += delta_tau
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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)
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# Enforce positivity of tau_tilde. Even though this is guaranteed for logconcave sites, it is still possible
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# to get negative values due to numerical errors. Moreover, the value of tau_tilde should be positive in order to
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# update the marginal likelihood without inestability issues.
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if tau_tilde[i] < np.finfo(float).eps:
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tau_tilde[i] = np.finfo(float).eps
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delta_tau = tau_tilde[i] - tau_tilde_prev
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v_tilde[i] += delta_v
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#Site parameters update
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delta_tau, delta_v = ga_approx._update_i(self.eta, self.delta, post_params, marg_moments, i)
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if self.parallel_updates == False:
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#Posterior distribution parameters update
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ci = delta_tau/(1.+ delta_tau*Sigma[i,i])
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DSYR(Sigma, Sigma[:,i].copy(), -ci)
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mu = np.dot(Sigma, v_tilde)
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post_params._update_rank1(delta_tau, ga_approx, i)
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#(re) compute Sigma and mu using full Cholesky decompy
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mu, Sigma, _ = self._ep_compute_posterior(K, tau_tilde, v_tilde)
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def _log_Z_tilde(self, marg_moments, ga_approx, cav_params):
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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))
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+ 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)))))
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#monitor convergence
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if iterations > 0:
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tau_diff = np.mean(np.square(tau_tilde-tau_tilde_old))
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v_diff = np.mean(np.square(v_tilde-v_tilde_old))
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tau_tilde_old = tau_tilde.copy()
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v_tilde_old = v_tilde.copy()
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iterations += 1
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mu_tilde = v_tilde/tau_tilde
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mu_cav = v_cav/tau_cav
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sigma2_sigma2tilde = 1./tau_cav + 1./tau_tilde
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# Z_tilde after removing the terms that can lead to infinite terms due to tau_tilde close to zero.
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# This terms cancel with the coreresponding terms in the marginal loglikelihood
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log_Z_tilde = (np.log(Z_hat) + 0.5*np.log(2*np.pi) + 0.5*np.log(1+tau_tilde/tau_cav)
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- 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))))
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# - 0.5*np.log(tau_tilde) + 0.5*(v_tilde*v_tilde*1./tau_tilde)
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self.old_mutilde = mu_tilde
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self.old_vtilde = v_tilde
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return mu, Sigma, mu_tilde, tau_tilde, log_Z_tilde
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def _ep_compute_posterior(self, K, tau_tilde, v_tilde):
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num_data = len(tau_tilde)
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tau_tilde_root = np.sqrt(tau_tilde)
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Sroot_tilde_K = tau_tilde_root[:,None] * K
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B = np.eye(num_data) + Sroot_tilde_K * tau_tilde_root[None,:]
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L = jitchol(B)
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V, _ = dtrtrs(L, Sroot_tilde_K, lower=1)
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Sigma = K - np.dot(V.T,V) #K - KS^(1/2)BS^(1/2)K = (K^(-1) + \Sigma^(-1))^(-1)
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mu = np.dot(Sigma,v_tilde)
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return (mu, Sigma, L)
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def _ep_marginal(self, K, tau_tilde, v_tilde, Z_tilde):
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mu, Sigma, L = self._ep_compute_posterior(K, tau_tilde, v_tilde)
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def _ep_marginal(self, K, ga_approx, Z_tilde):
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post_params = posteriorParams._recompute(K, ga_approx)
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# Gaussian log marginal excluding terms that can go to infinity due to arbitrarily small tau_tilde.
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# These terms cancel out with the terms excluded from Z_tilde
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B_logdet = np.sum(2.0*np.log(np.diag(L)))
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log_marginal = 0.5*(-len(tau_tilde) * log_2_pi - B_logdet + np.sum(v_tilde * np.dot(Sigma,v_tilde)))
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B_logdet = np.sum(2.0*np.log(np.diag(post_params.L)))
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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)))
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log_marginal += Z_tilde
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return log_marginal, mu, Sigma, L
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return log_marginal, post_params
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def _inference(self, K, tau_tilde, v_tilde, likelihood, Z_tilde, Y_metadata=None):
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log_marginal, mu, Sigma, L = self._ep_marginal(K, tau_tilde, v_tilde, Z_tilde)
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def _inference(self, Y, K, ga_approx, cav_params, likelihood, Z_tilde, Y_metadata=None):
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log_marginal, post_params = self._ep_marginal(K, ga_approx, Z_tilde)
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|
||||
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)
|
||||
|
|
|
|||
|
|
@ -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):
|
||||
"""
|
||||
|
|
|
|||
|
|
@ -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):
|
||||
|
|
|
|||
|
|
@ -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
|
||||
|
|
|
|||
147
GPy/testing/ep_likelihood_tests.py
Normal file
147
GPy/testing/ep_likelihood_tests.py
Normal file
|
|
@ -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()
|
||||
|
|
@ -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']),
|
||||
|
|
|
|||
39
GPy/testing/quadrature_tests.py
Normal file
39
GPy/testing/quadrature_tests.py
Normal file
|
|
@ -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))
|
||||
|
|
@ -17,3 +17,4 @@ from . import multioutput
|
|||
from . import parallel
|
||||
from . import functions
|
||||
from . import cluster_with_offset
|
||||
from . import quad_integrate
|
||||
|
|
|
|||
119
GPy/util/quad_integrate.py
Normal file
119
GPy/util/quad_integrate.py
Normal file
|
|
@ -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
|
||||
|
||||
|
||||
Loading…
Add table
Add a link
Reference in a new issue