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Fixing bernoulli likelihood for Laplace, fixing Zep for EP, and starting working on quadrature limits

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This commit is contained in:
Alan Saul 2015-10-19 19:29:57 +01:00
parent 6b6938bd11
commit 5b4abf4c34
8 changed files with 70 additions and 39 deletions
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2
GPy/examples/regression.py
View file

@ -275,7 +275,7 @@ def toy_rbf_1d_50(optimize=True, plot=True):
def toy_poisson_rbf_1d_laplace(optimize=True, plot=True):
"""Run a simple demonstration of a standard Gaussian process fitting it to data sampled from an RBF covariance."""
optimizer='scg'
x_len = 30
x_len = 100
X = np.linspace(0, 10, x_len)[:, None]
f_true = np.random.multivariate_normal(np.zeros(x_len), GPy.kern.RBF(1).K(X))
Y = np.array([np.random.poisson(np.exp(f)) for f in f_true])[:,None]

10
GPy/inference/latent_function_inference/exact_gaussian_inference.py
View file

@ -22,7 +22,7 @@ class ExactGaussianInference(LatentFunctionInference):
def __init__(self):
pass#self._YYTfactor_cache = caching.cache()
def inference(self, kern, X, likelihood, Y, mean_function=None, Y_metadata=None, K=None, precision=None):
def inference(self, kern, X, likelihood, Y, mean_function=None, Y_metadata=None, K=None, precision=None, Z=None):
"""
Returns a Posterior class containing essential quantities of the posterior
"""
@ -49,9 +49,15 @@ class ExactGaussianInference(LatentFunctionInference):
log_marginal = 0.5*(-Y.size * log_2_pi - Y.shape[1] * W_logdet - np.sum(alpha * YYT_factor))
if Z is not None:
# This is a correction term for the log marginal likelihood
# In EP this is log Z_tilde, which is the difference between the
# Gaussian marginal and Z_EP
log_marginal += Z
dL_dK = 0.5 * (tdot(alpha) - Y.shape[1] * Wi)
dL_dthetaL = likelihood.exact_inference_gradients(np.diag(dL_dK),Y_metadata)
dL_dthetaL = likelihood.exact_inference_gradients(np.diag(dL_dK), Y_metadata)
return Posterior(woodbury_chol=LW, woodbury_vector=alpha, K=K), log_marginal, {'dL_dK':dL_dK, 'dL_dthetaL':dL_dthetaL, 'dL_dm':alpha}

50
GPy/inference/latent_function_inference/expectation_propagation.py
View file

@ -39,26 +39,25 @@ class EPBase(object):
class EP(EPBase, ExactGaussianInference):
def inference(self, kern, X, likelihood, Y, mean_function=None, Y_metadata=None, precision=None, K=None):
num_data, output_dim = Y.shape
assert output_dim ==1, "ep in 1D only (for now!)"
assert output_dim == 1, "ep in 1D only (for now!)"
if K is None:
K = kern.K(X)
if self._ep_approximation 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, Z_hat = self._ep_approximation = self.expectation_propagation(K, Y, likelihood, Y_metadata)
mu, Sigma, mu_tilde, tau_tilde, 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, Z_hat = self._ep_approximation
mu, Sigma, mu_tilde, tau_tilde, Z_tilde = self._ep_approximation
return super(EP, self).inference(kern, X, likelihood, mu_tilde[:,None], mean_function=mean_function, Y_metadata=Y_metadata, precision=1./tau_tilde, K=K)
return super(EP, self).inference(kern, X, likelihood, mu_tilde[:,None], mean_function=mean_function, Y_metadata=Y_metadata, precision=1./tau_tilde, K=K, Z=np.log(Z_tilde).sum())
def expectation_propagation(self, K, Y, likelihood, Y_metadata):
num_data, data_dim = Y.shape
assert data_dim == 1, "This EP methods only works for 1D outputs"
#Initial values - Posterior distribution parameters: q(f|X,Y) = N(f|mu,Sigma)
mu = np.zeros(num_data)
Sigma = K.copy()
@ -69,6 +68,9 @@ class EP(EPBase, ExactGaussianInference):
mu_hat = np.empty(num_data,dtype=np.float64)
sigma2_hat = np.empty(num_data,dtype=np.float64)
tau_cav = np.empty(num_data,dtype=np.float64)
v_cav = np.empty(num_data,dtype=np.float64)
#initial values - Gaussian factors
if self.old_mutilde is None:
tau_tilde, mu_tilde, v_tilde = np.zeros((3, num_data))
@ -80,15 +82,17 @@ class EP(EPBase, ExactGaussianInference):
#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):
update_order = np.random.permutation(num_data)
for i in update_order:
#Cavity distribution parameters
tau_cav = 1./Sigma[i,i] - self.eta*tau_tilde[i]
v_cav = mu[i]/Sigma[i,i] - self.eta*v_tilde[i]
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]
#Marginal moments
Z_hat[i], mu_hat[i], sigma2_hat[i] = likelihood.moments_match_ep(Y[i], tau_cav, v_cav)#, Y_metadata=None)#=(None if Y_metadata is None else Y_metadata[i]))
Z_hat[i], mu_hat[i], sigma2_hat[i] = likelihood.moments_match_ep(Y[i], tau_cav[i], v_cav[i])#, Y_metadata=None)#=(None if Y_metadata is None else 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])
@ -108,7 +112,7 @@ class EP(EPBase, ExactGaussianInference):
mu = np.dot(Sigma,v_tilde)
#monitor convergence
if iterations>0:
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()
@ -117,7 +121,11 @@ class EP(EPBase, ExactGaussianInference):
iterations += 1
mu_tilde = v_tilde/tau_tilde
return mu, Sigma, mu_tilde, tau_tilde, Z_hat
mu_cav = v_cav/tau_cav
sigma2_sigma2tilde = 1./tau_cav + 1./tau_tilde
Z_tilde = np.exp(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))
return mu, Sigma, mu_tilde, tau_tilde, Z_tilde
class EPDTC(EPBase, VarDTC):
def inference(self, kern, X, Z, likelihood, Y, mean_function=None, Y_metadata=None, Lm=None, dL_dKmm=None, psi0=None, psi1=None, psi2=None):
@ -133,16 +141,16 @@ class EPDTC(EPBase, VarDTC):
Kmn = psi1.T
if self._ep_approximation is None:
mu, Sigma, mu_tilde, tau_tilde, Z_hat = self._ep_approximation = self.expectation_propagation(Kmm, Kmn, Y, likelihood, Y_metadata)
mu, Sigma, mu_tilde, tau_tilde, Z_tilde = self._ep_approximation = self.expectation_propagation(Kmm, Kmn, Y, likelihood, Y_metadata)
else:
mu, Sigma, mu_tilde, tau_tilde, Z_hat = self._ep_approximation
mu, Sigma, mu_tilde, tau_tilde, Z_tilde = self._ep_approximation
return super(EPDTC, self).inference(kern, X, Z, likelihood, mu_tilde,
mean_function=mean_function,
Y_metadata=Y_metadata,
precision=tau_tilde,
Lm=Lm, dL_dKmm=dL_dKmm,
psi0=psi0, psi1=psi1, psi2=psi2)
psi0=psi0, psi1=psi1, psi2=psi2, Z=Z_tilde)
def expectation_propagation(self, Kmm, Kmn, Y, likelihood, Y_metadata):
num_data, output_dim = Y.shape
@ -167,6 +175,9 @@ class EPDTC(EPBase, VarDTC):
mu_hat = np.zeros(num_data,dtype=np.float64)
sigma2_hat = np.zeros(num_data,dtype=np.float64)
tau_cav = np.empty(num_data,dtype=np.float64)
v_cav = np.empty(num_data,dtype=np.float64)
#initial values - Gaussian factors
if self.old_mutilde is None:
tau_tilde, mu_tilde, v_tilde = np.zeros((3, num_data))
@ -186,10 +197,10 @@ class EPDTC(EPBase, VarDTC):
while (tau_diff > self.epsilon) or (v_diff > self.epsilon):
for i in update_order:
#Cavity distribution parameters
tau_cav = 1./Sigma_diag[i] - self.eta*tau_tilde[i]
v_cav = mu[i]/Sigma_diag[i] - self.eta*v_tilde[i]
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]
#Marginal moments
Z_hat[i], mu_hat[i], sigma2_hat[i] = likelihood.moments_match_ep(Y[i], tau_cav, v_cav)#, Y_metadata=None)#=(None if Y_metadata is None else Y_metadata[i]))
Z_hat[i], mu_hat[i], sigma2_hat[i] = likelihood.moments_match_ep(Y[i], tau_cav[i], v_cav[i])#, Y_metadata=None)#=(None if Y_metadata is None else 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])
@ -233,5 +244,8 @@ class EPDTC(EPBase, VarDTC):
iterations += 1
mu_tilde = v_tilde/tau_tilde
return mu, Sigma, ObsAr(mu_tilde[:,None]), tau_tilde, Z_hat
mu_cav = v_cav/tau_cav
sigma2_sigma2tilde = 1./tau_cav + 1./tau_tilde
Z_tilde = np.exp(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))
return mu, Sigma, ObsAr(mu_tilde[:,None]), tau_tilde, Z_tilde

10
GPy/likelihoods/bernoulli.py
View file

@ -140,7 +140,7 @@ class Bernoulli(Likelihood):
Each y_i must be in {0, 1}
"""
#objective = (inv_link_f**y) * ((1.-inv_link_f)**(1.-y))
return np.where(y, inv_link_f, 1.-inv_link_f)
return np.where(y==1, inv_link_f, 1.-inv_link_f)
def logpdf_link(self, inv_link_f, y, Y_metadata=None):
"""
@ -179,7 +179,7 @@ class Bernoulli(Likelihood):
#grad = (y/inv_link_f) - (1.-y)/(1-inv_link_f)
#grad = np.where(y, 1./inv_link_f, -1./(1-inv_link_f))
ff = np.clip(inv_link_f, 1e-9, 1-1e-9)
denom = np.where(y, ff, -(1-ff))
denom = np.where(y==1, ff, -(1-ff))
return 1./denom
def d2logpdf_dlink2(self, inv_link_f, y, Y_metadata=None):
@ -205,7 +205,7 @@ class Bernoulli(Likelihood):
"""
#d2logpdf_dlink2 = -y/(inv_link_f**2) - (1-y)/((1-inv_link_f)**2)
#d2logpdf_dlink2 = np.where(y, -1./np.square(inv_link_f), -1./np.square(1.-inv_link_f))
arg = np.where(y, inv_link_f, 1.-inv_link_f)
arg = np.where(y==1, inv_link_f, 1.-inv_link_f)
ret = -1./np.square(np.clip(arg, 1e-9, 1e9))
if np.any(np.isinf(ret)):
stop
@ -230,7 +230,7 @@ class Bernoulli(Likelihood):
#d3logpdf_dlink3 = 2*(y/(inv_link_f**3) - (1-y)/((1-inv_link_f)**3))
state = np.seterr(divide='ignore')
# TODO check y \in {0, 1} or {-1, 1}
d3logpdf_dlink3 = np.where(y, 2./(inv_link_f**3), -2./((1.-inv_link_f)**3))
d3logpdf_dlink3 = np.where(y==1, 2./(inv_link_f**3), -2./((1.-inv_link_f)**3))
np.seterr(**state)
return d3logpdf_dlink3
@ -243,8 +243,6 @@ class Bernoulli(Likelihood):
p = self.predictive_mean(mu, var)
return [np.asarray(p>(q/100.), dtype=np.int32) for q in quantiles]
def samples(self, gp, Y_metadata=None):
"""
Returns a set of samples of observations based on a given value of the latent variable.

2
GPy/likelihoods/gaussian.py
View file

@ -67,7 +67,7 @@ class Gaussian(Likelihood):
"""
return Y
def _moments_match_ep(self, data_i, tau_i, v_i):
def moments_match_ep(self, data_i, tau_i, v_i):
"""
Moments match of the marginal approximation in EP algorithm

21
GPy/likelihoods/likelihood.py
View file

@ -49,8 +49,8 @@ class Likelihood(Parameterized):
"""
return Y.shape[1]
def _gradients(self,partial):
return np.zeros(0)
def exact_inference_gradients(self, dL_dKdiag,Y_metadata=None):
return np.zeros(self.size)
def update_gradients(self, partial):
if self.size > 0:
@ -176,8 +176,10 @@ class Likelihood(Parameterized):
log_p_ystar = np.array(log_p_ystar).reshape(*y_test.shape)
return log_p_ystar
def quad_limits(self):
return -np.inf, np.inf
def _moments_match_ep(self,obs,tau,v):
def moments_match_ep(self,obs,tau,v):
"""
Calculation of moments using quadrature
@ -188,20 +190,27 @@ class Likelihood(Parameterized):
#Compute first integral for zeroth moment.
#NOTE constant np.sqrt(2*pi/tau) added at the end of the function
mu = v/tau
sigma2 = 1./tau
#Lets do these for now based on the same idea as Gaussian quadrature
# i.e. multiply anything by close to zero, and its zero.
f_min = mu - 8*np.sqrt(sigma2)
f_max = mu + 8*np.sqrt(sigma2)
# f_min, f_max = self.quad_limits()
def int_1(f):
return self.pdf(f, obs)*np.exp(-0.5*tau*np.square(mu-f))
z_scaled, accuracy = quad(int_1, -np.inf, np.inf)
z_scaled, accuracy = quad(int_1, f_min, f_max)
#Compute second integral for first moment
def int_2(f):
return f*self.pdf(f, obs)*np.exp(-0.5*tau*np.square(mu-f))
mean, accuracy = quad(int_2, -np.inf, np.inf)
mean, accuracy = quad(int_2, f_min, f_max)
mean /= z_scaled
#Compute integral for variance
def int_3(f):
return (f**2)*self.pdf(f, obs)*np.exp(-0.5*tau*np.square(mu-f))
Ef2, accuracy = quad(int_3, -np.inf, np.inf)
Ef2, accuracy = quad(int_3, f_min, f_max)
Ef2 /= z_scaled
variance = Ef2 - mean**2

5
GPy/likelihoods/poisson.py
View file

@ -28,7 +28,7 @@ class Poisson(Likelihood):
"""
the expected value of y given a value of f
"""
return self.gp_link.transf(gp)
return self.gp_link.transf(f)
def pdf_link(self, link_f, y, Y_metadata=None):
"""
@ -46,7 +46,8 @@ class Poisson(Likelihood):
:rtype: float
"""
assert np.atleast_1d(link_f).shape == np.atleast_1d(y).shape
return np.prod(stats.poisson.pmf(y,link_f))
return np.exp(self.logpdf_link(link_f, y, Y_metadata))
# return np.prod(stats.poisson.pmf(y,link_f))
def logpdf_link(self, link_f, y, Y_metadata=None):
"""

9
GPy/testing/likelihood_tests.py
View file

@ -113,6 +113,7 @@ class TestNoiseModels(object):
self.Y = (np.sin(self.X[:, 0]*2*np.pi) + noise)[:, None]
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())
tmp = np.round(self.X[:, 0]*3-3)[:, None] + np.random.randint(0,3, self.X.shape[0])[:, None]
self.integer_Y = np.where(tmp > 0, tmp, 0)
@ -561,12 +562,14 @@ class TestNoiseModels(object):
print("\n{}".format(inspect.stack()[0][3]))
np.random.seed(111)
#Normalize
Y = Y/Y.max()
# Y = Y/Y.max()
white_var = 1e-5
kernel = GPy.kern.RBF(X.shape[1]) + GPy.kern.White(X.shape[1])
laplace_likelihood = GPy.inference.latent_function_inference.Laplace()
m = GPy.core.GP(X.copy(), Y.copy(), kernel, likelihood=model, Y_metadata=Y_metadata, inference_method=laplace_likelihood)
m['.*white'].constrain_fixed(white_var)
m.randomize()
#Set constraints
@ -591,7 +594,7 @@ class TestNoiseModels(object):
print("\n{}".format(inspect.stack()[0][3]))
#Normalize
Y = Y/Y.max()
white_var = 1e-6
white_var = 1e-5
kernel = GPy.kern.RBF(X.shape[1]) + GPy.kern.White(X.shape[1])
ep_inf = GPy.inference.latent_function_inference.EP()