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various fixes in likelihoods, esp studentT and plotting
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James Hensman
parent
cc96f5b3d5
commit
b7508ce12b
5 changed files with 50 additions and 56 deletions
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@ -123,7 +123,7 @@ class GP(Model):
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mean, var = self.likelihood.predictive_values(mu, var, full_cov, Y_metadata)
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return mean, var
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def predict_quantiles(self, X, quantiles=(0.025, 0.975), Y_metadata=None):
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def predict_quantiles(self, X, quantiles=(2.5, 97.5), Y_metadata=None):
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m, v = self._raw_predict(X, full_cov=False)
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return self.likelihood.predictive_quantiles(m, v, quantiles, Y_metadata)
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@ -93,7 +93,6 @@ class Bernoulli(Likelihood):
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return 0.
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else:
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return np.nan
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#raise NotImplementedError
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def pdf_link(self, link_f, y, extra_data=None):
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"""
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@ -58,6 +58,18 @@ class Likelihood(Parameterized):
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"""
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return Y
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def conditional_mean(self, gp):
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"""
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The mean of the random variable conditioned on one value of the GP
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"""
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raise NotImplementedError
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def conditional_variance(self, gp):
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"""
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The variance of the random variable conditioned on one value of the GP
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"""
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raise NotImplementedError
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def log_predictive_density(self, y_test, mu_star, var_star):
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"""
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Calculation of the log predictive density
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@ -120,7 +132,7 @@ class Likelihood(Parameterized):
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return z, mean, variance
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def _predictive_mean(self,mu,variance):
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def predictive_mean(self, mu, variance, Y_metadata=None):
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"""
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Quadrature calculation of the predictive mean: E(Y_star|Y) = E( E(Y_star|f_star, Y) )
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@ -128,14 +140,15 @@ class Likelihood(Parameterized):
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:param sigma: standard deviation of posterior
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"""
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#conditional_mean: the edpected value of y given some f, under this likelihood
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def int_mean(f,m,v):
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return self._mean(f)*np.exp(-(0.5/v)*np.square(f - m))
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return self.conditional_mean(f)*np.exp(-(0.5/v)*np.square(f - m))
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scaled_mean = [quad(int_mean, -np.inf, np.inf,args=(mj,s2j))[0] for mj,s2j in zip(mu,variance)]
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mean = np.array(scaled_mean)[:,None] / np.sqrt(2*np.pi*(variance))
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return mean
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def _predictive_variance(self, mu,variance, predictive_mean=None):
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def predictive_variance(self, mu,variance, predictive_mean=None, Y_metadata=None):
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"""
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Numerical approximation to the predictive variance: V(Y_star)
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@ -152,7 +165,7 @@ class Likelihood(Parameterized):
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# E( V(Y_star|f_star) )
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def int_var(f,m,v):
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return self._variance(f)*np.exp(-(0.5/v)*np.square(f - m))
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return self.conditional_variance(f)*np.exp(-(0.5/v)*np.square(f - m))
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scaled_exp_variance = [quad(int_var, -np.inf, np.inf,args=(mj,s2j))[0] for mj,s2j in zip(mu,variance)]
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exp_var = np.array(scaled_exp_variance)[:,None] / normalizer
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@ -165,13 +178,14 @@ class Likelihood(Parameterized):
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#E( E(Y_star|f_star)**2 )
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def int_pred_mean_sq(f,m,v,predictive_mean_sq):
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return self._mean(f)**2*np.exp(-(0.5/v)*np.square(f - m))
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return self.conditional_mean(f)**2*np.exp(-(0.5/v)*np.square(f - m))
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scaled_exp_exp2 = [quad(int_pred_mean_sq, -np.inf, np.inf,args=(mj,s2j,pm2j))[0] for mj,s2j,pm2j in zip(mu,variance,predictive_mean_sq)]
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exp_exp2 = np.array(scaled_exp_exp2)[:,None] / normalizer
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var_exp = exp_exp2 - predictive_mean_sq
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# V(Y_star) = E( V(Y_star|f_star) ) + V( E(Y_star|f_star) )
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# V(Y_star) = E[ V(Y_star|f_star) ] + V[ E(Y_star|f_star) ]
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# V(Y_star) = E[ V(Y_star|f_star) ] + E(Y_star**2|f_star) - E[Y_star|f_star]**2
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return exp_var + var_exp
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def pdf_link(self, link_f, y, extra_data=None):
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@ -373,6 +387,15 @@ class Likelihood(Parameterized):
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return pred_mean, pred_var
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def predictive_quantiles(self, mu, var, quantiles, Y_metadata):
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#compute the quantiles by sampling!!!
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N_samp = 1000
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s = np.random.randn(mu.shape[0], N_samp)*np.sqrt(var) + mu
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ss_f = s.flatten()
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ss_y = self.samples(ss_f)
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ss_y = ss_y.reshape(mu.shape[0], N_samp)
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return [np.percentile(ss_y ,q, axis=1)[:,None] for q in quantiles]
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def samples(self, gp):
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"""
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@ -381,23 +404,3 @@ class Likelihood(Parameterized):
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:param gp: latent variable
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"""
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raise NotImplementedError
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if sampling:
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#Get gp_samples f* using posterior mean and variance
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if not full_cov:
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gp_samples = np.random.multivariate_normal(mu.flatten(), np.diag(var.flatten()),
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size=num_samples).T
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else:
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gp_samples = np.random.multivariate_normal(mu.flatten(), var,
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size=num_samples).T
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#Push gp samples (f*) through likelihood to give p(y*|f*)
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samples = self.samples(gp_samples)
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axis=-1
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#Calculate mean, variance and precentiles from samples
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print "WARNING: Using sampling to calculate mean, variance and predictive quantiles."
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pred_mean = np.mean(samples, axis=axis)[:,None]
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pred_var = np.var(samples, axis=axis)[:,None]
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q1 = np.percentile(samples, 2.5, axis=axis)[:,None]
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q3 = np.percentile(samples, 97.5, axis=axis)[:,None]
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@ -131,15 +131,15 @@ class Poisson(Likelihood):
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d3lik_dlink3 = 2*y/(link_f)**3
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return d3lik_dlink3
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def _mean(self,gp):
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def conditional_mean(self,gp):
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"""
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Mass (or density) function
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The mean of the random variable conditioned on one value of the GP
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"""
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return self.gp_link.transf(gp)
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def _variance(self,gp):
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def conditional_variance(self,gp):
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"""
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Mass (or density) function
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The variance of the random variable conditioned on one value of the GP
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"""
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return self.gp_link.transf(gp)
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@ -9,6 +9,7 @@ from scipy import stats, integrate
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from scipy.special import gammaln, gamma
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from likelihood import Likelihood
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from ..core.parameterization import Param
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from ..core.parameterization.transformations import Logexp
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class StudentT(Likelihood):
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"""
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@ -28,7 +29,7 @@ class StudentT(Likelihood):
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self.sigma2 = Param('t_noise', float(sigma2))
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self.v = Param('deg_free', float(deg_free))
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self.add_parameter(self.sigma2)
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self.add_parameter(self.sigma2, Logexp())
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self.add_parameter(self.v)
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self.v.constrain_fixed()
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@ -244,32 +245,23 @@ class StudentT(Likelihood):
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d2logpdf_dlink2_dv = np.zeros_like(d2logpdf_dlink2_dvar) #FIXME: Not done yet
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return np.hstack((d2logpdf_dlink2_dvar, d2logpdf_dlink2_dv))
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def predictive_variance(self, mu, sigma, predictive_mean=None):
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"""
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Compute predictive variance of student_t*normal p(y*|f*)p(f*)
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Need to find what the variance is at the latent points for a student t*normal p(y*|f*)p(f*)
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(((g((v+1)/2))/(g(v/2)*s*sqrt(v*pi)))*(1+(1/v)*((y-f)/s)^2)^(-(v+1)/2))
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*((1/(s*sqrt(2*pi)))*exp(-(1/(2*(s^2)))*((y-f)^2)))
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"""
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#FIXME: Not correct
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#We want the variance around test points y which comes from int p(y*|f*)p(f*) df*
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#Var(y*) = Var(E[y*|f*]) + E[Var(y*|f*)]
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#Since we are given f* (mu) which is our mean (expected) value of y*|f* then the variance is the variance around this
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#Which was also given to us as (var)
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#We also need to know the expected variance of y* around samples f*, this is the variance of the student t distribution
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#However the variance of the student t distribution is not dependent on f, only on sigma and the degrees of freedom
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true_var = 1/(1/sigma**2 + 1/self.variance)
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return true_var
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def predictive_mean(self, mu, sigma):
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def predictive_mean(self, mu, sigma, Y_metadata=None):
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"""
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Compute mean of the prediction
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"""
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#FIXME: Not correct
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return mu
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return self.gp_link.transf(mu) # only true in link is monotoci, which it is.
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def predictive_variance(self, mu,variance, predictive_mean=None, Y_metadata=None):
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if self.deg_free <2.:
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return np.empty(mu.shape)*np.nan #not defined for small degress fo freedom
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else:
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return super(StudentT, self).predictive_variance(mu, variance, predictive_mean, Y_metadata)
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def conditional_mean(self, gp):
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return self.gp_link.transf(gp)
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def conditional_variance(self, gp):
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return self.deg_free/(self.deg_free - 2.)
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def samples(self, gp):
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"""
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