class CBayesPosterior:
# Class attributes:
# - p_obs: the observed density
# - p_prior: the prior
# - p_prior_pf: the push-forward of the prior
# - p_post: the posterior
# - p_post_pf: the push-forward of the posterior
# - r: the ratio between p_obs and p_prior_pf evaluations
# - acc_rate: the acceptance rate of the sampling algorithm
# Constructor
def __init__(self, p_obs, p_prior, p_prior_pf):
assert type(
p_obs
) is Distribution, "p_obs is not of type Distribution: %r" % p_obs
assert type(
p_prior
) is Distribution, "p_prior is not of type Distribution: %r" % p_prior
assert type(
p_prior_pf
) is Distribution, "p_prior_pf is not of type Distribution: %r" % p_prior_pf
if p_obs.n_dim > 3:
print('Framework has only been tested with up to 3 QoIs.')
exit()
self.p_obs = p_obs
self.p_prior = p_prior
self.p_prior_pf = p_prior_pf
self.p_post = None
self.p_post_pf = None
self.r = None
self.acc_rate = None
self.acc_idx = None
# Perform accept/reject sampling on a set of proposal samples using the weights r associated with the set of
# samples and return the indices idx of the proposal sample set that are accepted.
def generate_posterior_samples(self):
# Calculate the weights
r = np.divide(
self.p_obs.kernel_density(np.squeeze(self.p_prior_pf.samples).T) +
1e-10,
self.p_prior_pf.kernel_density(
np.squeeze(self.p_prior_pf.samples).T))
# Check against
check = np.random.uniform(low=0, high=1, size=r.size)
# Normalize weights
r_scaled = r / np.max(r)
# Evaluate criterion
idx = np.where(r_scaled >= check)[0]
self.r = r
self.acc_rate = idx.size / r.shape[0]
if self.acc_rate < 1.0e-2:
warnings.warn('Small acceptance rate: %f / %d accepted samples.' %
(self.acc_rate, idx.size))
return self.p_prior.samples[idx], self.p_prior_pf.samples[idx]
# Create the posterior and its push-forward
def setup_posterior_and_pf(self):
# Sample the posterior
post_samples, post_pf_samples = self.generate_posterior_samples()
# Create a posterior distribution
self.p_post = Distribution(samples=post_samples,
rv_name=self.p_prior.rv_name,
rv_transform=self.p_prior.rv_transform,
label='Updated',
kde=False)
# Create the posterior push-forward distribution
self.p_post_pf = Distribution(samples=post_pf_samples,
rv_name=self.p_obs.rv_name,
rv_transform=self.p_obs.rv_transform,
label='PF Updated')
# Get the KL between prior and posterior
def get_prior_post_kl(self):
return np.mean(self.r * np.log(self.r))
# Print a bunch of output diagnostics
def print_stats(self):
print('')
print('########### CBayes statistics ##########')
print('')
# The rejection sampling acceptance rate
print('Acceptance rate:\t\t\t\t%f' % self.acc_rate)
# The posterior push-forward mean and std
# (these should match the observed density)
print('Posterior push-forward mean:\t%s' % self.p_post_pf.mean())
print('Posterior push-forward std:\t\t%s' % self.p_post_pf.std())
# The KL between the push-forward of the posterior and the observed density
# (this should be very close to zero)
print('Posterior-PF-Obs KL:\t\t\t%f' %
self.p_post_pf.calculate_kl_divergence(self.p_obs))
# The posterior integral
# (this should be very close to 1.0)
print('Posterior integral:\t\t\t\t%f' % np.mean(self.r))
# The KL between posterior and prior (i.e. how informative is the data?)
# This is done via r / doing KDE for the prior and posterior densities is infeasible when the number of
# random variables is large.
print('Posterior-Prior KL:\t\t\t\t%f' %
np.mean(self.r * np.log(self.r)))
print('')
print('########################################')
print('')
# Plot results
def plot_results(self, model_tag='hf'):
# Determine bounds
xmin = np.min(
[np.min(self.p_prior_pf.samples),
np.min(self.p_obs.samples)])
xmax = np.max(
[np.max(self.p_prior_pf.samples),
np.max(self.p_obs.samples)])
# Plot
if self.p_obs.n_dim == 1:
self.p_prior_pf.plot_kde(color='C0', xmin=xmin, xmax=xmax)
self.p_obs.plot_kde(color='C1', xmin=xmin, xmax=xmax)
self.p_post_pf.plot_kde(color='C2',
linestyle='--',
xmin=xmin,
xmax=xmax)
elif self.p_obs.n_dim == 2:
sns.kdeplot(self.p_prior_pf.samples[:, 0],
self.p_prior_pf.samples[:, 1],
shade=True,
shade_lowest=False,
cmap='Blues',
label='PF-initial',
color='C0')
sns.kdeplot(self.p_obs.samples[:, 0],
self.p_obs.samples[:, 1],
shade=True,
shade_lowest=False,
cmap='Reds',
label='Observed density',
color='C3')
sns.kdeplot(self.p_post_pf.samples[:, 0],
self.p_post_pf.samples[:, 1],
cmap='Greys',
alpha=1.0,
label='PF-updated',
color='Black')
plt.legend(loc='upper right')
plt.xlabel('$Q_1$')
plt.ylabel('$Q_2$')
else:
return
plt.grid(b=True)
plt.gcf().savefig('output/cbayes_dists_%s.pdf' % model_tag, dpi=300)
# Plot some bivariate distributions
if self.p_obs.n_dim == 2 and model_tag == 'hf':
self.p_obs.plot_kde()
plt.grid(b=True)
plt.gcf().savefig('output/cbayes_dists_obs.pdf', dpi=300)
plt.clf()
self.p_post_pf.plot_kde()
plt.grid(b=True)
plt.gcf().savefig('output/cbayes_dists_hf_post_pf.pdf', dpi=300)
plt.clf()
# Plot posterior
def plot_posterior(self,
fignum=1,
color='C0',
linestyle='-',
label='Posterior',
save_fig=False):
if self.p_obs.n_dim == 1 and self.p_post.n_dim == 1:
self.p_post.create_kernel_density()
xmin = np.min(self.p_prior.samples, axis=0)
xmax = np.max(self.p_prior.samples, axis=0)
self.p_post.plot_kde(fignum=fignum,
color=color,
linestyle=linestyle,
label=label,
xmin=xmin,
xmax=xmax)
if save_fig:
plt.grid(b=True)
plt.gcf().savefig('output/cbayes_post_densities.pdf', dpi=300)
elif self.p_post.n_dim == 2:
self.p_post.create_kernel_density()
self.p_post.plot_kde(fignum=fignum,
color=color,
linestyle=linestyle,
label=label)
if save_fig:
xmin = np.min(self.p_prior.samples[:, 0], axis=0)
xmax = np.max(self.p_prior.samples[:, 0], axis=0)
ymin = np.min(self.p_prior.samples[:, 1], axis=0)
ymax = np.max(self.p_prior.samples[:, 1], axis=0)
plt.xlim([xmin, xmax])
plt.ylim([ymin, ymax])
plt.grid(b=True)
plt.gcf().savefig('output/cbayes_post_densities.pdf', dpi=300)
plt.clf()
