def test_get_statistics(self):
univariate_variables = [
stats.uniform(2, 4), stats.beta(1, 1, -1, 2), stats.norm(0, 1)]
variable = IndependentMultivariateRandomVariable(univariate_variables)
mean = variable.get_statistics('mean')
assert np.allclose(mean.squeeze(), [4, 0, 0])
intervals = variable.get_statistics('interval', alpha=1)
assert np.allclose(intervals, np.array(
[[2, 6], [-1, 1], [-np.inf, np.inf]]))
def test_bayesian_importance_sampling_avar(self):
np.random.seed(1)
nrandom_vars = 2
Amat = np.array([[-0.5, 1]])
noise_std = 0.1
prior_variable = IndependentMultivariateRandomVariable(
[stats.norm(0, 1)] * nrandom_vars)
prior_mean = prior_variable.get_statistics('mean')
prior_cov = np.diag(prior_variable.get_statistics('var')[:, 0])
prior_cov_inv = np.linalg.inv(prior_cov)
noise_cov_inv = np.eye(Amat.shape[0]) / noise_std**2
true_sample = np.array([.4] * nrandom_vars)[:, None]
collected_obs = Amat.dot(true_sample)
collected_obs += np.random.normal(0, noise_std, (collected_obs.shape))
exact_post_mean, exact_post_cov = \
laplace_posterior_approximation_for_linear_models(
Amat, prior_mean, prior_cov_inv, noise_cov_inv,
collected_obs)
chol_factor = np.linalg.cholesky(exact_post_cov)
chol_factor_inv = np.linalg.inv(chol_factor)
def g_model(samples):
return np.exp(
np.sum(chol_factor_inv.dot(samples - exact_post_mean),
axis=0))[:, None]
nsamples = int(1e6)
prior_samples = generate_independent_random_samples(
prior_variable, nsamples)
posterior_samples = chol_factor.dot(
np.random.normal(0, 1, (nrandom_vars, nsamples))) + exact_post_mean
g_mu, g_sigma = 0, np.sqrt(nrandom_vars)
f, f_cdf, f_pdf, VaR, CVaR, ssd, ssd_disutil = \
get_lognormal_example_exact_quantities(g_mu, g_sigma)
beta = .1
cvar_exact = CVaR(beta)
cvar_mc = conditional_value_at_risk(g_model(posterior_samples), beta)
prior_pdf = prior_variable.pdf
post_pdf = stats.multivariate_normal(mean=exact_post_mean[:, 0],
cov=exact_post_cov).pdf
weights = post_pdf(prior_samples.T) / prior_pdf(prior_samples)[:, 0]
weights /= weights.sum()
cvar_im = conditional_value_at_risk(g_model(prior_samples), beta,
weights)
# print(cvar_exact, cvar_mc, cvar_im)
assert np.allclose(cvar_exact, cvar_mc, rtol=1e-3)
assert np.allclose(cvar_exact, cvar_im, rtol=2e-3)
def help_compare_prediction_based_oed(self, deviation_fun,
gauss_deviation_fun,
use_gauss_quadrature,
ninner_loop_samples, ndesign_vars,
tol):
ncandidates_1d = 5
design_candidates = cartesian_product(
[np.linspace(-1, 1, ncandidates_1d)] * ndesign_vars)
ncandidates = design_candidates.shape[1]
# Define model used to predict likely observable data
indices = compute_hyperbolic_indices(ndesign_vars, 1)[:, 1:]
Amat = monomial_basis_matrix(indices, design_candidates)
obs_fun = partial(linear_obs_fun, Amat)
# Define model used to predict unobservable QoI
qoi_fun = exponential_qoi_fun
# Define the prior PDF of the unknown variables
nrandom_vars = indices.shape[1]
prior_variable = IndependentMultivariateRandomVariable(
[stats.norm(0, 0.5)] * nrandom_vars)
# Define the independent observational noise
noise_std = 1
# Define initial design
init_design_indices = np.array([ncandidates // 2])
# Define OED options
nouter_loop_samples = 100
if use_gauss_quadrature:
# 301 needed for cvar deviation
# only 31 needed for variance deviation
ninner_loop_samples_1d = ninner_loop_samples
var_trans = AffineRandomVariableTransformation(prior_variable)
x_quad, w_quad = gauss_hermite_pts_wts_1D(ninner_loop_samples_1d)
x_quad = cartesian_product([x_quad] * nrandom_vars)
w_quad = outer_product([w_quad] * nrandom_vars)
x_quad = var_trans.map_from_canonical_space(x_quad)
ninner_loop_samples = x_quad.shape[1]
def generate_inner_prior_samples(nsamples):
assert nsamples == x_quad.shape[1], (nsamples, x_quad.shape)
return x_quad, w_quad
else:
# use default Monte Carlo sampling
generate_inner_prior_samples = None
# Define initial design
init_design_indices = np.array([ncandidates // 2])
# Setup OED problem
oed = BayesianBatchDeviationOED(design_candidates,
obs_fun,
noise_std,
prior_variable,
qoi_fun,
nouter_loop_samples,
ninner_loop_samples,
generate_inner_prior_samples,
deviation_fun=deviation_fun)
oed.populate()
oed.set_collected_design_indices(init_design_indices)
prior_mean = oed.prior_variable.get_statistics('mean')
prior_cov = np.diag(prior_variable.get_statistics('var')[:, 0])
prior_cov_inv = np.linalg.inv(prior_cov)
selected_indices = init_design_indices
# Generate experimental design
nexperiments = 3
for step in range(len(init_design_indices), nexperiments):
# Copy current state of OED before new data is determined
# This copy will be used to compute Laplace based utility and
# evidence values for testing
oed_copy = copy.deepcopy(oed)
# Update the design
utility_vals, selected_indices = oed.update_design()
utility, deviations, evidences, weights = \
oed_copy.compute_expected_utility(
oed_copy.collected_design_indices, selected_indices, True)
exact_deviations = np.empty(nouter_loop_samples)
for jj in range(nouter_loop_samples):
# only test intermediate quantities associated with design
# chosen by the OED step
idx = oed.collected_design_indices
obs_jj = oed_copy.outer_loop_obs[jj:jj + 1, idx]
noise_cov_inv_jj = np.eye(idx.shape[0]) / noise_std**2
exact_post_mean_jj, exact_post_cov_jj = \
laplace_posterior_approximation_for_linear_models(
Amat[idx, :],
prior_mean, prior_cov_inv, noise_cov_inv_jj, obs_jj.T)
exact_deviations[jj] = gauss_deviation_fun(
exact_post_mean_jj, exact_post_cov_jj)
print('d',
np.absolute(exact_deviations - deviations[:, 0]).max(), tol)
# print(exact_deviations, deviations[:, 0])
assert np.allclose(exact_deviations, deviations[:, 0], atol=tol)
assert np.allclose(utility_vals[selected_indices],
-np.mean(exact_deviations),
atol=tol)
def test_sequential_kl_oed(self):
"""
Observations collected ARE used to inform subsequent designs
"""
nrandom_vars = 1
noise_std = 1
ndesign = 5
nouter_loop_samples = int(1e4)
ninner_loop_samples = 31
ncandidates = 6
design_candidates = np.linspace(-1, 1, ncandidates)[None, :]
def obs_fun(samples):
assert design_candidates.ndim == 2
assert samples.ndim == 2
Amat = design_candidates.T
return Amat.dot(samples).T
prior_variable = IndependentMultivariateRandomVariable(
[stats.norm(0, 1)] * nrandom_vars)
true_sample = np.array([.4] * nrandom_vars)[:, None]
def obs_process(new_design_indices):
obs = obs_fun(true_sample)[:, new_design_indices]
obs += oed.noise_fun(obs)
return obs
x_quad, w_quad = gauss_hermite_pts_wts_1D(ninner_loop_samples)
def generate_inner_prior_samples_gauss(n):
# use precomputed samples so to avoid cost of regenerating
assert n == x_quad.shape[0]
return x_quad[None, :], w_quad
generate_inner_prior_samples = generate_inner_prior_samples_gauss
# Define initial design
init_design_indices = np.array([ncandidates // 2])
oed = BayesianSequentialKLOED(design_candidates, obs_fun, noise_std,
prior_variable, obs_process,
nouter_loop_samples, ninner_loop_samples,
generate_inner_prior_samples)
oed.populate()
oed.set_collected_design_indices(init_design_indices)
prior_mean = oed.prior_variable.get_statistics('mean')
prior_cov = np.diag(prior_variable.get_statistics('var')[:, 0])
prior_cov_inv = np.linalg.inv(prior_cov)
exact_post_mean_prev = prior_mean
exact_post_cov_prev = prior_cov
post_var_prev = stats.multivariate_normal(mean=exact_post_mean_prev[:,
0],
cov=exact_post_cov_prev)
selected_indices = init_design_indices
# Because of Monte Carlo error set step tols individually
# It is too expensive to up the number of outer_loop samples to
# reduce errors
step_tols = [7.3e-3, 6.5e-2, 3.3e-2, 1.6e-1]
for step in range(len(init_design_indices), ndesign):
current_design = design_candidates[:, oed.collected_design_indices]
noise_cov_inv = np.eye(current_design.shape[1]) / noise_std**2
# Compute posterior moving from previous posterior and using
# only the most recently collected data
noise_cov_inv_incr = np.eye(
selected_indices.shape[0]) / noise_std**2
exact_post_mean, exact_post_cov = \
laplace_posterior_approximation_for_linear_models(
design_candidates[:, selected_indices].T,
exact_post_mean_prev, np.linalg.inv(exact_post_cov_prev),
noise_cov_inv_incr, oed.collected_obs[:, -1:].T)
# check using current posteior as prior and only using new
# data (above) produces the same posterior as using original prior
# and all collected data (from_prior approach). The posteriors
# should be the same but the evidences will be difference.
# This is tested below
exact_post_mean_from_prior, exact_post_cov_from_prior = \
laplace_posterior_approximation_for_linear_models(
current_design.T, prior_mean, prior_cov_inv, noise_cov_inv,
oed.collected_obs.T)
assert np.allclose(exact_post_mean, exact_post_mean_from_prior)
assert np.allclose(exact_post_cov, exact_post_cov_from_prior)
# Compute PDF of current posterior that uses all collected data
post_var = stats.multivariate_normal(
mean=exact_post_mean[:, 0].copy(), cov=exact_post_cov.copy())
# Compute evidence moving from previous posterior to
# new posterior (not initial prior to posterior).
# Values can be computed exactly for Gaussian prior and noise
gauss_evidence = laplace_evidence(
lambda x: np.exp(
gaussian_loglike_fun(
oed.collected_obs[:, -1:],
obs_fun(x)[:, oed.collected_design_indices[-1:]],
noise_std))[:, 0],
lambda y: np.atleast_2d(post_var_prev.pdf(y.T)).T,
exact_post_cov, exact_post_mean)
# Compute evidence using Gaussian quadrature rule. This
# is possible for this low-dimensional example.
quad_loglike_vals = np.exp(
gaussian_loglike_fun(
oed.collected_obs[:, -1:],
obs_fun(
x_quad[None, :])[:, oed.collected_design_indices[-1:]],
noise_std))[:, 0]
# we must divide integarnd by initial prior_pdf since it is
# already implicilty included via the quadrature weights
integrand_vals = quad_loglike_vals * post_var_prev.pdf(
x_quad[:, None]) / prior_variable.pdf(x_quad[None, :])[:, 0]
quad_evidence = integrand_vals.dot(w_quad)
# print(quad_evidence, gauss_evidence)
assert np.allclose(gauss_evidence, quad_evidence), step
# print('G', gauss_evidence, oed.evidence)
assert np.allclose(gauss_evidence, oed.evidence), step
# compute the evidence of moving from the initial prior
# to the current posterior. This will be used for testing later
gauss_evidence_from_prior = laplace_evidence(
lambda x: np.exp(
gaussian_loglike_fun(
oed.collected_obs,
obs_fun(x)[:, oed.collected_design_indices], noise_std)
)[:, 0], prior_variable.pdf, exact_post_cov, exact_post_mean)
# Copy current state of OED before new data is determined
# This copy will be used to compute Laplace based utility and
# evidence values for testing
oed_copy = copy.deepcopy(oed)
# Update the design
utility_vals, selected_indices = oed.update_design()
new_obs = oed.obs_process(selected_indices)
oed.update_observations(new_obs)
utility = utility_vals[selected_indices]
# Re-compute the evidences that were used to update the design
# above. This will be used for testing later
# print('D', oed_copy.evidence)
evidences = oed_copy.compute_expected_utility(
oed_copy.collected_design_indices, selected_indices, True)[1]
# print('Collected plus selected indices',
# oed.collected_design_indices,
# oed_copy.collected_design_indices, selected_indices)
# For all outer loop samples compute the posterior exactly
# and compute intermediate values for testing. While OED
# considers all possible candidate design indices
# Here we just test the one that was chosen last when
# design was updated
exact_evidences = np.empty(nouter_loop_samples)
exact_kl_divs = np.empty_like(exact_evidences)
for jj in range(nouter_loop_samples):
# Fill obs with those predicted by outer loop sample
idx = oed.collected_design_indices
obs_jj = oed_copy.outer_loop_obs[jj:jj + 1, idx]
# Overwrite the previouly simulated obs with collected obs.
# Do not ovewrite the last value which is the potential
# data used to compute expected utility
obs_jj[:, :oed_copy.collected_obs.shape[1]] = \
oed_copy.collected_obs
# Compute the posterior obtained by using predicted value
# of outer loop sample
noise_cov_inv_jj = np.eye(
selected_indices.shape[0]) / noise_std**2
exact_post_mean_jj, exact_post_cov_jj = \
laplace_posterior_approximation_for_linear_models(
design_candidates[:, selected_indices].T,
exact_post_mean, np.linalg.inv(exact_post_cov),
noise_cov_inv_jj, obs_jj[:, -1].T)
# Use post_pdf so measure change from current posterior (prior)
# to new posterior
gauss_evidence_jj = laplace_evidence(
lambda x: np.exp(
gaussian_loglike_fun(obs_jj[:, -1:],
obs_fun(x)[:, selected_indices],
noise_std))[:, 0],
lambda y: np.atleast_2d(post_var.pdf(y.T)).T,
exact_post_cov_jj, exact_post_mean_jj)
exact_evidences[jj] = gauss_evidence_jj
# Check quadrature gets the same answer
quad_loglike_vals = np.exp(
gaussian_loglike_fun(
obs_jj[:, -1:],
obs_fun(x_quad[None, :])[:, selected_indices],
noise_std))[:, 0]
integrand_vals = quad_loglike_vals * post_var.pdf(
x_quad[:, None]) / prior_variable.pdf(x_quad[None, :])[:,
0]
quad_evidence = integrand_vals.dot(w_quad)
# print(quad_evidence, gauss_evidence_jj)
assert np.allclose(gauss_evidence_jj, quad_evidence), step
# Check that evidence of moving from current posterior
# to new posterior with (potential data from outer-loop sample)
# is equal to the evidence of moving from
# intitial prior to new posterior divide by the evidence
# from moving from the initial prior to the current posterior
gauss_evidence_jj_from_prior = laplace_evidence(
lambda x: np.exp(
gaussian_loglike_fun(obs_jj,
obs_fun(x)[:, idx], noise_std)
)[:, 0], prior_variable.pdf, exact_post_cov_jj,
exact_post_mean_jj)
# print(gauss_evidence_jj_from_prior/gauss_evidence_from_prior,
# gauss_evidence_jj)
# print('gauss_evidence_from_prior', gauss_evidence_from_prior)
assert np.allclose(
gauss_evidence_jj_from_prior / gauss_evidence_from_prior,
gauss_evidence_jj)
gauss_kl_div = gaussian_kl_divergence(exact_post_mean_jj,
exact_post_cov_jj,
exact_post_mean,
exact_post_cov)
# gauss_kl_div = gaussian_kl_divergence(
# exact_post_mean, exact_post_cov,
# exact_post_mean_jj, exact_post_cov_jj)
exact_kl_divs[jj] = gauss_kl_div
# print(evidences[:, 0], exact_evidences)
assert np.allclose(evidences[:, 0], exact_evidences)
# Outer loop samples are from prior. Use importance reweighting
# to sample from previous posterior. This step is only relevant
# for open loop design (used here)
# where observed data informs current estimate
# of parameters. Closed loop design (not used here)
# never collects data and so it always samples from the prior.
post_weights = post_var.pdf(
oed.outer_loop_prior_samples.T) / post_var_prev.pdf(
oed.outer_loop_prior_samples.T) / oed.nouter_loop_samples
laplace_utility = np.sum(exact_kl_divs * post_weights)
# print('u', (utility-laplace_utility)/laplace_utility, step)
assert np.allclose(utility,
laplace_utility,
rtol=step_tols[step - 1])
exact_post_mean_prev = exact_post_mean
exact_post_cov_prev = exact_post_cov
post_var_prev = post_var
def test_compute_expected_kl_utility_monte_carlo(self):
nrandom_vars = 1
noise_std = .3
design = np.linspace(-1, 1, 2)[None, :]
Amat = design.T
def obs_fun(x):
return (Amat.dot(x)).T
def noise_fun(values):
return np.random.normal(0, noise_std, (values.shape))
# specify the first design point
collected_design_indices = np.array([0])
prior_variable = IndependentMultivariateRandomVariable(
[stats.norm(0, 1)] * nrandom_vars)
prior_mean = prior_variable.get_statistics('mean')
prior_cov = np.diag(prior_variable.get_statistics('var')[:, 0])
prior_cov_inv = np.linalg.inv(prior_cov)
noise_cov_inv = np.eye(Amat.shape[0]) / noise_std**2
def generate_random_prior_samples(n):
return (generate_independent_random_samples(prior_variable,
n), np.ones(n) / n)
def generate_inner_prior_samples_mc(n):
return generate_random_prior_samples(n), np.ones(n) / n
ninner_loop_samples = 300
x, w = gauss_hermite_pts_wts_1D(ninner_loop_samples)
def generate_inner_prior_samples_gauss(n):
# use precomputed samples so to avoid cost of regenerating
assert n == x.shape[0]
return x[None, :], w
generate_inner_prior_samples = generate_inner_prior_samples_gauss
nouter_loop_samples = 10000
outer_loop_obs, outer_loop_pred_obs, inner_loop_pred_obs, \
inner_loop_weights, __, __ = \
precompute_compute_expected_kl_utility_data(
generate_random_prior_samples, nouter_loop_samples, obs_fun,
noise_fun, ninner_loop_samples,
generate_inner_prior_samples=generate_inner_prior_samples)
new_design_indices = np.array([1])
outer_loop_weights = np.ones(
(nouter_loop_samples, 1)) / nouter_loop_samples
def log_likelihood_fun(obs, pred_obs, active_indices=None):
return gaussian_loglike_fun(obs, pred_obs, noise_std,
active_indices)
utility = compute_expected_kl_utility_monte_carlo(
log_likelihood_fun, outer_loop_obs, outer_loop_pred_obs,
inner_loop_pred_obs, inner_loop_weights, outer_loop_weights,
collected_design_indices, new_design_indices, False)
kl_divs = []
# overwrite subset of obs with previously collected data
# make copy so that outerloop obs can be used again
outer_loop_obs_copy = outer_loop_obs.copy()
for ii in range(nouter_loop_samples):
idx = np.hstack((collected_design_indices, new_design_indices))
obs_ii = outer_loop_obs_copy[ii:ii + 1, idx]
idx = np.hstack((collected_design_indices, new_design_indices))
exact_post_mean, exact_post_cov = \
laplace_posterior_approximation_for_linear_models(
Amat[idx, :], prior_mean, prior_cov_inv,
noise_cov_inv[np.ix_(idx, idx)], obs_ii.T)
kl_div = gaussian_kl_divergence(exact_post_mean, exact_post_cov,
prior_mean, prior_cov)
kl_divs.append(kl_div)
print(utility - np.mean(kl_divs), utility, np.mean(kl_divs))
assert np.allclose(utility, np.mean(kl_divs), rtol=2e-2)