def test_batch_kl_oed(self):
"""
No observations collected to inform subsequent designs
"""
np.random.seed(1)
nrandom_vars = 1
noise_std = 1
ndesign = 4
nouter_loop_samples = 10000
ninner_loop_samples = 31
ncandidates = 11
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)
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 = BayesianBatchKLOED(design_candidates, obs_fun, noise_std,
prior_variable, nouter_loop_samples,
ninner_loop_samples,
generate_inner_prior_samples)
oed.populate()
oed.set_collected_design_indices(init_design_indices)
for ii in range(len(init_design_indices), ndesign):
# loop must be before oed.updated design because
# which updates oed.collected_design_indices and thus
# changes problem
d_utility_vals = np.zeros(ncandidates)
for kk in range(ncandidates):
if kk not in oed.collected_design_indices:
new_design = np.hstack(
(design_candidates[:, oed.collected_design_indices],
design_candidates[:, kk:kk + 1]))
Amat = new_design.T
d_utility_vals[kk] = d_optimal_utility(Amat, noise_std)
utility_vals, selected_indices = oed.update_design()
# ignore entries of previously collected data
II = np.where(d_utility_vals > 0)
print((np.absolute(d_utility_vals[II] - utility_vals[II]) /
d_utility_vals[II]).max())
assert np.allclose(d_utility_vals[II], utility_vals[II], rtol=4e-2)
def test_tensor_product_lagrange_interpolation(self):
nvars = 5
level = 10
x = gauss_hermite_pts_wts_1D(level + 1)[0]
# active_vars = np.arange(nvars)
active_vars = np.hstack([np.arange(2), np.arange(3, nvars)])
nactive_vars = active_vars.shape[0]
abscissa_1d = [x] * nactive_vars
power = x.shape[0] - 2
def fun(samples):
return np.sum(samples[active_vars, :]**power, axis=0)[:, None]
nsamples = 1000
validation_samples = np.random.normal(0, 1, (nvars, nsamples))
zz = abscissa_1d.copy()
zz.insert(2, np.zeros(1))
train_samples = cartesian_product(zz)
values = fun(train_samples)
approx_values = tensor_product_lagrange_interpolation(
validation_samples, abscissa_1d, active_vars, values)
barycentric_weights_1d = [
compute_barycentric_weights_1d(x) for x in abscissa_1d
]
poly_vals = multivariate_barycentric_lagrange_interpolation(
validation_samples, abscissa_1d, barycentric_weights_1d, values,
active_vars)
assert np.allclose(approx_values, fun(validation_samples))
assert np.allclose(poly_vals, fun(validation_samples))
def get_tensor_product_points(level, var_trans, quad_type):
abscissa_1d = []
num_vars = var_trans.num_vars()
if quad_type == 'CC':
x, w = clenshaw_curtis_pts_wts_1D(level)
elif quad_type == 'GH':
x, w = gauss_hermite_pts_wts_1D(level)
for dd in range(num_vars):
abscissa_1d.append(x)
pts = cartesian_product(abscissa_1d, 1)
pts = var_trans.map_from_canonical_space(pts)
return pts
def test_gaussian_loglike_fun(self):
nvars = 1
def fun(design, samples):
assert design.ndim == 2
assert samples.ndim == 2
Amat = design.T
return Amat.dot(samples).T
noise_std = 0.3
prior_mean = np.zeros((nvars, 1))
prior_cov = np.eye(nvars)
design = np.linspace(-1, 1, 4)[None, :]
true_sample = np.ones((nvars, 1)) * 0.4
obs = fun(design, true_sample)
obs += np.random.normal(0, noise_std, obs.shape)
noise_cov_inv = np.eye(obs.shape[1]) / (noise_std**2)
obs_matrix = design.T
exact_post_mean, exact_post_cov = \
laplace_posterior_approximation_for_linear_models(
obs_matrix, prior_mean, np.linalg.inv(prior_cov),
noise_cov_inv, obs.T)
lb, ub = stats.norm(0, 1).interval(0.99)
xx = np.linspace(lb, ub, 101)
true_pdf_vals = stats.norm(exact_post_mean[0],
np.sqrt(exact_post_cov[0])).pdf(xx)[:, None]
prior_pdf = stats.norm(prior_mean[0], np.sqrt(prior_cov[0])).pdf
pred_obs = fun(design, xx[None, :])
lvals = np.exp(gaussian_loglike_fun(
obs, pred_obs, noise_std)) * prior_pdf(xx)[:, None]
xx_gauss, ww_gauss = gauss_hermite_pts_wts_1D(300)
pred_obs = fun(design, xx_gauss[None, :])
evidence = np.exp(
gaussian_loglike_fun(obs, pred_obs, noise_std)[:, 0]).dot(ww_gauss)
post_pdf_vals = lvals / evidence
gauss_evidence = laplace_evidence(
lambda x: np.exp(
gaussian_loglike_fun(obs, fun(design, x), noise_std)[:, 0]),
prior_pdf, exact_post_cov, exact_post_mean)
assert np.allclose(evidence, gauss_evidence)
# accuracy depends on quadrature rule and size of noise
# print(post_pdf_vals - true_pdf_vals)
assert np.allclose(post_pdf_vals, true_pdf_vals)
def test_gauss_hermite_quadrature(self):
"""
integrate x^2 1/sqrt(2*pi)exp(-x**2/2) dx from x=-inf..inf
"""
true_mean = 0.
true_variance = 1.
x, w = gauss_hermite_pts_wts_1D(2)
def function(x):
return x**2
assert np.allclose(
np.dot(function(x), w) - true_mean**2, true_variance)
def test_interpolation_gaussian_leja_sequence(self):
def f(x):
return np.exp(-np.sum(x**2, axis=0))
level = 30
# abscissa_leja, __ = gaussian_leja_quadrature_rule(
# level, return_weights_for_all_levels=False)
# abscissa = abscissa_leja
abscissa_gauss = gauss_hermite_pts_wts_1D(level + 1)[0]
abscissa = abscissa_gauss
# print(abscissa_leja.shape,abscissa_gauss.shape)
abscissa_1d = [abscissa]
barycentric_weights_1d = [
compute_barycentric_weights_1d(abscissa_1d[0])
]
# print(barycentric_weights_1d[0])
barycentric_weights_1d[0] /= barycentric_weights_1d[0].max()
# print(barycentric_weights_1d[0])
fn_vals = f(np.array(abscissa).reshape(1,
abscissa.shape[0]))[:,
np.newaxis]
# print(fn_vals.shape)
samples = np.random.normal(0, 1, (1, 1000))
poly_vals = multivariate_barycentric_lagrange_interpolation(
samples, abscissa_1d, barycentric_weights_1d, fn_vals,
np.array([0]))[:, 0]
l2_error = np.linalg.norm(poly_vals-f(samples)) / \
np.sqrt(samples.shape[1])
# print('l2_error',l2_error)
# pts = np.linspace(abscissa.min(),abscissa.max(),101).reshape(1,101)
# poly_vals = multivariate_barycentric_lagrange_interpolation(
# pts,abscissa_1d,barycentric_weights_1d,fn_vals,np.array([0]))
# import matplotlib.pyplot as plt
# plt.plot(pts[0,:],poly_vals.squeeze())
# plt.plot(abscissa_1d[0],fn_vals.squeeze(),'r*')
# plt.plot(abscissa_leja,abscissa_leja*0,'ro')
# plt.plot(abscissa_gauss,abscissa_gauss*0,'ks',ms=3)
# plt.ylim(-1,2)
# plt.show()
assert l2_error < 1e-2
def test_gaussian_kl_divergence(self):
nvars = 1
mean1, sigma1 = np.zeros((nvars, 1)), np.eye(nvars) * 2
mean2, sigma2 = np.ones((nvars, 1)), np.eye(nvars) * 3
kl_div = gaussian_kl_divergence(mean1, sigma1, mean2, sigma2)
rv1 = stats.multivariate_normal(mean1, sigma1)
rv2 = stats.multivariate_normal(mean2, sigma2)
xx, ww = gauss_hermite_pts_wts_1D(300)
xx = xx * np.sqrt(sigma1[0, 0]) + mean1[0]
kl_div_quad = np.log(rv1.pdf(xx) / rv2.pdf(xx)).dot(ww)
assert np.allclose(kl_div, kl_div_quad)
xx = np.random.normal(mean1[0], np.sqrt(sigma1[0, 0]), (int(1e6)))
ww = np.ones(xx.shape[0]) / xx.shape[0]
kl_div_quad = np.log(rv1.pdf(xx) / rv2.pdf(xx)).dot(ww)
assert np.allclose(kl_div, kl_div_quad, rtol=1e-2)
def test_predictor_corrector_function_of_independent_variables(self):
"""
Test 1: Sum of Gaussians is a Gaussian
Test 2: Product of uniforms on [0,1]
"""
nvars, nterms = 2, 5
nquad_samples_1d = 50
quad_rules = [gauss_hermite_pts_wts_1D(nquad_samples_1d)] * nvars
def fun(x):
return x.sum(axis=0)
ab = predictor_corrector_function_of_independent_variables(
nterms, quad_rules, fun)
rv = stats.norm(0, np.sqrt(nvars))
lb, ub = rv.interval(1)
ab_full = predictor_corrector(nterms, rv.pdf, lb, ub)
assert np.allclose(ab_full, ab)
nvars = 2
def measure(x):
return (-1)**(nvars - 1) * np.log(x)**(nvars -
1) / factorial(nvars - 1)
def fun(x):
return x.prod(axis=0)
quad_opts = {}
ab_full = predictor_corrector(nterms, measure, 0, 1, quad_opts)
xx, ww = gauss_jacobi_pts_wts_1D(nquad_samples_1d, 0, 0)
xx = (xx + 1) / 2
quad_rules = [(xx, ww)] * nvars
ab = predictor_corrector_function_of_independent_variables(
nterms, quad_rules, fun)
assert np.allclose(ab_full, ab)
def test_barycentric_weights_1d(self):
eps = 1e-12
# test barycentric weights for uniform points using direct calculation
abscissa = np.linspace(-1, 1., 5)
weights = compute_barycentric_weights_1d(abscissa,
normalize_weights=False)
n = abscissa.shape[0] - 1
h = 2. / n
true_weights = np.empty((n + 1), np.double)
for j in range(n + 1):
true_weights[j] = (-1.)**(n - j) * nchoosek(
n, j) / (h**n * factorial(n))
assert np.allclose(true_weights, weights, eps)
# test barycentric weights for uniform points using analytical formula
# and with scaling on
weights = compute_barycentric_weights_1d(abscissa,
interval_length=1,
normalize_weights=False)
weights_analytical = equidistant_barycentric_weights(5)
ratio = weights / weights_analytical
# assert the two weights array differ by only a constant factor
assert np.allclose(np.min(ratio), np.max(ratio))
# test barycentric weights for clenshaw curtis points
level = 7
abscissa, tmp = clenshaw_curtis_pts_wts_1D(level)
n = abscissa.shape[0]
weights = compute_barycentric_weights_1d(abscissa,
normalize_weights=False,
interval_length=2)
true_weights = np.empty((n), np.double)
true_weights[0] = true_weights[n - 1] = 0.5
true_weights[1:n - 1] = [(-1)**ii for ii in range(1, n - 1)]
factor = true_weights[1] / weights[1]
assert np.allclose(true_weights / factor, weights, atol=eps)
# check barycentric weights are correctly computed regardless of
# order of points. Eventually ordering can effect numerical stability
# but not until very high level
abscissa, tmp = clenshaw_curtis_in_polynomial_order(level)
II = np.argsort(abscissa)
n = abscissa.shape[0]
weights = compute_barycentric_weights_1d(
abscissa,
normalize_weights=False,
interval_length=abscissa.max() - abscissa.min())
true_weights = np.empty((n), np.double)
true_weights[0] = true_weights[n - 1] = 0.5
true_weights[1:n - 1] = [(-1)**ii for ii in range(1, n - 1)]
factor = true_weights[1] / weights[II][1]
assert np.allclose(true_weights / factor, weights[II], eps)
num_samples = 65
abscissa, tmp = gauss_hermite_pts_wts_1D(num_samples)
weights = compute_barycentric_weights_1d(
abscissa,
normalize_weights=False,
interval_length=abscissa.max() - abscissa.min())
print(weights)
print(np.absolute(weights).max(), np.absolute(weights).min())
print(np.absolute(weights).max() / np.absolute(weights).min())
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 help_compare_sequential_kl_oed_econ(self, use_gauss_quadrature):
"""
Use the same inner loop samples for all outer loop samples
"""
nrandom_vars = 1
noise_std = 1
ndesign = 5
nouter_loop_samples = int(1e1)
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
generate_random_prior_samples = partial(
generate_independent_random_samples, prior_variable)
def generate_inner_prior_samples_mc(n):
# fix seed that when econ is False we are still creating
# the samples each time. This is just for testing purposes
# to make sure that econ is True does this in effect
np.random.seed(1)
return generate_random_prior_samples(n), np.ones(n) / n
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
if use_gauss_quadrature:
generate_inner_prior_samples = generate_inner_prior_samples_gauss
else:
generate_inner_prior_samples = generate_inner_prior_samples_mc
# Define initial design
init_design_indices = np.array([ncandidates // 2])
np.random.seed(1)
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)
# randomness only enters during populate and when collecting
# real observations, i.e. evaluating obs model so setting seed
# here is sufficient when below we just evaluate obs model once
# and use same value for both oed instances
np.random.seed(1)
oed_econ = BayesianSequentialKLOED(design_candidates,
obs_fun,
noise_std,
prior_variable,
obs_process,
nouter_loop_samples,
ninner_loop_samples,
generate_inner_prior_samples,
econ=True)
oed_econ.populate()
oed_econ.set_collected_design_indices(init_design_indices)
for step in range(len(init_design_indices), ndesign):
utility_vals, selected_indices = oed.update_design()
new_obs = oed.obs_process(selected_indices)
oed.update_observations(new_obs)
econ_utility_vals, econ_selected_indices = oed_econ.update_design()
assert np.allclose(econ_utility_vals, utility_vals)
assert np.allclose(econ_selected_indices, selected_indices)
# use same data as non econ version do not call model
# as different noise will be added
oed_econ.update_observations(new_obs)
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)
