def test_linear_gaussian_inference(self):
# set random seed, so the data is reproducible each time
np.random.seed(1)
nobs = 10 # number of observations
noise_stdev = .1 # standard deviation of noise
x = np.linspace(0., 9., nobs)
Amatrix = np.hstack([np.ones((nobs, 1)), x[:, np.newaxis]])
univariate_variables = [norm(1, 1), norm(0, 4)]
variables = IndependentMultivariateRandomVariable(univariate_variables)
mtrue = 0.4 # true gradient
ctrue = 2. # true y-intercept
true_sample = np.array([[ctrue, mtrue]]).T
model = LinearModel(Amatrix)
# make data
data = noise_stdev * np.random.randn(nobs) + model(true_sample)[0, :]
loglike = GaussianLogLike(model, data, noise_stdev**2)
loglike = PYMC3LogLikeWrapper(loglike)
# number of draws from the distribution
ndraws = 5000
# number of "burn-in points" (which we'll discard)
nburn = min(1000, int(ndraws * 0.1))
# number of parallel chains
njobs = 4
#algorithm='nuts'
algorithm = 'metropolis'
samples, effective_sample_size, map_sample = \
run_bayesian_inference_gaussian_error_model(
loglike,variables,ndraws,nburn,njobs,
algorithm=algorithm,get_map=True,print_summary=False)
prior_mean = np.asarray(
[rv.mean() for rv in variables.all_variables()])
prior_hessian = np.diag(
[1. / rv.var() for rv in variables.all_variables()])
noise_covariance_inv = 1. / noise_stdev**2 * np.eye(nobs)
from pyapprox.bayesian_inference.laplace import \
laplace_posterior_approximation_for_linear_models
exact_mean, exact_covariance = \
laplace_posterior_approximation_for_linear_models(
Amatrix, prior_mean, prior_hessian,
noise_covariance_inv, data)
print('mcmc mean error', samples.mean(axis=1) - exact_mean)
print('mcmc cov error', np.cov(samples) - exact_covariance)
print('MAP sample', map_sample)
print('exact mean', exact_mean.squeeze())
print('exact cov', exact_covariance)
assert np.allclose(map_sample, exact_mean)
assert np.allclose(exact_mean.squeeze(),
samples.mean(axis=1),
atol=1e-2)
assert np.allclose(exact_covariance, np.cov(samples), atol=1e-2)
def test_marginalize_polynomial_chaos_expansions(self):
univariate_variables = [uniform(-1, 2), norm(0, 1), uniform(-1, 2)]
variable = IndependentMultivariateRandomVariable(univariate_variables)
var_trans = AffineRandomVariableTransformation(variable)
num_vars = len(univariate_variables)
poly = PolynomialChaosExpansion()
poly_opts = define_poly_options_from_variable_transformation(var_trans)
poly.configure(poly_opts)
degree = 2
indices = compute_hyperbolic_indices(num_vars, degree, 1)
poly.set_indices(indices)
poly.set_coefficients(np.ones((indices.shape[1], 1)))
for ii in range(num_vars):
# Marginalize out 2 variables
xx = np.linspace(-1, 1, 101)
inactive_idx = np.hstack(
(np.arange(ii), np.arange(ii + 1, num_vars)))
marginalized_pce = marginalize_polynomial_chaos_expansion(
poly, inactive_idx)
mvals = marginalized_pce(xx[None, :])
variable_ii = variable.all_variables()[ii:ii + 1]
var_trans_ii = AffineRandomVariableTransformation(variable_ii)
poly_ii = PolynomialChaosExpansion()
poly_opts_ii = define_poly_options_from_variable_transformation(
var_trans_ii)
poly_ii.configure(poly_opts_ii)
indices_ii = compute_hyperbolic_indices(1, degree, 1.)
poly_ii.set_indices(indices_ii)
poly_ii.set_coefficients(np.ones((indices_ii.shape[1], 1)))
pvals = poly_ii(xx[None, :])
# import matplotlib.pyplot as plt
# plt.plot(xx, pvals)
# plt.plot(xx, mvals, '--')
# plt.show()
assert np.allclose(mvals, pvals)
# Marginalize out 1 variable
xx = cartesian_product([xx] * 2)
inactive_idx = np.array([ii])
marginalized_pce = marginalize_polynomial_chaos_expansion(
poly, inactive_idx)
mvals = marginalized_pce(xx)
variable_ii = variable.all_variables()[:ii]+\
variable.all_variables()[ii+1:]
var_trans_ii = AffineRandomVariableTransformation(variable_ii)
poly_ii = PolynomialChaosExpansion()
poly_opts_ii = define_poly_options_from_variable_transformation(
var_trans_ii)
poly_ii.configure(poly_opts_ii)
indices_ii = compute_hyperbolic_indices(2, degree, 1.)
poly_ii.set_indices(indices_ii)
poly_ii.set_coefficients(np.ones((indices_ii.shape[1], 1)))
pvals = poly_ii(xx)
assert np.allclose(mvals, pvals)
name, scales, shapes = get_distribution_info(rv)
if ii not in [0, 2]:
opts = {'rv_type': name, 'shapes': shapes,
'var_nums': re_variable.unique_variable_indices[ii]}
basis_opts['basis%d' % ii] = opts
continue
#identity_map_indices += re_variable.unique_variable_indices[ii] # wrong
identity_map_indices += list(re_variable.unique_variable_indices[ii]) # right
quad_rules = []
inds = index_product[cnt]
nquad_samples_1d = 50
for jj in inds:
a, b = variable.all_variables()[jj].interval(1)
x, w = gauss_jacobi_pts_wts_1D(nquad_samples_1d, 0, 0)
x = (x+1)/2 # map to [0, 1]
x = (b-a)*x+a # map to [a,b]
quad_rules.append((x, w))
funs = [identity_fun]*len(inds)
basis_opts['basis%d' % ii] = {'poly_type': 'product_indpnt_vars',
'var_nums': [ii], 'funs': funs,
'quad_rules': quad_rules}
cnt += 1
poly_opts = {'var_trans': re_var_trans}
poly_opts['poly_types'] = basis_opts
#var_trans.set_identity_maps(identity_map_indices) #wrong
re_var_trans.set_identity_maps(identity_map_indices) #right