def test_logitnormal_moments(self):
# global parameters for computing lognormals
gh_loc, gh_weights = hermgauss(4)
# log normal parameters
lognorm_means = np.random.random((5, 3)) # should work for arrays now
lognorm_infos = np.random.random((5, 3))**2 + 1
alpha = 2 # dp parameter
# draw samples
num_draws = 10**5
samples = np.random.normal(lognorm_means,
1/np.sqrt(lognorm_infos), size = (num_draws, 5, 3))
logit_norm_samples = sp.special.expit(samples)
# test lognormal means
np_test.assert_allclose(
np.mean(logit_norm_samples, axis = 0),
ef.get_e_logitnormal(
lognorm_means, lognorm_infos, gh_loc, gh_weights),
atol = 3 * np.std(logit_norm_samples) / np.sqrt(num_draws))
# test Elog(x) and Elog(1-x)
log_logistic_norm = np.mean(np.log(logit_norm_samples), axis = 0)
log_1m_logistic_norm = np.mean(np.log(1 - logit_norm_samples), axis = 0)
tol1 = 3 * np.std(np.log(logit_norm_samples))/ np.sqrt(num_draws)
tol2 = 3 * np.std(np.log(1 - logit_norm_samples))/ np.sqrt(num_draws)
np_test.assert_allclose(
log_logistic_norm,
ef.get_e_log_logitnormal(
lognorm_means, lognorm_infos, gh_loc, gh_weights)[0],
atol = tol1)
np_test.assert_allclose(
log_1m_logistic_norm,
ef.get_e_log_logitnormal(
lognorm_means, lognorm_infos, gh_loc, gh_weights)[1],
atol = tol2)
# test prior
prior_samples = np.mean((alpha - 1) *
np.log(1 - logit_norm_samples), axis = 0)
tol3 = 3 * np.std((alpha - 1) * np.log(1 - logit_norm_samples)) \
/np.sqrt(num_draws)
np_test.assert_allclose(
prior_samples,
ef.get_e_dp_prior_logitnorm_approx(
alpha, lognorm_means, lognorm_infos, gh_loc, gh_weights),
atol = tol3)
x = np.random.normal(0, 1e2, size = 10)
def e_log_v(x):
return np.sum(ef.get_e_log_logitnormal(\
x[0:5], np.abs(x[5:10]), gh_loc, gh_weights)[0])
check_grads(e_log_v, order=2)(x)
def get_stick_breaking_entropy(stick_propn_mean, stick_propn_info,
gh_loc, gh_weights):
# return the entropy of logitnormal distriibution on the sticks whose
# logit has mean stick_propn_mean and information stick_propn_info
# Integration is done on the real line with respect to the Lesbegue measure
# integration is done numerical with Gauss Hermite quadrature.
# gh_loc and gh_weights specifiy the location and weights of the
# quadrature points
# we seek E[log q(V)], where q is the density of a logit-normal, and
# V ~ logit-normal. Let W := logit(V), so W ~ Normal. Hence,
# E[log q(W)]; we can then decompose log q(x) into the terms of a normal
# distribution and the jacobian term. The expectation of the normal term
# evaluates to the normal entropy, and we add the jacobian term to it.
# The jacobian term is 1/(x(1-x)), so we simply add -EV - E(1-V) to the normal
# entropy.
assert np.all(gh_weights > 0)
assert stick_propn_mean.shape == stick_propn_info.shape
assert np.all(stick_propn_info) > 0
e_log_v, e_log_1mv =\
ef.get_e_log_logitnormal(
lognorm_means = stick_propn_mean,
lognorm_infos = stick_propn_info,
gh_loc = gh_loc,
gh_weights = gh_weights)
return np.sum(ef.univariate_normal_entropy(stick_propn_info)) + \
np.sum(e_log_v + e_log_1mv)
def get_e_logitnorm_dp_prior(stick_propn_mean, stick_propn_info, alpha,
gh_loc, gh_weights):
# expected log prior for the stick breaking proportions under the
# logitnormal variational distribution
# integration is done numerical with Gauss Hermite quadrature.
# gh_loc and gh_weights specifiy the location and weights of the
# quadrature points
assert np.all(gh_weights > 0)
assert stick_propn_mean.shape == stick_propn_info.shape
assert np.all(stick_propn_info) > 0
e_log_v, e_log_1mv = \
ef.get_e_log_logitnormal(
lognorm_means = stick_propn_mean,
lognorm_infos = stick_propn_info,
gh_loc = gh_loc,
gh_weights = gh_weights)
return (alpha - 1) * np.sum(e_log_1mv)
def get_e_log_cluster_probabilities(stick_propn_mean, stick_propn_info,
gh_loc, gh_weights):
# the expected log mixture weights
# stick_propn_mean is of shape ... x k_approx
assert np.all(gh_weights > 0)
assert stick_propn_mean.shape == stick_propn_info.shape
if len(stick_propn_mean.shape) == 1:
stick_propn_mean = stick_propn_mean[None, :]
stick_propn_info = stick_propn_info[None, :]
assert np.all(stick_propn_info) > 0
e_log_v, e_log_1mv = \
ef.get_e_log_logitnormal(
lognorm_means = stick_propn_mean,
lognorm_infos = stick_propn_info,
gh_loc = gh_loc,
gh_weights = gh_weights)
return get_e_log_cluster_probabilities_from_e_log_stick(e_log_v, e_log_1mv)
def e_log_v(x):
return np.sum(ef.get_e_log_logitnormal(\
x[0:5], np.abs(x[5:10]), gh_loc, gh_weights)[0])