def test_noise_variance_posterior_matches_expected(self):
# Generate a synthetic regression task.
num_features = 5
num_outputs = 20
design_matrix, _, targets = self.evaluate(
self._random_regression_task(num_features=num_features,
num_outputs=num_outputs,
batch_shape=[2],
seed=test_util.test_seed()))
observation_noise_variance_prior_concentration = 0.03
observation_noise_variance_prior_scale = 0.015
# Posterior on noise variance if all weights are zero.
naive_posterior = tfd.InverseGamma(
concentration=(observation_noise_variance_prior_concentration +
num_outputs / 2.),
scale=(observation_noise_variance_prior_scale +
tf.reduce_sum(tf.square(targets), axis=-1) / 2.))
# Compare to sampler with weights constrained to near-zero.
# We can do this by reducing the width of the slab (here),
# or by reducing the probability of the slab (below). Both should give
# equivalent noise posteriors.
tight_slab_sampler = spike_and_slab.SpikeSlabSampler(
design_matrix,
weights_prior_precision=tf.eye(num_features) * 1e6,
observation_noise_variance_prior_concentration=(
observation_noise_variance_prior_concentration),
observation_noise_variance_prior_scale=(
observation_noise_variance_prior_scale))
self.assertAllClose(
tight_slab_sampler.
observation_noise_variance_posterior_concentration,
naive_posterior.concentration)
self.assertAllClose(tight_slab_sampler._initialize_sampler_state(
targets=targets, nonzeros=tf.ones(
[num_features],
dtype=tf.bool)).observation_noise_variance_posterior_scale,
naive_posterior.scale,
atol=1e-2)
downweighted_slab_sampler = spike_and_slab.SpikeSlabSampler(
design_matrix,
observation_noise_variance_prior_concentration=(
observation_noise_variance_prior_concentration),
observation_noise_variance_prior_scale=(
observation_noise_variance_prior_scale))
self.assertAllClose(
(downweighted_slab_sampler.
observation_noise_variance_posterior_concentration),
naive_posterior.concentration)
self.assertAllClose(
downweighted_slab_sampler._initialize_sampler_state(
targets=targets,
nonzeros=tf.zeros(
[num_features],
dtype=tf.bool)).observation_noise_variance_posterior_scale,
naive_posterior.scale)
def test_updated_state_matches_initial_computation(self):
design_matrix, _, targets = self._random_regression_task(
num_outputs=2,
num_features=3,
batch_shape=[],
seed=test_util.test_seed())
sampler = spike_and_slab.SpikeSlabSampler(design_matrix=design_matrix,
nonzero_prior_prob=0.3)
all_nonzero_sampler_state = sampler._initialize_sampler_state(
targets=targets, nonzeros=tf.convert_to_tensor([True, True, True]))
# Flipping a weight from nonzero to zero (slab to spike) should result in
# the same state as if we'd initialized with that sparsity pattern.
flipped_state_from_update = sampler._flip_feature(
all_nonzero_sampler_state, idx=0)
flipped_state_from_scratch = sampler._initialize_sampler_state(
targets=targets,
nonzeros=tf.convert_to_tensor([False, True, True]))
self.assertAllCloseNested(flipped_state_from_update,
flipped_state_from_scratch)
# Reverse direction (spike to slab).
double_flipped_state_from_update = sampler._flip_feature(
flipped_state_from_update, idx=0)
self.assertAllCloseNested(double_flipped_state_from_update,
all_nonzero_sampler_state,
atol=1e-4)
def test_sampler_respects_pseudo_observations(self):
design_matrix = self.evaluate(
samplers.uniform([2, 20, 5], seed=test_util.test_seed()))
first_obs = 2.
second_obs = 10.
first_sampler = spike_and_slab.SpikeSlabSampler(
design_matrix,
default_pseudo_observations=first_obs)
second_sampler = spike_and_slab.SpikeSlabSampler(
design_matrix,
default_pseudo_observations=second_obs)
self.assertNotAllClose(
first_sampler.weights_prior_precision,
second_sampler.weights_prior_precision)
self.assertAllClose(
first_sampler.weights_prior_precision / first_obs,
second_sampler.weights_prior_precision / second_obs)
def test_posterior_on_nonzero_subset_matches_bayesian_regression(self):
# Generate a synthetic regression task.
design_matrix, _, targets = self.evaluate(
self._random_regression_task(num_features=5,
num_outputs=20,
batch_shape=[2],
seed=test_util.test_seed()))
# Utilities to extract values for nonzero-weight features.
nonzeros = tf.convert_to_tensor([True, False, True, False, True])
nonzero_subvector = (
lambda x: tf.boolean_mask(x, nonzeros, axis=ps.rank(x) - 1))
nonzero_submatrix = lambda x: tf.boolean_mask( # pylint: disable=g-long-lambda
tf.boolean_mask(x, nonzeros, axis=ps.rank(x) - 2),
nonzeros,
axis=ps.rank(x) - 1)
# Compute the weight posterior mean and precision for these nonzeros.
sampler = spike_and_slab.SpikeSlabSampler(design_matrix)
initial_state = sampler._initialize_sampler_state(targets=targets,
nonzeros=nonzeros)
# Compute the analytic posterior for the regression problem restricted to
# only the selected features. Note that by slicing a submatrix of the
# prior precision we are implicitly *conditioning* on having observed the
# other weights to be zero (which is sensible in this case), versus slicing
# into the covariance which would give the marginal (unconditional) prior
# on the selected weights.
(restricted_weights_posterior_mean,
restricted_weights_posterior_prec) = tfd.mvn_conjugate_linear_update(
prior_scale=tf.linalg.cholesky(
tf.linalg.inv(
nonzero_submatrix(sampler.weights_prior_precision))),
linear_transformation=nonzero_subvector(design_matrix),
likelihood_scale=tf.eye(20),
observation=targets)
# The sampler's posterior should match the posterior from the restricted
# problem.
self.assertAllClose(
nonzero_subvector(initial_state.conditional_weights_mean),
restricted_weights_posterior_mean)
self.assertAllClose(
nonzero_submatrix(
initial_state.conditional_posterior_precision_chol),
tf.linalg.cholesky(restricted_weights_posterior_prec.to_dense()))
def test_samples_from_weights_prior(self):
nonzero_prior_prob = 0.7
num_outputs, num_features = 200, 4
# Setting the design matrix to zero, the targets provide no information
# about weights, so the sampler should sample from the prior.
design_matrix = tf.zeros([num_outputs, num_features])
targets = 0.42 * samplers.normal([num_outputs],
seed=test_util.test_seed())
sampler = spike_and_slab.SpikeSlabSampler(
design_matrix=design_matrix,
weights_prior_precision=tf.eye(num_features),
nonzero_prior_prob=nonzero_prior_prob)
# Draw 100 posterior samples. Since all state needed for the
# internal feature sweep is a function of the sparsity pattern, it's
# sufficient to pass the sparsity pattern (by way of the weights) as
# the outer-loop state.
@tf.function(autograph=False)
def loop_body(var_weights_seed, _):
_, weights, seed = var_weights_seed
seed, next_seed = samplers.split_seed(seed, n=2)
variance, weights = sampler.sample_noise_variance_and_weights(
initial_nonzeros=tf.not_equal(weights, 0.),
targets=targets,
seed=seed)
return variance, weights, next_seed
init_seed = test_util.test_seed(sampler_type='stateless')
variance_samples, weight_samples, _ = tf.scan(
fn=loop_body,
initializer=(1., tf.ones([num_features]), init_seed),
elems=tf.range(100))
# With the default (relatively uninformative) prior, the noise variance
# posterior mean should be close to the most-likely value.
self.assertAllClose(tf.reduce_mean(variance_samples),
tf.math.reduce_std(targets)**2,
atol=0.03)
# Since there is no evidence for the weights, the sparsity of our samples
# should match the prior.
nonzero_weight_samples = tf.cast(tf.not_equal(weight_samples, 0.),
tf.float32)
self.assertAllClose(nonzero_prior_prob,
tf.reduce_mean(nonzero_weight_samples),
atol=0.03)
def test_sanity_check_sweep_over_features(self):
num_outputs = 100
num_features = 3
batch_shape = [2]
design_matrix, true_weights, targets = self.evaluate(
self._random_regression_task(
num_outputs=num_outputs,
num_features=num_features,
batch_shape=batch_shape,
# Specify weights with a clear sparsity pattern.
weights=tf.convert_to_tensor([[10., 0., -10.],
[0., 0., 0.5]]),
seed=test_util.test_seed()))
sampler = spike_and_slab.SpikeSlabSampler(
design_matrix,
# Ensure the probability of keeping an irrelevant feature is tiny.
nonzero_prior_prob=1e-6)
initial_state = sampler._initialize_sampler_state(
targets=targets,
nonzeros=tf.convert_to_tensor([True, True, True]),
observation_noise_variance=1.)
final_state = self.evaluate(
sampler._resample_all_features(
initial_state, seed=test_util.test_seed()))
# Check that we recovered the true sparsity pattern and approximate weights.
self.assertAllEqual(final_state.nonzeros, [[True, False, True],
[False, False, True]])
self.assertAllClose(final_state.conditional_weights_mean,
true_weights, rtol=0.05, atol=0.15)
# Check shapes of other components.
self.assertAllEqual(final_state.conditional_prior_precision_chol.shape,
batch_shape + [num_features, num_features])
self.assertAllEqual(final_state.conditional_posterior_precision_chol.shape,
batch_shape + [num_features, num_features])
self.assertAllEqual(
final_state.observation_noise_variance_posterior_scale.shape,
batch_shape)
posterior = sampler._get_conditional_posterior(final_state)
posterior_variances, posterior_weights = self.evaluate(
posterior.sample(10, seed=test_util.test_seed()))
self.assertAllFinite(posterior_variances)
self.assertAllFinite(posterior_weights)
def test_deterministic_given_seed(self, use_xla):
design_matrix, _, targets = self.evaluate(
self._random_regression_task(
num_outputs=3, num_features=4, batch_shape=[],
seed=test_util.test_seed()))
sampler = spike_and_slab.SpikeSlabSampler(design_matrix)
initial_nonzeros = tf.convert_to_tensor([True, False, False, True])
seed = test_util.test_seed(sampler_type='stateless')
@tf.function(jit_compile=use_xla)
def do_sample(seed):
return sampler.sample_noise_variance_and_weights(
targets, initial_nonzeros, seed=seed)
variance1, weights1 = self.evaluate(do_sample(seed))
variance2, weights2 = self.evaluate(do_sample(seed))
self.assertAllFinite(variance1)
self.assertAllClose(variance1, variance2)
self.assertAllFinite(weights1)
self.assertAllClose(weights1, weights2)
def test_updated_state_matches_initial_computation(
self, num_outputs, num_features, num_flips, batch_shape, use_xla):
rng = test_util.test_np_rng()
initial_nonzeros = rng.randint(
low=0, high=2, size=batch_shape + [num_features]).astype(np.bool)
flip_idxs = rng.choice(
num_features, size=num_flips, replace=False).astype(np.int32)
if batch_shape:
should_flip = rng.randint(
low=0, high=2, size=[num_flips] + batch_shape).astype(np.bool)
else:
should_flip = np.array([True] * num_flips)
nonzeros = initial_nonzeros.copy()
for i in range(num_flips):
nonzeros[..., flip_idxs[i]] = (
nonzeros[..., flip_idxs[i]] != should_flip[i])
design_matrix, _, targets = self._random_regression_task(
num_outputs=num_outputs, num_features=num_features,
batch_shape=batch_shape, seed=test_util.test_seed())
sampler = spike_and_slab.SpikeSlabSampler(design_matrix=design_matrix,
nonzero_prior_prob=0.3)
@tf.function(autograph=False, jit_compile=use_xla)
def _do_flips():
state = sampler._initialize_sampler_state(
targets=targets,
nonzeros=initial_nonzeros,
observation_noise_variance=1.)
def _do_flip(state, i):
new_state = sampler._flip_feature(state, tf.gather(flip_idxs, i))
return mcmc_util.choose(tf.gather(should_flip, i), new_state, state)
return tf.foldl(_do_flip, elems=tf.range(num_flips), initializer=state)
self.assertAllCloseNested(
sampler._initialize_sampler_state(targets, nonzeros, 1.),
_do_flips(),
atol=num_outputs * 2e-4, rtol=num_outputs * 2e-4)
def _build_sampler_loop_body(model,
observed_time_series,
is_missing=None):
"""Builds a Gibbs sampler for the given model and observed data.
Args:
model: A `tf.sts.StructuralTimeSeries` model instance. This must be of the
form constructed by `build_model_for_gibbs_sampling`.
observed_time_series: Float `Tensor` time series of shape
`[..., num_timesteps]`.
is_missing: Optional `bool` `Tensor` of shape `[..., num_timesteps]`. A
`True` value indicates that the observation for that timestep is missing.
Returns:
sampler_loop_body: Python callable that performs a single cycle of Gibbs
sampling. Its first argument is a `GibbsSamplerState`, and it returns a
new `GibbsSamplerState`. The second argument (passed by `tf.scan`) is
ignored.
"""
level_component = model.components[0]
if not (isinstance(level_component, sts.LocalLevel) or
isinstance(level_component, sts.LocalLinearTrend)):
raise ValueError('Expected the first model component to be an instance of '
'`tfp.sts.LocalLevel` or `tfp.sts.LocalLinearTrend`; '
'instead saw {}'.format(level_component))
model_has_slope = isinstance(level_component, sts.LocalLinearTrend)
regression_component = model.components[1]
if not (isinstance(regression_component, sts.LinearRegression) or
isinstance(regression_component, SpikeAndSlabSparseLinearRegression)):
raise ValueError('Expected the second model component to be an instance of '
'`tfp.sts.LinearRegression` or '
'`SpikeAndSlabSparseLinearRegression`; '
'instead saw {}'.format(regression_component))
model_has_spike_slab_regression = isinstance(
regression_component, SpikeAndSlabSparseLinearRegression)
if is_missing is not None: # Ensure series does not contain NaNs.
observed_time_series = tf.where(is_missing,
tf.zeros_like(observed_time_series),
observed_time_series)
num_observed_steps = prefer_static.shape(observed_time_series)[-1]
design_matrix = _get_design_matrix(model).to_dense()[:num_observed_steps]
if is_missing is not None:
# Replace design matrix with zeros at unobserved timesteps. This ensures
# they will not affect the posterior on weights.
design_matrix = tf.where(is_missing[..., tf.newaxis],
tf.zeros_like(design_matrix),
design_matrix)
# Untransform scale priors -> variance priors by reaching thru Sqrt bijector.
observation_noise_param = model.parameters[0]
if 'observation_noise' not in observation_noise_param.name:
raise ValueError('Model parameters {} do not match the expected sampler '
'state.'.format(model.parameters))
observation_noise_variance_prior = observation_noise_param.prior.distribution
if model_has_slope:
level_scale_variance_prior, slope_scale_variance_prior = [
p.prior.distribution for p in level_component.parameters]
else:
level_scale_variance_prior = (
level_component.parameters[0].prior.distribution)
if model_has_spike_slab_regression:
spike_and_slab_sampler = spike_and_slab.SpikeSlabSampler(
design_matrix,
weights_prior_precision=regression_component._weights_prior_precision, # pylint: disable=protected-access
nonzero_prior_prob=regression_component._sparse_weights_nonzero_prob, # pylint: disable=protected-access
observation_noise_variance_prior_concentration=(
observation_noise_variance_prior.concentration),
observation_noise_variance_prior_scale=(
observation_noise_variance_prior.scale),
observation_noise_variance_upper_bound=(
observation_noise_variance_prior.upper_bound
if hasattr(observation_noise_variance_prior, 'upper_bound')
else None))
else:
weights_prior_scale = (
regression_component.parameters[0].prior.scale)
def sampler_loop_body(previous_sample, _):
"""Runs one sampler iteration, resampling all model variables."""
(weights_seed,
level_seed,
observation_noise_scale_seed,
level_scale_seed,
loop_seed) = samplers.split_seed(
previous_sample.seed, n=5, salt='sampler_loop_body')
# Preserve backward-compatible seed behavior by splitting slope separately.
slope_scale_seed, = samplers.split_seed(
previous_sample.seed, n=1, salt='sampler_loop_body_slope')
# We encourage a reasonable initialization by sampling the weights first,
# so at the first step they are regressed directly against the observed
# time series. If we instead sampled the level first it might 'explain away'
# some observed variation that we would ultimately prefer to explain through
# the regression weights, because the level can represent arbitrary
# variation, while the weights are limited to representing variation in the
# subspace given by the design matrix.
if model_has_spike_slab_regression:
(observation_noise_variance,
weights) = spike_and_slab_sampler.sample_noise_variance_and_weights(
initial_nonzeros=tf.not_equal(previous_sample.weights, 0.),
targets=observed_time_series - previous_sample.level,
seed=weights_seed)
observation_noise_scale = tf.sqrt(observation_noise_variance)
else:
weights = _resample_weights(
design_matrix=design_matrix,
target_residuals=observed_time_series - previous_sample.level,
observation_noise_scale=previous_sample.observation_noise_scale,
weights_prior_scale=weights_prior_scale,
seed=weights_seed)
# Noise scale will be resampled below.
observation_noise_scale = previous_sample.observation_noise_scale
regression_residuals = observed_time_series - tf.linalg.matvec(
design_matrix, weights)
latents = _resample_latents(
observed_residuals=regression_residuals,
level_scale=previous_sample.level_scale,
slope_scale=previous_sample.slope_scale if model_has_slope else None,
observation_noise_scale=observation_noise_scale,
initial_state_prior=level_component.initial_state_prior,
is_missing=is_missing,
seed=level_seed)
level = latents[..., 0]
level_residuals = level[..., 1:] - level[..., :-1]
if model_has_slope:
slope = latents[..., 1]
level_residuals -= slope[..., :-1]
slope_residuals = slope[..., 1:] - slope[..., :-1]
# Estimate level scale from the empirical changes in level.
level_scale = _resample_scale(
prior=level_scale_variance_prior,
observed_residuals=level_residuals,
is_missing=None,
seed=level_scale_seed)
if model_has_slope:
slope_scale = _resample_scale(
prior=slope_scale_variance_prior,
observed_residuals=slope_residuals,
is_missing=None,
seed=slope_scale_seed)
if not model_has_spike_slab_regression:
# Estimate noise scale from the residuals.
observation_noise_scale = _resample_scale(
prior=observation_noise_variance_prior,
observed_residuals=regression_residuals - level,
is_missing=is_missing,
seed=observation_noise_scale_seed)
return GibbsSamplerState(
observation_noise_scale=observation_noise_scale,
level_scale=level_scale,
slope_scale=(slope_scale if model_has_slope
else previous_sample.slope_scale),
weights=weights,
level=level,
slope=(slope if model_has_slope
else previous_sample.slope),
seed=loop_seed)
return sampler_loop_body
