def test_gamma_gaussian_hmm_log_prob(sample_shape, batch_shape, num_steps,
hidden_dim, obs_dim):
init_dist = random_mvn(batch_shape, hidden_dim)
trans_mat = torch.randn(batch_shape + (num_steps, hidden_dim, hidden_dim))
trans_dist = random_mvn(batch_shape + (num_steps, ), hidden_dim)
obs_mat = torch.randn(batch_shape + (num_steps, hidden_dim, obs_dim))
obs_dist = random_mvn(batch_shape + (num_steps, ), obs_dim)
scale_dist = random_gamma(batch_shape)
d = dist.GammaGaussianHMM(scale_dist, init_dist, trans_mat, trans_dist,
obs_mat, obs_dist)
obs_mvn = obs_dist
data = obs_dist.sample(sample_shape)
assert data.shape == sample_shape + d.shape()
actual_log_prob = d.log_prob(data)
# Compare against hand-computed density.
# We will construct enormous unrolled joint gaussian-gammas with shapes:
# t | 0 1 2 3 1 2 3 T = 3 in this example
# ------+-----------------------------------------
# init | H
# trans | H H H H H = hidden
# obs | H H H O O O O = observed
# and then combine these using gamma_gaussian_tensordot().
T = num_steps
init = gamma_and_mvn_to_gamma_gaussian(scale_dist, init_dist)
trans = matrix_and_mvn_to_gamma_gaussian(trans_mat, trans_dist)
obs = matrix_and_mvn_to_gamma_gaussian(obs_mat, obs_mvn)
unrolled_trans = reduce(
operator.add,
[
trans[..., t].event_pad(left=t * hidden_dim,
right=(T - t - 1) * hidden_dim)
for t in range(T)
],
)
unrolled_obs = reduce(
operator.add,
[
obs[..., t].event_pad(left=t * obs.dim(),
right=(T - t - 1) * obs.dim())
for t in range(T)
],
)
# Permute obs from HOHOHO to HHHOOO.
perm = torch.cat(
[torch.arange(hidden_dim) + t * obs.dim() for t in range(T)] +
[torch.arange(obs_dim) + hidden_dim + t * obs.dim() for t in range(T)])
unrolled_obs = unrolled_obs.event_permute(perm)
unrolled_data = data.reshape(data.shape[:-2] + (T * obs_dim, ))
assert init.dim() == hidden_dim
assert unrolled_trans.dim() == (1 + T) * hidden_dim
assert unrolled_obs.dim() == T * (hidden_dim + obs_dim)
logp = gamma_gaussian_tensordot(init, unrolled_trans, hidden_dim)
logp = gamma_gaussian_tensordot(logp, unrolled_obs, T * hidden_dim)
# compute log_prob of the joint student-t distribution
expected_log_prob = logp.compound().log_prob(unrolled_data)
assert_close(actual_log_prob, expected_log_prob)
def test_gamma_gaussian_tensordot(dot_dims, x_batch_shape, x_dim, x_rank,
y_batch_shape, y_dim, y_rank):
x_rank = min(x_rank, x_dim)
y_rank = min(y_rank, y_dim)
x = random_gamma_gaussian(x_batch_shape, x_dim, x_rank)
y = random_gamma_gaussian(y_batch_shape, y_dim, y_rank)
na = x_dim - dot_dims
nb = dot_dims
nc = y_dim - dot_dims
try:
torch.linalg.cholesky(x.precision[..., na:, na:] +
y.precision[..., :nb, :nb])
except RuntimeError:
pytest.skip(
"Cannot marginalize the common variables of two Gaussians.")
z = gamma_gaussian_tensordot(x, y, dot_dims)
assert z.dim() == x_dim + y_dim - 2 * dot_dims
# We make these precision matrices positive definite to test the math
x.precision = x.precision + 3 * torch.eye(x.dim())
y.precision = y.precision + 3 * torch.eye(y.dim())
z = gamma_gaussian_tensordot(x, y, dot_dims)
# compare against broadcasting, adding, and marginalizing
precision = pad(x.precision, (0, nc, 0, nc)) + pad(y.precision,
(na, 0, na, 0))
info_vec = pad(x.info_vec, (0, nc)) + pad(y.info_vec, (na, 0))
covariance = torch.inverse(precision)
loc = (covariance.matmul(info_vec.unsqueeze(-1)).squeeze(-1)
if info_vec.size(-1) > 0 else info_vec)
z_covariance = torch.inverse(z.precision)
z_loc = z_covariance.matmul(
z.info_vec.view(z.info_vec.shape + (int(z.dim() > 0), ))).sum(-1)
assert_close(loc[..., :na], z_loc[..., :na])
assert_close(loc[..., x_dim:], z_loc[..., na:])
assert_close(covariance[..., :na, :na], z_covariance[..., :na, :na])
assert_close(covariance[..., :na, x_dim:], z_covariance[..., :na, na:])
assert_close(covariance[..., x_dim:, :na], z_covariance[..., na:, :na])
assert_close(covariance[..., x_dim:, x_dim:], z_covariance[..., na:, na:])
s = torch.randn(z.batch_shape).exp()
# Assume a = c = 0, integrate out b
num_samples = 200000
scale = 10
# generate samples in [-10, 10]
value_b = torch.rand((num_samples, ) + z.batch_shape +
(nb, )) * scale - scale / 2
value_x = pad(value_b, (na, 0))
value_y = pad(value_b, (0, nc))
expect = torch.logsumexp(x.log_density(value_x, s) +
y.log_density(value_y, s),
dim=0)
expect += math.log(scale**nb / num_samples)
actual = z.log_density(torch.zeros(z.batch_shape + (z.dim(), )), s)
assert_close(actual.clamp(max=10.0),
expect.clamp(max=10.0),
atol=0.1,
rtol=0.1)
def test_sequential_gamma_gaussian_tensordot(batch_shape, state_dim,
num_steps):
g = random_gamma_gaussian(batch_shape + (num_steps, ),
state_dim + state_dim)
actual = _sequential_gamma_gaussian_tensordot(g)
assert actual.dim() == g.dim()
assert actual.batch_shape == batch_shape
# Check against hand computation.
expected = g[..., 0]
for t in range(1, num_steps):
expected = gamma_gaussian_tensordot(expected, g[..., t], state_dim)
assert_close_gamma_gaussian(actual, expected)
def log_prob(self, value):
# Combine observation and transition factors.
result = self._trans + self._obs.condition(value).event_pad(
left=self.hidden_dim)
# Eliminate time dimension.
result = _sequential_gamma_gaussian_tensordot(
result.expand(result.batch_shape))
# Combine initial factor.
result = gamma_gaussian_tensordot(self._init,
result,
dims=self.hidden_dim)
# Marginalize out final state.
result = result.event_logsumexp()
# Marginalize out multiplier.
result = result.logsumexp()
return result
def _sequential_gamma_gaussian_tensordot(gamma_gaussian):
"""
Integrates a GammaGaussian ``x`` whose rightmost batch dimension is time, computes::
x[..., 0] @ x[..., 1] @ ... @ x[..., T-1]
"""
assert isinstance(gamma_gaussian, GammaGaussian)
assert gamma_gaussian.dim() % 2 == 0, "dim is not even"
batch_shape = gamma_gaussian.batch_shape[:-1]
state_dim = gamma_gaussian.dim() // 2
while gamma_gaussian.batch_shape[-1] > 1:
time = gamma_gaussian.batch_shape[-1]
even_time = time // 2 * 2
even_part = gamma_gaussian[..., :even_time]
x_y = even_part.reshape(batch_shape + (even_time // 2, 2))
x, y = x_y[..., 0], x_y[..., 1]
contracted = gamma_gaussian_tensordot(x, y, state_dim)
if time > even_time:
contracted = GammaGaussian.cat(
(contracted, gamma_gaussian[..., -1:]), dim=-1)
gamma_gaussian = contracted
return gamma_gaussian[..., 0]
def filter(self, value):
"""
Compute posteriors over the multiplier and the final state
given a sequence of observations. The posterior is a pair of
Gamma and MultivariateNormal distributions (i.e. a GammaGaussian
instance).
:param ~torch.Tensor value: A sequence of observations.
:return: A pair of posterior distributions over the mixing and the latent
state at the final time step.
:rtype: a tuple of ~pyro.distributions.Gamma and ~pyro.distributions.MultivariateNormal
"""
# Combine observation and transition factors.
logp = self._trans + self._obs.condition(value).event_pad(
left=self.hidden_dim)
# Eliminate time dimension.
logp = _sequential_gamma_gaussian_tensordot(
logp.expand(logp.batch_shape))
# Combine initial factor.
logp = gamma_gaussian_tensordot(self._init, logp, dims=self.hidden_dim)
# Posterior of the scale
gamma_dist = logp.event_logsumexp()
scale_post = Gamma(gamma_dist.concentration,
gamma_dist.rate,
validate_args=self._validate_args)
# Conditional of last state on unit scale
scale_tril = logp.precision.cholesky()
loc = logp.info_vec.unsqueeze(-1).cholesky_solve(scale_tril).squeeze(
-1)
mvn = MultivariateNormal(loc,
scale_tril=scale_tril,
validate_args=self._validate_args)
return scale_post, mvn