def _m_step_nu(self, expectations, datas, inputs, masks, tags, optimizer,
num_iters, **kwargs):
K, D = self.K, self.D
E_taus = np.zeros(K)
E_logtaus = np.zeros(K)
weights = np.zeros(K)
for (
Ez,
_,
_,
), data, input, mask, tag in zip(expectations, datas, inputs, masks,
tags):
# nu: (K,) mus: (K, D) sigmas: (K, D) y: (T, D) -> w: (T, K, D)
mus = self._compute_mus(data, input, mask, tag)
sigmas = self._compute_sigmas(data, input, mask, tag)
nus = np.exp(self.inv_nus[:, None])
alpha = nus / 2 + 1 / 2
beta = nus / 2 + 1 / 2 * (data[:, None, :] - mus)**2 / sigmas
E_taus += np.sum(Ez[:, :, None] * alpha / beta, axis=(0, 2))
E_logtaus += np.sum(Ez[:, :, None] *
(digamma(alpha) - np.log(beta)),
axis=(0, 2))
weights += np.sum(Ez, axis=0) * D
E_taus /= weights
E_logtaus /= weights
for k in range(K):
self.inv_nus[k] = np.log(
generalized_newton_studentst_dof(E_taus[k], E_logtaus[k]))
def _m_step_nu(self, expectations, datas, inputs, masks, tags):
"""
The shape parameter nu determines a gamma prior. We have
tau_n ~ Gamma(nu/2, nu/2)
y_n ~ N(mu, sigma^2 / tau_n)
To update nu, we do EM and optimize the expected log likelihood using
a generalized Newton's method. See the notebook in doc/students_t for
complete details.
"""
K, D = self.K, self.D
# Compute the precisions w for each data point
E_taus = np.zeros(K)
E_logtaus = np.zeros(K)
weights = np.zeros(K)
for y, (Ez, _, _) in zip(datas, expectations):
# nu: (K,) mus: (K, D) sigmas: (K, D) y: (T, D) -> alpha/beta: (T, K, D)
nus = np.exp(self.inv_nus[:, None])
alpha = nus/2 + 1/2
beta = nus/2 + 1/2 * (y[:, None, :] - self.mus)**2 / np.exp(self.inv_sigmas)
E_taus += np.sum(Ez[:, :, None] * alpha / beta, axis=(0, 2))
E_logtaus += np.sum(Ez[:, :, None] * (digamma(alpha) - np.log(beta)), axis=(0, 2))
weights += np.sum(Ez, axis=0) * D
E_taus /= weights
E_logtaus /= weights
for k in range(K):
self.inv_nus[k] = np.log(generalized_newton_studentst_dof(E_taus[k], E_logtaus[k]))