def build_score_loss_entropy(self):
"""Build loss function. Its automatic differentiation
is a stochastic gradient of
.. math::
-ELBO = - ( E_{q(z; \lambda)} [ \log p(x, z) ]
+ H(q(z; \lambda)) )
based on the score function estimator. (Paisley et al., 2012)
It assumes the entropy is analytic.
Computed by sampling from :math:`q(z;\lambda)` and evaluating the
expectation using Monte Carlo sampling.
"""
x = self.data
z = self.variational.sample(self.n_samples)
q_log_prob = self.variational.log_prob(stop_gradient(z))
p_log_prob = self.model.log_prob(x, z)
q_entropy = self.variational.entropy()
self.loss = tf.reduce_mean(p_log_prob) + q_entropy
return -(tf.reduce_mean(q_log_prob * stop_gradient(p_log_prob)) +
q_entropy)
def build_score_loss_kl(self):
"""Build loss function. Its automatic differentiation
is a stochastic gradient of
.. math::
-ELBO = - ( E_{q(z; \lambda)} [ \log p(x | z) ]
+ KL(q(z; \lambda) || p(z)) )
based on the score function estimator. (Paisley et al., 2012)
It assumes the KL is analytic.
It assumes the prior is :math:`p(z) = \mathcal{N}(z; 0, 1)`.
Computed by sampling from :math:`q(z;\lambda)` and evaluating the
expectation using Monte Carlo sampling.
"""
x = self.data
z = self.variational.sample(self.n_samples)
q_log_prob = self.variational.log_prob(stop_gradient(z))
p_log_lik = self.model.log_lik(x, z)
mu = tf.pack([layer.loc for layer in self.variational.layers])
sigma = tf.pack([layer.scale for layer in self.variational.layers])
kl = kl_multivariate_normal(mu, sigma)
self.loss = tf.reduce_mean(p_log_lik) - kl
return -(tf.reduce_mean(q_log_prob * stop_gradient(p_log_lik)) - kl)
def build_score_loss(self):
"""Build loss function. Its automatic differentiation
is a stochastic gradient of
.. math::
-E_{q(z^1; \lambda), ..., q(z^K; \lambda)} [
\log 1/K \sum_{k=1}^K p(x, z^k)/q(z^k; \lambda) ]
based on the score function estimator. (Paisley et al., 2012)
Computed by sampling from :math:`q(z;\lambda)` and evaluating
the expectation using Monte Carlo sampling. Note there is a
difference between the number of samples to approximate the
expectations (`n_samples`) and the number of importance
samples to determine how many expectations (`K`).
"""
x = self.data
losses = []
for s in range(self.n_samples):
z = self.variational.sample(self.K)
p_log_prob = self.model.log_prob(x, z)
q_log_prob = self.variational.log_prob(stop_gradient(z))
log_w = p_log_prob - q_log_prob
losses += [log_mean_exp(log_w)]
losses = tf.pack(losses)
self.loss = tf.reduce_mean(losses)
return -tf.reduce_mean(q_log_prob * stop_gradient(losses))
def build_score_loss(self):
"""Build loss function. Its automatic differentiation
is a stochastic gradient of
.. math::
-E_{q(z^1; \lambda), ..., q(z^K; \lambda)} [
\log 1/K \sum_{k=1}^K p(x, z^k)/q(z^k; \lambda) ]
based on the score function estimator. (Paisley et al., 2012)
Computed by sampling from :math:`q(z;\lambda)` and evaluating
the expectation using Monte Carlo sampling. Note there is a
difference between the number of samples to approximate the
expectations (`n_samples`) and the number of importance
samples to determine how many expectations (`K`).
"""
x = self.data
losses = []
for s in range(self.n_samples):
z = self.variational.sample(self.K)
p_log_prob = self.model.log_prob(x, z)
q_log_prob = self.variational.log_prob(stop_gradient(z))
log_w = p_log_prob - q_log_prob
losses += [log_mean_exp(log_w)]
losses = tf.pack(losses)
self.loss = tf.reduce_mean(losses)
return -tf.reduce_mean(q_log_prob * stop_gradient(losses))
def build_loss(self):
"""Build loss function. Its automatic differentiation
is a stochastic gradient of
.. math::
KL( p(z |x) || q(z) )
=
E_{p(z | x)} [ \log p(z | x) - \log q(z; \lambda) ]
based on importance sampling.
Computed as
.. math::
1/B \sum_{b=1}^B [ w_{norm}(z^b; \lambda) *
(\log p(x, z^b) - \log q(z^b; \lambda) ]
where
.. math::
z^b \sim q(z^b; \lambda)
w_{norm}(z^b; \lambda) = w(z^b; \lambda) / \sum_{b=1}^B ( w(z^b; \lambda) )
w(z^b; \lambda) = p(x, z^b) / q(z^b; \lambda)
which gives a gradient
.. math::
- 1/B \sum_{b=1}^B
w_{norm}(z^b; \lambda) \partial_{\lambda} \log q(z^b; \lambda)
"""
x = self.data
z = self.variational.sample(self.n_samples)
# normalized importance weights
q_log_prob = self.variational.log_prob(stop_gradient(z))
log_w = self.model.log_prob(x, z) - q_log_prob
log_w_norm = log_w - log_sum_exp(log_w)
w_norm = tf.exp(log_w_norm)
self.loss = tf.reduce_mean(w_norm * log_w)
return -tf.reduce_mean(q_log_prob * stop_gradient(w_norm))
def build_score_loss(self):
"""Build loss function. Its automatic differentiation
is a stochastic gradient of
.. math::
-ELBO = -E_{q(z; \lambda)} [ \log p(x, z) - \log q(z; \lambda) ]
based on the score function estimator. (Paisley et al., 2012)
Computed by sampling from :math:`q(z;\lambda)` and evaluating the
expectation using Monte Carlo sampling.
"""
x = self.data
z = self.variational.sample(self.n_samples)
q_log_prob = self.variational.log_prob(stop_gradient(z))
losses = self.model.log_prob(x, z) - q_log_prob
self.loss = tf.reduce_mean(losses)
return -tf.reduce_mean(q_log_prob * stop_gradient(losses))