def kl_normal1_normal1(mean1, var1, mean2, var2, eps=0.0):
"""
Compute closed-form solution to the KL-divergence between two Gaussians parameterized
with diagonal variance.
Parameters
----------
mean1 : torch tensor
Mean of the q Gaussian.
var1 : torch tensor
Variance of the q Gaussian.
mean2 : torch tensor
Mean of the p Gaussian.
var2 : torch tensor
Variance of the p Gaussian.
eps : float
Small number added to variances to avoid NaNs.
Returns
-------
torch tensor
Element-wise KL-divergence, this has to be summed when the Gaussian distributions are multi-variate.
See also
--------
kl_normal2_normal2 : using log variance parameterization
"""
var1 += eps
var2 += eps
return 0.5 * T.log(var2 / var1) + (var1 +
(mean1 - mean2)**2) / (2 * var2) - 0.5
def log_normal(x, mean, std, eps=0.0):
"""
Compute log pdf of a Gaussian distribution with diagonal covariance, at values x.
Variance is parameterized as standard deviation.
.. math:: \log p(x) = \log \mathcal{N}(x; \mu, \sigma^2I)
Parameters
----------
x : torch tensor
Values at which to evaluate pdf.
mean : torch tensor
Mean of the Gaussian distribution.
std : torch tensor
Standard deviation of the diagonal covariance Gaussian.
eps : float
Small number added to standard deviation to avoid NaNs.
Returns
-------
torch tensor
Element-wise log probability, this has to be summed for multi-variate distributions.
See also
--------
log_normal1 : using variance parameterization
log_normal2 : using log variance parameterization
"""
std += eps
return c - T.log(T.abs_(std)) - (x - mean)**2 / (2 * std**2)
def kl_normal1_stdnormal(mean, var, eps=0.0):
"""
Closed-form solution of the KL-divergence between a Gaussian parameterized
with diagonal variance and a standard Gaussian.
.. math::
D_{KL}[\mathcal{N}(\mu, \sigma^2 I) || \mathcal{N}(0, I)]
Parameters
----------
mean : torch tensor
Mean of the diagonal covariance Gaussian.
var : torch tensor
Variance of the diagonal covariance Gaussian.
eps : float
Small number added to variance to avoid NaNs.
Returns
-------
torch tensor
Element-wise KL-divergence, this has to be summed when the Gaussian distributions are multi-variate.
See also
--------
kl_normal2_stdnormal : using log variance parameterization
"""
var += eps
return -0.5 * (1 + T.log(var) - mean**2 - var)