def normal_lccdf(mu, sigma, x):
z = (x - mu) / sigma
return aet.switch(
aet.gt(z, 1.0),
aet.log(aet.erfcx(z / aet.sqrt(2.0)) / 2.0) - aet.sqr(z) / 2.0,
aet.log1p(-aet.erfc(-z / aet.sqrt(2.0)) / 2.0),
)
def normal_lcdf(mu, sigma, x):
"""Compute the log of the cumulative density function of the normal."""
z = (x - mu) / sigma
return aet.switch(
aet.lt(z, -1.0),
aet.log(aet.erfcx(-z / aet.sqrt(2.0)) / 2.0) - aet.sqr(z) / 2.0,
aet.log1p(-aet.erfc(z / aet.sqrt(2.0)) / 2.0),
)
def log_diff_normal_cdf(mu, sigma, x, y):
"""
Compute :math:`\\log(\\Phi(\frac{x - \\mu}{\\sigma}) - \\Phi(\frac{y - \\mu}{\\sigma}))` safely in log space.
Parameters
----------
mu: float
mean
sigma: float
std
x: float
y: float
must be strictly less than x.
Returns
-------
log (\\Phi(x) - \\Phi(y))
"""
x = (x - mu) / sigma / aet.sqrt(2.0)
y = (y - mu) / sigma / aet.sqrt(2.0)
# To stabilize the computation, consider these three regions:
# 1) x > y > 0 => Use erf(x) = 1 - e^{-x^2} erfcx(x) and erf(y) =1 - e^{-y^2} erfcx(y)
# 2) 0 > x > y => Use erf(x) = e^{-x^2} erfcx(-x) and erf(y) = e^{-y^2} erfcx(-y)
# 3) x > 0 > y => Naive formula log( (erf(x) - erf(y)) / 2 ) works fine.
return aet.log(0.5) + aet.switch(
aet.gt(y, 0),
-aet.square(y) + aet.log(
aet.erfcx(y) -
aet.exp(aet.square(y) - aet.square(x)) * aet.erfcx(x)),
aet.switch(
aet.lt(x, 0), # 0 > x > y
-aet.square(x) + aet.log(
aet.erfcx(-x) -
aet.exp(aet.square(x) - aet.square(y)) * aet.erfcx(-y)),
aet.log(aet.erf(x) - aet.erf(y)), # x >0 > y
),
)
