def _quantile(self, p, loc=None, scale=None, low=None, high=None):
loc, scale, low, high = self._loc_scale_low_high(loc, scale, low, high)
std_low, std_high = self._standardized_low_and_high(low=low,
high=high,
loc=loc,
scale=scale)
# Use the sum of tangents formula.
# First, the quantile of the cauchy distribution is tan(pi * (x - 0.5)).
# and the cdf of the cauchy distribution is 0.5 + arctan(x) / np.pi
# WLOG, we will assume loc = 0 , scale = 1 (these can be taken in to account
# by rescaling and shifting low and high, and then scaling the output).
# We would like to compute quantile(p * (cdf(high) - cdf(low)) + cdf(low))
# This is the same as:
# tan(pi * (cdf(low) + (cdf(high) - cdf(low)) * p - 0.5))
# Let a = pi * (cdf(low) - 0.5), b = pi * (cdf(high) - cdf(low)) * u
# By using the formula for the cdf we have:
# a = arctan(low), b = arctan_difference(high, low) * u
# Thus the quantile is now tan(a + b).
# By appealing to the sum of tangents formula we have:
# tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a) * tan(b)) =
# (low + tan(b)) / (1 - low * tan(b))
# Thus for a 'standard' truncated cauchy we have the quantile as:
# quantile(p) = (low + tan(b)) / (1 - low * tan(b)) where
# b = arctan_difference(high, low) * p.
tanb = tf.math.tan(tfp_math.atan_difference(std_high, std_low) * p)
x = (std_low + tanb) / (1 - std_low * tanb)
return x * scale + loc
def _mean(self):
loc, scale, low, high = self._loc_scale_low_high()
std_low, std_high = self._standardized_low_and_high(
low=low, high=high, loc=loc, scale=scale)
# Formula from David Olive, "Applied Robust Statistics" --
# see http://parker.ad.siu.edu/Olive/ch4.pdf .
t = (tf.math.log1p(tf.math.square(std_high))
- tf.math.log1p(tf.math.square(std_low)))
t = t / (2 * tfp_math.atan_difference(std_high, std_low))
return loc + scale * t
def _variance(self):
loc, scale, low, high = self._loc_scale_low_high()
std_low, std_high = self._standardized_low_and_high(
low=low, high=high, loc=loc, scale=scale)
# Formula from David Olive, "Applied Robust Statistics" --
# see http://parker.ad.siu.edu/Olive/ch4.pdf .
atan_diff = tfp_math.atan_difference(std_high, std_low)
t = (std_high - std_low - atan_diff) / atan_diff
std_mean = ((tf.math.log1p(tf.math.square(std_high))
- tf.math.log1p(tf.math.square(std_low))) / (2 * atan_diff))
return tf.math.square(scale) * (t - tf.math.square(std_mean))
def _cauchy_cdf_diff(x, y):
return tfp_math.atan_difference(x, y) / np.pi
