"""
Some basic Shannon information quantities.
"""
import numpy as np
from ..math import LogOperations
__all__ = (
"conditional_entropy",
"entropy",
"entropy_pmf",
"mutual_information",
)
def entropy_pmf(pmf):
"""
Returns the entropy of the probability mass function.
Assumption: Linearly distributed probabilities.
Parameters
----------
pmf : NumPy array, shape (k,) or (n,k)
Returns the entropy over the last index.
"""
pmf = np.asarray(pmf)
return np.nansum(-pmf * np.log2(pmf), axis=-1)
[docs]
def entropy(dist, rvs=None):
"""
Returns the entropy H[X] over the random variables in `rvs`.
If the distribution represents linear probabilities, then the entropy
is calculated with units of 'bits' (base-2). Otherwise, the entropy is
calculated in whatever base that matches the distribution's pmf.
Parameters
----------
dist : Distribution or float
The distribution from which the entropy is calculated. If a float,
then we calculate the binary entropy.
rvs : list, None
The indexes of the random variable used to calculate the entropy.
If None, then the entropy is calculated over all random variables.
This should remain `None` for scalar distributions.
Returns
-------
H : float
The entropy of the distribution.
"""
try:
# Handle binary entropy.
float(dist)
except TypeError:
pass
else:
# Assume linear probability for binary entropy.
import dit
dist = dit.Distribution([dist, 1 - dist])
d = dist.marginal(rvs) if rvs is not None else dist
pmf = d.pmf
if d.is_symbolic():
return _symbolic_entropy(pmf)
if d.is_log():
base = d.get_base(numerical=True)
terms = -(base**pmf) * pmf
else:
# Calculate entropy in bits.
log = LogOperations(2).log
terms = -pmf * log(pmf)
H = np.nansum(terms)
return H
def _symbolic_entropy(pmf):
"""Shannon entropy (base 2) of a symbolic pmf, as a sympy expression.
Uses the convention ``0 * log(0) = 0``: any probability that is literally
zero contributes nothing.
"""
import sympy
terms = []
for p in pmf:
p = sympy.sympify(p)
if p == 0:
continue
terms.append(-p * sympy.log(p, 2))
return sympy.Add(*terms)
[docs]
def conditional_entropy(dist, rvs_X, rvs_Y):
"""
Returns the conditional entropy of H[X|Y].
If the distribution represents linear probabilities, then the entropy
is calculated with units of 'bits' (base-2).
Parameters
----------
dist : Distribution
The distribution from which the conditional entropy is calculated.
rvs_X : list, None
The indexes of the random variables defining X.
rvs_Y : list, None
The indexes of the random variables defining Y.
Returns
-------
H_XgY : float
The conditional entropy H[X|Y].
"""
if set(rvs_X).issubset(rvs_Y):
# This is not necessary, but it makes the answer *exactly* zero,
# instead of 1e-12 or something smaller.
return 0.0
MI_XY = mutual_information(dist, rvs_X, rvs_Y)
H_X = entropy(dist, rvs_X)
H_XgY = H_X - MI_XY
return H_XgY