Source code for dit.multivariate.caekl_mutual_information

"""
The CAEKL mutual information, as define [Chan, Chung, et al. "Multivariate Mutual Information Inspired by Secret-Key Agreement." Proceedings of the IEEE 103.10 (2015): 1883-1913].
"""

from __future__ import division

from ..helpers import normalize_rvs
from ..utils import partitions, unitful
from .entropy import entropy


__all__ = [
    'caekl_mutual_information',
]


@unitful
def caekl_mutual_information(dist, rvs=None, crvs=None, rv_mode=None):
    """
    Calculates the Chan-AlBashabsheh-Ebrahimi-Kaced-Liu mutual information.

    Parameters
    ----------
    dist : Distribution
        The distribution from which the CAEKL mutual information is calculated.
    rvs : list, None
        A list of lists. Each inner list specifies the indexes of the random
        variables used to calculate the total correlation. If None, then the
        total correlation is calculated over all random variables, which is
        equivalent to passing `rvs=dist.rvs`.
    crvs : list, None
        A single list of indexes specifying the random variables to condition
        on. If None, then no variables are conditioned on.
    rv_mode : str, None
        Specifies how to interpret `rvs` and `crvs`. Valid options are:
        {'indices', 'names'}. If equal to 'indices', then the elements of
        `crvs` and `rvs` are interpreted as random variable indices. If equal
        to 'names', the the elements are interpreted as random variable names.
        If `None`, then the value of `dist._rv_mode` is consulted, which
        defaults to 'indices'.

    Returns
    -------
    J : float
        The CAEKL mutual information.

    Examples
    --------
    >>> d = dit.example_dists.Xor()
    >>> dit.multivariate.caekl_mutual_information(d)
    0.5
    >>> dit.multivariate.caekl_mutual_information(d, rvs=[[0], [1]])
    0.0

    Raises
    ------
    ditException
        Raised if `dist` is not a joint distribution or if `rvs` or `crvs`
        contain non-existant random variables.
    """
    rvs, crvs, rv_mode = normalize_rvs(dist, rvs, crvs, rv_mode)

    H = entropy(dist, rvs, crvs, rv_mode)

    def I_P(part):
        a = sum(entropy(dist, rvs=p, crvs=crvs, rv_mode=rv_mode) for p in part)
        return (a - H)/(len(part) - 1)

    J = min(I_P(p) for p in partitions(map(tuple, rvs)) if len(p) > 1)

    return J