CAEKL Mutual Information

The Chan-AlBashabsheh-Ebrahimi-Kaced-Liu mutual information [CABE+15] is one possible generalization of the Mutual Information.

\(\J{X_{0:n}}\) is the smallest \(\gamma\) such that:

\[\H{X_{0:n}} - \gamma = \sum_{C \in \mathcal{P}} \left[ \H{X_C} - \gamma \right]\]

for some non-trivial partition \(\mathcal{P}\) of \(\left\{0:n\right\}\). For example, the CAEKL mutual information for the xor distribution is \(\frac{1}{2}\), because the joint entropy is 2 bits, each of the three marginals is 1 bit, and \(2 - \frac{1}{2} = 3 (1 - \frac{1}{2})\).

In [1]: from dit.multivariate import caekl_mutual_information as J

In [2]: d = dit.example_dists.Xor()

In [3]: J(d)
Out[3]: 0.5

A more concrete way of defining the CAEKL mutual information is:

\[\J{X_{0:n}} = \min_{\mathcal{P} \in \Pi} ~ \operatorname{I}_\mathcal{P}\left[X_{0:n}\right]\]

where \(\operatorname{I}_\mathcal{P}\) is the Total Correlation of the partition:

\[\operatorname{I}_\mathcal{P}\left[X_{0:n}\right] = \sum_{C \in \mathcal{P}} \H{X_C} - \H{X_{0:n}}\]

and \(\Pi\) is the set of all non-trivial partitions of \(\left\{0:n\right\}\).

dit evaluates this minimum using the agglomerative principal-sequence-of-partitions algorithm of Chan and Liu [CL17] (minimum-norm-base subroutines per [CJK14]), which is exact and faster than enumerating partitions in practice.

API

caekl_mutual_information(dist, rvs=None, crvs=None)[source]

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.

Returns:

J – The CAEKL mutual information.

Return type:

float

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.