APPROX-MIC, :math:`MIC_e`, :math:`TIC_e` and parameters: a practical guide
==========================================================================

MIC_* is the population value of the maximal information coefficient (MIC)
MIC_e is a consistent estimator of MIC_*
MIC_e has better bias/variance properties than the heuristic algorithm Approx-MIC
The main goal of MIC_e is equitability rather than power against a null hypothesis of independence

One could imagine using TIC_e to filter out insignificant relationships, and then ranking the remaining ones using MIC_e

MIC_e: ``equicharacteristic matrix`` which we denote [M]. The difference between
[M] and the characteristic matrix M is as follows: while the k,l-th entry of M
is computed from the maximal achievable mutual information using any k-by-l
grid,  the k,l-th entry of [M] is computed from the maximal achievable mutual information
using any k-by-l grid that equipartitions the dimension with more rows/columns.

Both MIC and MIC e are consistent estimators of MIC. The difference between them is that
while MIC can currently be computed efficiently only via a heuristic approximation, MIC_e
can be computed exactly in polynomial time.

The complexity of the search procedure in MIC_e is O(n^{5\alpha/2}) (O(n) when \alpha=0.4)whereas the complexity of
the search procedure in the APPROX-MIC algorithm used to compute MIC is O(n^{4\alpha}).

alpha
-----
when using MIC_e (or, more likely, TIC_e) to generate tests for statistical
dependence one should use a lower value of \alpha, while if one is interested in
equitability, a larger \alpha is required.

As the relationship class of interest grows to include more complex structure
relative to sample size, the value of should be increased accordingly
