Maximum Correlation

The Hirschfeld–Gebelein–Rényi maximal correlation of a pair of random variables is

\[\rho_m(X:Y) = \max_{f,g} \mathbb{E}[f(X)g(Y)]\]

subject to zero-mean, unit-variance \(f\) and \(g\). It is 1 if the variables are a deterministic function of each other (a giant bit) and 0 if they are independent.

In [1]: from dit.divergences import maximum_correlation

In [2]: from dit.example_dists import giant_bit, Xor

In [3]: abs(maximum_correlation(giant_bit(2, 2), [[0], [1]]) - 1.0) < 1e-8
Out[3]: True

In [4]: abs(maximum_correlation(Xor(), [[0], [1]])) < 1e-10
Out[4]: True

API

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

Compute the (conditional) maximum or Renyi correlation between two variables:

\[\rho^{*} = \max_{f, g} \rho(f(X,Z), g(Y,Z) | Z)\]
Parameters:

  • dist (Distribution) – The distribution for which the maximum correlation is to computed.

  • rvs (list, None; len(rvs) == 2) – A list of lists. Each inner list specifies the indexes of the random variables for which the maximum correlation is to be computed. If None, then all random variables are used, 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:

rho_max – The conditional maximum correlation between rvs given crvs.

Return type:

float; -1 <= rho_max <= 1