Mutual information between two variables

The problem, from first principles

03-probability's correlation measures whether two variables move together LINEARLY, and can be exactly zero even when two variables are strongly, deterministically related in a curved way (y = x^2, say). What you actually want, much of the time, is a measure of "how much does knowing X tell you about Y," with no assumption about linearity at all. That's mutual information: it's zero if and only if X and Y are genuinely, completely independent (knowing one tells you literally nothing about the other), and it grows the more knowing X narrows down what Y could be, no matter what shape that relationship takes.

The elegant part, and the reason this question sits at the very end of both the Probability and Information Theory tracks: mutual information is built entirely from tools you've already implemented, it is literally 03-kl-divergence's KL divergence, applied to compare the real joint distribution against what that same joint distribution WOULD look like if X and Y had no relationship at all.

From theory to code

Theory defines mutual information as KL(P(X,Y) || P(X)*P(Y)): the real joint distribution, compared against the "independent" joint you'd get by multiplying the two marginals together (03-probability/04-conditional-probability's marginal_x/marginal_y). Build that hypothetical independent joint via an outer product, then measure the KL divergence between the two, flattened into matching 1D distributions.

Implement mutual_information(joint, base=2.0) against that reasoning. The signature and docstring are already in the editor.

Constraints

• joint is a 2D array, joint[i, j] = P(X=i, Y=j), summing to 1.0.
• Must use marginal_x/marginal_y (already imported) to build the independent joint, not compute it any other way.
• Must reuse kl_divergence (already imported), not reimplement its formula.

Hints

Open one at a time. Each gives away a little more than the last.