joint[i][j] is $P(X=i, Y=j)$. Return the mutual information between X and Y, in bits.

$$I(X;Y) = \sum_{i,j} p_{ij}\log_2\frac{p_{ij}}{p_i\,p_j}$$

Recover the marginals by summing the rows and columns. The result is 0 exactly when X and Y are independent, and it measures how much knowing one tells you about the other.

Input

[[0.25, 0.25], [0.25, 0.25]]

Output

np.float64(0.0)