Leave $ P ( mathbf {X}, Y) $ be a *discreet* trivariate distribution where $ mathbf {X} $ it is a bivariate random variable $ mathbf {X} = { mathbf {x} _1, dots, mathbf {x} _I } = {(x_1 ^ 1, x_1 ^ 2), dots, (x_I ^ 1, x_I ^ 2) } $Y $ Y = {y_1, dots dots y_J } $ A univariate variable.

We know

$$

I (X ^ n; Y) triangleq sum_ {i, j} p ( mathbf {x} _i, y_j) cdot frac {p (x_i ^ n, y_j)} {p (x_i ^ n) cdot p (y_j)} ge 0,

qquad text {para} quad n = 1,2

$$

Now let's build $ bar {n} (i) $, which, for each $ i = 1, points, I $, can take the courage $ 1 $ or $ 2 $. We can define

$$

I (X ^ bar {n}; Y) triangleq sum_ {i, j} p ( mathbf {x} _i, y_j) cdot frac {p (x_i ^ bar {n}, y_j)} {p (x_i ^ bar {n}) cdot p (y_j)} ge 0,

$$

**Question:** Can we say that $ I (X ^ bar {n}; Y) ge 0 $ For any $ bar {n} $?