def test_gaussian_with_condition_equal_to_y(self) -> None:
# MI(X;Y | Y) should be equal to 0
rng = np.random.default_rng(4)
cov = np.array([[1, 0.6], [0.6, 1]])
data = rng.multivariate_normal([0, 0], cov, size=314)
x = data[:, 0]
y = data[:, 1]
actual = _estimate_conditional_mi(x, y, y, k=4)
self.assertAlmostEqual(actual, 0.0, delta=0.001)
def test_three_gaussians(self) -> None:
# First example in doi:10.1103/PhysRevLett.99.204101,
# we know the analytic expression for conditional MI of a three-
# dimensional Gaussian random variable. Here, the covariance matrix
# is not particularly interesting. The expected CMI expression
# contains determinants of submatrices.
rng = np.random.default_rng(5)
cov = np.array([[1, 1, 1], [1, 4, 1], [1, 1, 9]])
data = rng.multivariate_normal([0, 0, 0], cov, size=1000)
actual = _estimate_conditional_mi(data[:, 0], data[:, 1], data[:, 2])
expected = 0.5 * (log(8) + log(35) - log(9) - log(24))
self.assertAlmostEqual(actual, expected, delta=0.015)
def test_gaussian_with_independent_condition(self) -> None:
# In this case, the results should be same as in ordinary MI
cases = [ (0.5, 200, 3, 0.03),
(0.75, 400, 3, 0.01),
(-0.9, 4000, 5, 0.03), ]
for (rho, n, k, delta) in cases:
with self.subTest(rho=rho, n=n, k=k):
rng = np.random.default_rng(0)
cov = np.array([[1, rho], [rho, 1]])
data = rng.multivariate_normal([0, 0], cov, size=n)
x = data[:,0]
y = data[:,1]
cond = rng.uniform(0, 1, size=n)
actual = _estimate_conditional_mi(x, y, cond, k=k)
expected = -0.5 * log(1 - rho**2)
self.assertAlmostEqual(actual, expected, delta=delta)
def test_four_gaussians(self) -> None:
# As above, but now the condition is two-dimensional.
# The covariance matrix is defined by transforming a standard normal
# distribution (u1, u2, u3, u4) as follows:
# x = u1,
# y = u2 + u3 + 2*u4,
# z1 = 2*u1 + u3,
# z2 = u1 + u4.
# Unconditionally, x and y are independent, but conditionally they aren't.
rng = np.random.default_rng(25)
cov = np.array([[1, 0, 2, 1], [0, 6, 1, 2], [2, 1, 5, 2], [1, 2, 2,
2]])
# The data needs to be normalized for estimation accuracy,
# and the sample size must be quite large
data = rng.multivariate_normal([0, 0, 0, 0], cov, size=8000)
data = data / np.sqrt(np.var(data, axis=0))
actual = _estimate_conditional_mi(data[:, 0], data[:, 1], data[:, 2:])
expected = 0.64964
self.assertAlmostEqual(actual, expected, delta=0.04)