def test_probability_rewrite():
X = Normal('X', 2, 3)
Y = Normal('Y', 3, 4)
Z = Poisson('Z', 4)
W = Poisson('W', 3)
x, y, w, z = symbols('x, y, w, z')
assert Variance(w).rewrite(Expectation) == 0
assert Variance(X).rewrite(Expectation) == Expectation(
X**2) - Expectation(X)**2
assert Variance(X, condition=Y).rewrite(Expectation) == Expectation(
X**2, Y) - Expectation(X, Y)**2
assert Variance(X, Y) != Expectation(X**2) - Expectation(X)**2
assert Variance(X + z).rewrite(Expectation) == Expectation(
(X + z)**2) - Expectation(X + z)**2
assert Variance(X * Y).rewrite(Expectation) == Expectation(
X**2 * Y**2) - Expectation(X * Y)**2
assert Covariance(
w, X).rewrite(Expectation) == -w * Expectation(X) + Expectation(w * X)
assert Covariance(X, Y).rewrite(Expectation) == Expectation(
X * Y) - Expectation(X) * Expectation(Y)
assert Covariance(X, Y, condition=W).rewrite(Expectation) == Expectation(
X * Y, W) - Expectation(X, W) * Expectation(Y, W)
w, x, z = symbols("W, x, z")
px = Probability(Eq(X, x))
pz = Probability(Eq(Z, z))
assert Expectation(X).rewrite(Probability) == Integral(
x * px, (x, -oo, oo))
assert Expectation(Z).rewrite(Probability) == Sum(z * pz, (z, 0, oo))
assert Variance(X).rewrite(Probability) == Integral(
x**2 * px, (x, -oo, oo)) - Integral(x * px, (x, -oo, oo))**2
assert Variance(Z).rewrite(Probability) == Sum(z**2 * pz,
(z, 0, oo)) - Sum(
z * pz, (z, 0, oo))**2
assert Covariance(w, X).rewrite(Probability) == \
-w*Integral(x*Probability(Eq(X, x)), (x, -oo, oo)) + Integral(w*x*Probability(Eq(X, x)), (x, -oo, oo))
# To test rewrite as sum function
assert Variance(X).rewrite(Sum) == Variance(X).rewrite(Integral)
assert Expectation(X).rewrite(Sum) == Expectation(X).rewrite(Integral)
assert Covariance(w, X).rewrite(Sum) == 0
assert Covariance(w, X).rewrite(Integral) == 0
assert Variance(X, condition=Y).rewrite(Probability) == Integral(x**2*Probability(Eq(X, x), Y), (x, -oo, oo)) - \
Integral(x*Probability(Eq(X, x), Y), (x, -oo, oo))**2
def test_literal_probability():
X = Normal('X', 2, 3)
Y = Normal('Y', 3, 4)
Z = Poisson('Z', 4)
W = Poisson('W', 3)
x = symbols('x', real=True)
y, w, z = symbols('y, w, z')
assert Probability(X > 0).evaluate_integral() == probability(X > 0)
assert Probability(X > x).evaluate_integral() == probability(X > x)
assert Probability(X > 0).rewrite(Integral).doit() == probability(X > 0)
assert Probability(X > x).rewrite(Integral).doit() == probability(X > x)
assert Expectation(X).evaluate_integral() == expectation(X)
assert Expectation(X).rewrite(Integral).doit() == expectation(X)
assert Expectation(X**2).evaluate_integral() == expectation(X**2)
assert Expectation(x * X).args == (x * X, )
assert Expectation(x * X).doit() == x * Expectation(X)
assert Expectation(2 * X + 3 * Y + z * X * Y).doit(
) == 2 * Expectation(X) + 3 * Expectation(Y) + z * Expectation(X * Y)
assert Expectation(2 * X + 3 * Y + z * X * Y).args == (2 * X + 3 * Y +
z * X * Y, )
assert Expectation(sin(X)) == Expectation(sin(X)).doit()
assert Expectation(
2 * x * sin(X) * Y + y * X**2 +
z * X * Y).doit() == 2 * x * Expectation(sin(X) * Y) + y * Expectation(
X**2) + z * Expectation(X * Y)
assert Variance(w).args == (w, )
assert Variance(w).doit() == 0
assert Variance(X).evaluate_integral() == Variance(X).rewrite(
Integral).doit() == variance(X)
assert Variance(X + z).args == (X + z, )
assert Variance(X + z).doit() == Variance(X)
assert Variance(X * Y).args == (Mul(X, Y), )
assert type(Variance(X * Y)) == Variance
assert Variance(z * X).doit() == z**2 * Variance(X)
assert Variance(
X + Y).doit() == Variance(X) + Variance(Y) + 2 * Covariance(X, Y)
assert Variance(X + Y + Z +
W).doit() == (Variance(X) + Variance(Y) + Variance(Z) +
Variance(W) + 2 * Covariance(X, Y) +
2 * Covariance(X, Z) + 2 * Covariance(X, W) +
2 * Covariance(Y, Z) + 2 * Covariance(Y, W) +
2 * Covariance(W, Z))
assert Variance(X**2).evaluate_integral() == variance(X**2)
assert Variance(X**2) == Variance(X**2)
assert Variance(x * X**2).doit() == x**2 * Variance(X**2)
assert Variance(sin(X)).args == (sin(X), )
assert Variance(sin(X)).doit() == Variance(sin(X))
assert Variance(x * sin(X)).doit() == x**2 * Variance(sin(X))
assert Covariance(w, z).args == (w, z)
assert Covariance(w, z).doit() == 0
assert Covariance(X, w).doit() == 0
assert Covariance(w, X).doit() == 0
assert Covariance(X, Y).args == (X, Y)
assert type(Covariance(X, Y)) == Covariance
assert Covariance(z * X + 3, Y).doit() == z * Covariance(X, Y)
assert Covariance(X, X).args == (X, X)
assert Covariance(X, X).doit() == Variance(X)
assert Covariance(z * X + 3, w * Y + 4).doit() == w * z * Covariance(X, Y)
assert Covariance(X, Y) == Covariance(Y, X)
assert Covariance(X + Y, Z + W).doit() == Covariance(W, X) + Covariance(
W, Y) + Covariance(X, Z) + Covariance(Y, Z)
assert Covariance(
x * X + y * Y, z * Z +
w * W).doit() == (x * w * Covariance(W, X) + w * y * Covariance(W, Y) +
x * z * Covariance(X, Z) + y * z * Covariance(Y, Z))
assert Covariance(x * X**2 + y * sin(Y), z * Y * Z**2 +
w * W).doit() == (w * x * Covariance(W, X**2) +
w * y * Covariance(sin(Y), W) +
x * z * Covariance(Y * Z**2, X**2) +
y * z * Covariance(Y * Z**2, sin(Y)))
assert Covariance(X, X**2).doit() == Covariance(X, X**2)
assert Covariance(X, sin(X)).doit() == Covariance(sin(X), X)
assert Covariance(X**2, sin(X) * Y).doit() == Covariance(sin(X) * Y, X**2)
def test_multivariate_expectation():
expr = Expectation(a)
assert expr == Expectation(a) == ExpectationMatrix(a)
assert expr.expand() == a
expr = Expectation(X)
assert expr == Expectation(X) == ExpectationMatrix(X)
assert expr.shape == (k, 1)
assert expr.rows == k
assert expr.cols == 1
assert isinstance(expr, ExpectationMatrix)
expr = Expectation(A * X + b)
assert expr == ExpectationMatrix(A * X + b)
assert expr.expand() == A * ExpectationMatrix(X) + b
assert isinstance(expr, ExpectationMatrix)
assert expr.shape == (k, 1)
expr = Expectation(m1 * X2)
assert expr.expand() == expr
expr = Expectation(A2 * m1 * B2 * X2)
assert expr.args[0].args == (A2, m1, B2, X2)
assert expr.expand() == A2 * ExpectationMatrix(m1 * B2 * X2)
expr = Expectation((X + Y) * (X - Y).T)
assert expr.expand() == ExpectationMatrix(X*X.T) - ExpectationMatrix(X*Y.T) +\
ExpectationMatrix(Y*X.T) - ExpectationMatrix(Y*Y.T)
expr = Expectation(A * X + B * Y)
assert expr.expand() == A * ExpectationMatrix(X) + B * ExpectationMatrix(Y)
assert Expectation(m1).doit() == Matrix([[1, 2 * j], [0, 0]])
x1 = Matrix([[Normal('N11', 11, 1),
Normal('N12', 12, 1)],
[Normal('N21', 21, 1),
Normal('N22', 22, 1)]])
x2 = Matrix([[Normal('M11', 1, 1),
Normal('M12', 2, 1)],
[Normal('M21', 3, 1),
Normal('M22', 4, 1)]])
assert Expectation(
Expectation(x1 + x2)).doit(deep=False) == ExpectationMatrix(x1 + x2)
assert Expectation(Expectation(x1 + x2)).doit() == Matrix([[12, 14],
[24, 26]])