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 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_issue_12283():
x = symbols('x')
X = RandomSymbol(x)
Y = RandomSymbol('Y')
Z = RandomMatrixSymbol('Z', 2, 3)
RI = RandomIndexedSymbol(Indexed('RI', 3))
assert pspace(Z) == PSpace()
assert pspace(RI) == PSpace()
assert pspace(X) == PSpace()
assert E(X) == Expectation(X)
assert P(Y > 3) == Probability(Y > 3)
assert variance(X) == Variance(X)
assert variance(RI) == Variance(RI)
assert covariance(X, Y) == Covariance(X, Y)
assert covariance(X, Z) == Covariance(X, Z)
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 ** 2) - Expectation(X) ** 2
assert Variance(X * Y).rewrite(Expectation) == Expectation(X ** 2 * Y ** 2) - Expectation(X * Y) ** 2
assert Covariance(w, X).rewrite(Expectation) == 0
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)
def test_literal_probability():
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 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_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).expand() == x * Expectation(X)
assert Expectation(2 * X + 3 * Y + z * X * Y).expand(
) == 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)).expand()
assert Expectation(2*x*sin(X)*Y + y*X**2 + z*X*Y).expand() == 2*x*Expectation(sin(X)*Y) \
+ y*Expectation(X**2) + z*Expectation(X*Y)
assert Expectation(X + Y).expand() == Expectation(X) + Expectation(Y)
assert Expectation(
(X + Y) * (X - Y)).expand() == Expectation(X**2) - Expectation(Y**2)
assert Expectation((X + Y) * (X - Y)).expand().doit() == -12
assert Expectation(X + Y, evaluate=True).doit() == 5
assert Expectation(X + Expectation(Y)).doit() == 5
assert Expectation(X + Expectation(Y)).doit(
deep=False) == 2 + Expectation(Expectation(Y))
assert Expectation(X + Expectation(Y + Expectation(2*X))).doit(deep=False) == 2 \
+ Expectation(Expectation(Y + Expectation(2*X)))
assert Expectation(X + Expectation(Y + Expectation(2 * X))).doit() == 9
assert Expectation(Expectation(2 * X)).doit() == 4
assert Expectation(Expectation(2 * X)).doit(deep=False) == Expectation(2 *
X)
assert Expectation(
4 * Expectation(2 * X)).doit(deep=False) == 4 * Expectation(2 * X)
assert Expectation((X + Y)**3).expand() == 3*Expectation(X*Y**2) +\
3*Expectation(X**2*Y) + Expectation(X**3) + Expectation(Y**3)
assert Expectation((X - Y)**3).expand() == 3*Expectation(X*Y**2) -\
3*Expectation(X**2*Y) + Expectation(X**3) - Expectation(Y**3)
assert Expectation((X - Y)**2).expand() == -2*Expectation(X*Y) +\
Expectation(X**2) + Expectation(Y**2)
assert Variance(w).args == (w, )
assert Variance(w).expand() == 0
assert Variance(X).evaluate_integral() == Variance(X).rewrite(
Integral).doit() == variance(X)
assert Variance(X + z).args == (X + z, )
assert Variance(X + z).expand() == Variance(X)
assert Variance(X * Y).args == (Mul(X, Y), )
assert type(Variance(X * Y)) == Variance
assert Variance(z * X).expand() == z**2 * Variance(X)
assert Variance(
X + Y).expand() == Variance(X) + Variance(Y) + 2 * Covariance(X, Y)
assert Variance(X + Y + Z + W).expand() == (
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 unchanged(Variance, X**2)
assert Variance(x * X**2).expand() == x**2 * Variance(X**2)
assert Variance(sin(X)).args == (sin(X), )
assert Variance(sin(X)).expand() == Variance(sin(X))
assert Variance(x * sin(X)).expand() == x**2 * Variance(sin(X))
assert Covariance(w, z).args == (w, z)
assert Covariance(w, z).expand() == 0
assert Covariance(X, w).expand() == 0
assert Covariance(w, X).expand() == 0
assert Covariance(X, Y).args == (X, Y)
assert type(Covariance(X, Y)) == Covariance
assert Covariance(z * X + 3, Y).expand() == z * Covariance(X, Y)
assert Covariance(X, X).args == (X, X)
assert Covariance(X, X).expand() == Variance(X)
assert Covariance(z * X + 3,
w * Y + 4).expand() == w * z * Covariance(X, Y)
assert Covariance(X, Y) == Covariance(Y, X)
assert Covariance(X + Y, Z + W).expand() == Covariance(W, X) + Covariance(
W, Y) + Covariance(X, Z) + Covariance(Y, Z)
assert Covariance(x * X + y * Y,
z * Z + w * W).expand() == (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).expand() == (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).expand() == Covariance(X, X**2)
assert Covariance(X, sin(X)).expand() == Covariance(sin(X), X)
assert Covariance(X**2,
sin(X) * Y).expand() == Covariance(sin(X) * Y, X**2)
assert Covariance(w, X).evaluate_integral() == 0
def test_multivariate_variance():
raises(ShapeError, lambda: Variance(A))
expr = Variance(a) # type: VarianceMatrix
assert expr == Variance(a) == VarianceMatrix(a)
assert expr.expand() == ZeroMatrix(k, k)
expr = Variance(a.T)
assert expr == Variance(a.T) == VarianceMatrix(a.T)
assert expr.expand() == ZeroMatrix(k, k)
expr = Variance(X)
assert expr == Variance(X) == VarianceMatrix(X)
assert expr.shape == (k, k)
assert expr.rows == k
assert expr.cols == k
assert isinstance(expr, VarianceMatrix)
expr = Variance(A * X)
assert expr == VarianceMatrix(A * X)
assert expr.expand() == A * VarianceMatrix(X) * A.T
assert isinstance(expr, VarianceMatrix)
assert expr.shape == (k, k)
expr = Variance(A * B * X)
assert expr.expand() == A * B * VarianceMatrix(X) * B.T * A.T
expr = Variance(m1 * X2)
assert expr.expand() == expr
expr = Variance(A2 * m1 * B2 * X2)
assert expr.args[0].args == (A2, m1, B2, X2)
assert expr.expand() == expr
expr = Variance(A * X + B * Y)
assert expr.expand() == 2*A*CrossCovarianceMatrix(X, Y)*B.T +\
A*VarianceMatrix(X)*A.T + B*VarianceMatrix(Y)*B.T
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)