public void TransformedBetaMoments()
{
// For x ~ B(a,b), kth cumulant of \ln\left(\frac{x}{1-x}\right) = \psi^{(k-1)}(a) \pm \psi^{(k-1)}(b)
BetaDistribution D = new BetaDistribution(2.0, 3.0);
double m = D.ExpectationValue(x => Math.Log(x / (1.0 - x)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(m, AdvancedMath.Psi(D.Alpha) - AdvancedMath.Psi(D.Beta)));
double c2 = D.ExpectationValue(x => MoreMath.Pow(Math.Log(x / (1.0 - x)) - m, 2));
Assert.IsTrue(TestUtilities.IsNearlyEqual(c2, AdvancedMath.Psi(1, D.Alpha) + AdvancedMath.Psi(1, D.Beta)));
double c3 = D.ExpectationValue(x => MoreMath.Pow(Math.Log(x / (1.0 - x)) - m, 3));
Assert.IsTrue(TestUtilities.IsNearlyEqual(c3, AdvancedMath.Psi(2, D.Alpha) - AdvancedMath.Psi(2, D.Beta)));
double c4 = D.ExpectationValue(x => MoreMath.Pow(Math.Log(x / (1.0 - x)) - m, 4));
Assert.IsTrue(TestUtilities.IsNearlyEqual(c4 - 3.0 * c2 * c2, AdvancedMath.Psi(3, D.Alpha) + AdvancedMath.Psi(3, D.Beta)));
}
