/// <summary>
/// Gets the integral of the product of two Gaussians.
/// </summary>
/// <param name="that"></param>
/// <remarks>
/// <c>this = N(x;m1,v1)</c>.
/// <c>that = N(x;m2,v2)</c>.
/// <c>int_(-infinity)^(infinity) N(x;m1,v1) N(x;m2,v2) dx = N(m1; m2, v1+v2)</c>.
/// When improper, the density is redefined to be <c>exp(-0.5*x^2*(1/v) + x*(m/v))</c>,
/// i.e. we drop the terms <c>exp(-m^2/(2v))/sqrt(2*pi*v)</c>.
/// </remarks>
/// <returns>log(N(m1;m2,v1+v2)).</returns>
public double GetLogAverageOf(Gaussian that)
{
if (IsPointMass)
{
return(that.GetLogProb(Point));
}
else if (that.IsPointMass)
{
return(GetLogProb(that.Point));
}
else
{
// neither this nor that is a point mass.
// int_x N(x;m1,v1) N(x;m2,v2) dx = N(m1;m2,v1+v2)
// (m1-m2)^2/(v1+v2) = (m1-m2)^2/(v1*v2*product.Prec)
// (m1-m2)^2/(v1*v2) = (m1/(v1*v2) - m2/(v1*v2))^2 *v1*v2
Gaussian product = this * that;
//if (!product.IsProper()) throw new ArgumentException("The product is improper.");
return(product.GetLogNormalizer() - this.GetLogNormalizer() - that.GetLogNormalizer());
}
}
/// <summary>
/// Get the integral of this distribution times another distribution raised to a power.
/// </summary>
/// <param name="that"></param>
/// <param name="power"></param>
/// <returns></returns>
public double GetLogAverageOfPower(Gaussian that, double power)
{
if (IsPointMass)
{
return(power * that.GetLogProb(Point));
}
else if (that.IsPointMass)
{
if (power < 0)
{
throw new DivideByZeroException("The exponent is negative and the distribution is a point mass");
}
return(this.GetLogProb(that.Point));
}
else
{
var product = this * (that ^ power);
return(product.GetLogNormalizer() - this.GetLogNormalizer() - power * that.GetLogNormalizer());
}
}