// Use the large variance approximation to sigma(m,v) = \int N(x;m,v) logistic(x)
private static void BigvProposal(double m, double v, out double mf, out double vf)
{
double v2 = v + Math.PI * Math.PI / 3.0;
double Z = MMath.NormalCdf(m / Math.Sqrt(v2));
double s1 = Math.Exp(Gaussian.GetLogProb(0, m, v2));
double s2 = -s1 * m / v2;
mf = m + v * s1 / Z;
double Ex2 = v + m * m + 2 * m * v * s1 / Z + v * v * s2 / Z;
vf = Ex2 - mf * mf;
}
// Helper function for NormalCdfRatioConFrac
private static double NormalCdfRatioConFracNumer(double x, double y, double r, double scale, double sqrtomr2, double diff, double Rdiff)
{
double delta = (1 + r) * (y - x) / sqrtomr2;
double numer;
if (Math.Abs(delta) > 0.5)
{
numer = scale * r * Rdiff;
double diffy = (y - r * x) / sqrtomr2;
if (scale == 1)
{
numer += MMath.NormalCdfRatio(diffy);
}
else // this assumes scale = N((y-rx)/sqrt(1-r^2);0,1)
{
numer += MMath.NormalCdf(diffy);
}
}
else
{
numer = scale * (NormalCdfRatioDiff(diff, delta) + (1 + r) * Rdiff);
}
return(numer);
}
/// <summary>
/// Computes the cumulative bivariate normal distribution.
/// </summary>
/// <param name="x">First upper limit.</param>
/// <param name="y">Second upper limit.</param>
/// <param name="r">Correlation coefficient.</param>
/// <returns><c>phi(x,y,r)</c></returns>
/// <remarks>
/// The cumulative bivariate normal distribution is defined as
/// <c>int_(-inf)^x int_(-inf)^y N([x;y],[0;0],[1 r; r 1]) dx dy</c>
/// where <c>N([x;y],[0;0],[1 r; r 1]) = exp(-0.5*(x^2+y^2-2*x*y*r)/(1-r^2))/(2*pi*sqrt(1-r^2))</c>.
/// </remarks>
public static double NormalCdf(double x, double y, double r)
{
if (Double.IsNegativeInfinity(x) || Double.IsNegativeInfinity(y))
{
return(0.0);
}
else if (Double.IsPositiveInfinity(x))
{
return(NormalCdf(y));
}
else if (Double.IsPositiveInfinity(y))
{
return(NormalCdf(x));
}
else if (r == 0)
{
return(NormalCdf(x) * NormalCdf(y));
}
else if (r == 1)
{
return(NormalCdf(Math.Min(x, y)));
}
else if (r == -1)
{
return(Math.Max(0.0, NormalCdf(x) + NormalCdf(y) - 1));
}
// at this point, both x and y are finite.
// swap to ensure |x| > |y|
if (Math.Abs(y) > Math.Abs(x))
{
double t = x;
x = y;
y = t;
}
double offset = 0;
double scale = 1;
// ensure x <= 0
if (x > 0)
{
// phi(x,y,r) = phi(inf,y,r) - phi(-x,y,-r)
offset = MMath.NormalCdf(y);
scale = -1;
x = -x;
r = -r;
}
// ensure r <= 0
if (r > 0)
{
// phi(x,y,r) = phi(x,inf,r) - phi(x,-y,-r)
offset += scale * MMath.NormalCdf(x);
scale *= -1;
y = -y;
r = -r;
}
double omr2 = (1 - r) * (1 + r); // more accurate than 1-r*r
double ymrx = (y - r * x) / Math.Sqrt(omr2);
double exponent;
double result = NormalCdf_Helper(x, y, r, omr2, ymrx, out exponent);
return(offset + scale * result * Math.Exp(exponent));
}
