/// <summary>
/// Computes GammaLower(a, r*u) - GammaLower(a, r*l) to high accuracy.
/// </summary>
/// <param name="shape"></param>
/// <param name="rate"></param>
/// <param name="lowerBound"></param>
/// <param name="upperBound"></param>
/// <param name="regularized"></param>
/// <returns></returns>
public static double GammaProbBetween(double shape, double rate, double lowerBound, double upperBound, bool regularized = true)
{
double rl = rate * lowerBound;
// Use the criterion from Gautschi (1979) to determine whether GammaLower(a,x) or GammaUpper(a,x) is smaller.
bool lowerIsSmaller;
if (rl > 0.25)
{
lowerIsSmaller = (shape > rl + 0.25);
}
else
{
lowerIsSmaller = (shape > -MMath.Ln2 / Math.Log(rl));
}
if (!lowerIsSmaller)
{
double logl = Math.Log(lowerBound);
if (rate * upperBound < 1e-16 && shape < -1e-16 / (Math.Log(rate) + logl))
{
double logu = Math.Log(upperBound);
return(shape * (logu - logl));
}
else
{
// This is inaccurate when lowerBound is close to upperBound. In that case, use a Taylor expansion of lowerBound around upperBound.
return(MMath.GammaUpper(shape, rl, regularized) - MMath.GammaUpper(shape, rate * upperBound, regularized));
}
}
else
{
double diff = MMath.GammaLower(shape, rate * upperBound) - MMath.GammaLower(shape, rl);
return(regularized ? diff : (MMath.Gamma(shape) * diff));
}
}
/// <summary>
/// Computes <c>GammaUpper(s,x)/(x^(s-1)*exp(-x)) - 1</c> to high accuracy
/// </summary>
/// <param name="s"></param>
/// <param name="x">A real number gt;= 45 and gt; <paramref name="s"/>/0.99</param>
/// <param name="regularized"></param>
/// <returns></returns>
public static double GammaUpperRatio(double s, double x, bool regularized = true)
{
if (s >= x * 0.99)
{
throw new ArgumentOutOfRangeException(nameof(s), s, "s >= x*0.99");
}
if (x < 45)
{
throw new ArgumentOutOfRangeException(nameof(x), x, "x < 45");
}
double term = (s - 1) / x;
double sum = term;
for (int i = 2; i < 1000; i++)
{
term *= (s - i) / x;
double oldSum = sum;
sum += term;
if (MMath.AreEqual(sum, oldSum))
{
return(regularized ? sum / MMath.Gamma(s) : sum);
}
}
throw new Exception($"GammaUpperRatio not converging for s={s:g17}, x={x:g17}, regularized={regularized}");
}
private static void VarianceGammaTimesGaussianMoments(double a, double m, double v, out double sum, out double moment1, out double moment2)
{
int n = 1000000;
double lowerBound = -20;
double upperBound = 20;
double range = upperBound - lowerBound;
sum = 0;
double sumx = 0, sumx2 = 0;
for (int i = 0; i < n; i++)
{
double x = range * i / (n - 1) + lowerBound;
double absx = System.Math.Abs(x);
double diff = x - m;
double logp = -0.5 * diff * diff / v - absx;
double p = System.Math.Exp(logp);
if (a == 1)
{ // do nothing
}
else if (a == 2)
{
p *= 1 + absx;
}
else if (a == 3)
{
p *= 3 + 3 * absx + absx * absx;
}
else if (a == 4)
{
p *= 15 + 15 * absx + 6 * absx * absx + absx * absx * absx;
}
else
{
throw new ArgumentException("a is not in {1,2,3,4}");
}
sum += p;
sumx += x * p;
sumx2 += x * x * p;
}
moment1 = sumx / sum;
moment2 = sumx2 / sum;
sum /= MMath.Gamma(a) * System.Math.Pow(2, a);
sum /= MMath.Sqrt2PI * System.Math.Sqrt(v);
double inc = range / (n - 1);
sum *= inc;
}
public static double[] xBesselKDerivativesAt0(int n, double a)
{
double[] derivs = new double[n + 1];
derivs[0] = MMath.Gamma(System.Math.Abs(a)) * System.Math.Pow(2, a - 0.5) / System.Math.Sqrt(System.Math.PI);
if (n > 0)
{
derivs[1] = derivs[0];
if (n > 1)
{
double[] derivs2 = xBesselKDerivativesAt0(n - 2, a - 1);
for (int i = 0; i < n - 2; i++)
{
derivs[i + 2] = derivs[i + 1] - (i + 1) * derivs2[i];
}
}
}
return(derivs);
}
public void VarianceGammaTimesGaussianIntegralTest()
{
double logZ, mu, vu;
GaussianFromMeanAndVarianceOp.LaplacianTimesGaussianMoments(-100, 1, out logZ, out mu, out vu);
Console.WriteLine(logZ);
//GaussianFromMeanAndVarianceOp.VarianceGammaTimesGaussianMoments5(1, 1000, 1, out mu, out vu);
Console.WriteLine(mu);
Console.WriteLine(vu);
double[][] vgmoments = GaussianFromMeanAndVarianceOp.NormalVGMomentRatios(10, 1, -6, 1);
for (int i = 0; i < 10; i++)
{
double f = MMath.NormalCdfMomentRatio(i, -6) * MMath.Gamma(i + 1);
Console.WriteLine("R({0}) = {1}, Zp = {2}", i, f, vgmoments[0][i]);
}
// true values computed by tex/factors/matlab/test_variance_gamma2.m
double scale = 2 * System.Math.Exp(-Gaussian.GetLogProb(2 / System.Math.Sqrt(3), 0, 1));
Assert.True(MMath.AbsDiff(0.117700554409044, GaussianFromMeanAndVarianceOp.VarianceGammaTimesGaussianIntegral(1, 2, 3) / scale, 1e-20) < 1e-5);
Assert.True(MMath.AbsDiff(0.112933034747473, GaussianFromMeanAndVarianceOp.VarianceGammaTimesGaussianIntegral(2, 2, 3) / scale, 1e-20) < 1e-5);
Assert.True(MMath.AbsDiff(0.117331854901251, GaussianFromMeanAndVarianceOp.VarianceGammaTimesGaussianIntegral(1.1, 2, 3) / scale, 1e-20) < 1e-5);
Assert.True(MMath.AbsDiff(0.115563913123152, GaussianFromMeanAndVarianceOp.VarianceGammaTimesGaussianIntegral(1.5, 2, 3) / scale, 1e-20) < 2e-5);
scale = 2 * System.Math.Exp(-Gaussian.GetLogProb(20 / System.Math.Sqrt(0.3), 0, 1));
Assert.True(MMath.AbsDiff(1.197359429038085e-009, GaussianFromMeanAndVarianceOp.VarianceGammaTimesGaussianIntegral(1, 20, 0.3) / scale, 1e-20) < 1e-5);
Assert.True(MMath.AbsDiff(1.239267009054433e-008, GaussianFromMeanAndVarianceOp.VarianceGammaTimesGaussianIntegral(2, 20, 0.3) / scale, 1e-20) < 1e-5);
Assert.True(MMath.AbsDiff(1.586340098271600e-009, GaussianFromMeanAndVarianceOp.VarianceGammaTimesGaussianIntegral(1.1, 20, 0.3) / scale, 1e-20) < 1e-4);
Assert.True(MMath.AbsDiff(4.319412089896069e-009, GaussianFromMeanAndVarianceOp.VarianceGammaTimesGaussianIntegral(1.5, 20, 0.3) / scale, 1e-20) < 1e-4);
scale = 2 * System.Math.Exp(-Gaussian.GetLogProb(40 / System.Math.Sqrt(0.3), 0, 1));
scale = 2 * System.Math.Exp(40 - 0.3 / 2);
Assert.True(MMath.AbsDiff(2.467941724509690e-018, GaussianFromMeanAndVarianceOp.VarianceGammaTimesGaussianIntegral(1, 40, 0.3) / scale, 1e-20) < 1e-5);
Assert.True(MMath.AbsDiff(5.022261409377230e-017, GaussianFromMeanAndVarianceOp.VarianceGammaTimesGaussianIntegral(2, 40, 0.3) / scale, 1e-20) < 1e-5);
Assert.True(MMath.AbsDiff(3.502361147666615e-018, GaussianFromMeanAndVarianceOp.VarianceGammaTimesGaussianIntegral(1.1, 40, 0.3) / scale, 1e-20) < 1e-4);
Assert.True(MMath.AbsDiff(1.252310352551344e-017, GaussianFromMeanAndVarianceOp.VarianceGammaTimesGaussianIntegral(1.5, 40, 0.3) / scale, 1e-20) < 1e-4);
Assert.True(GaussianFromMeanAndVarianceOp.VarianceGammaTimesGaussianIntegral(1.138, 33.4, 0.187) > 0);
Assert.True(GaussianFromMeanAndVarianceOp.VarianceGammaTimesGaussianIntegral(1.138, -33.4, 0.187) > 0);
Assert.True(GaussianFromMeanAndVarianceOp.VarianceGammaTimesGaussianIntegral(1.138, 58.25, 0.187) > 0);
Assert.True(GaussianFromMeanAndVarianceOp.VarianceGammaTimesGaussianIntegral(1.138, 50, 0.187) > 0);
Assert.True(GaussianFromMeanAndVarianceOp.VarianceGammaTimesGaussianIntegral(1.138, 100, 0.187) > 0);
}
// returns int_0^Inf x^n VG(x;a) N(x;m,v) dx / (0.5*N(m;0,1))
public static double NormalVGMomentRatio(int n, int a, double m, double v)
{
if (a < 1)
{
throw new ArgumentException("a < 1", "a");
}
if (a == 1)
{
double sqrtV = Math.Sqrt(v);
return(MMath.Gamma(n + 1) * MMath.NormalCdfMomentRatio(n, m / sqrtV) * Math.Pow(sqrtV, n));
}
else if (a == 2)
{
double sqrtV = Math.Sqrt(v);
return(0.5 * NormalVGMomentRatio(n, 1, m, v) + 0.5 * MMath.Gamma(n + 2) * MMath.NormalCdfMomentRatio(n + 1, m / sqrtV) * Math.Pow(sqrtV, n + 1));
}
else // a > 2
{
return(0.25 / ((a - 2) * (a - 1)) * NormalVGMomentRatio(n + 2, a - 2, m, v) + (a - 1.5) / (a - 1) * NormalVGMomentRatio(n, a - 1, m, v));
}
}
/// <summary>
/// Computes int_0^Inf x^n N(x;m,v) dx / N(m/sqrt(v);0,1)
/// </summary>
/// <param name="nMax"></param>
/// <param name="m"></param>
/// <param name="v"></param>
/// <returns></returns>
public static double[] NormalCdfMomentRatios(int nMax, double m, double v)
{
double[] result = new double[nMax + 1];
double sqrtV = Math.Sqrt(v);
for (int i = 0; i <= nMax; i++)
{
// NormalCdfMomentRatio(0,37.66) = infinity
result[i] = MMath.NormalCdfMomentRatio(i, m / sqrtV) * Math.Pow(sqrtV, i) * MMath.Gamma(i + 1);
}
return(result);
}