// result is divided by n!! (odd n)
public static double NormalCdfMomentDy2(int n, double x, double y, double r)
{
double omr2 = 1 - r * r;
if (omr2 == 0)
{
double lfact;
if (n % 2 == 1)
{
lfact = DoubleFactorialLn(n);
}
else
{
lfact = DoubleFactorialLn(n - 1);
}
return(System.Math.Pow(x - r * y, n) * System.Math.Exp(Gaussian.GetLogProb(y, 0, 1) - lfact));
}
else
{
double diff = (x - r * y) / System.Math.Sqrt(omr2);
double lfact;
if (n % 2 == 1)
{
lfact = DoubleFactorialLn(n - 1); // n!/n!!
}
else
{
lfact = DoubleFactorialLn(n) - System.Math.Log(n + 1); // n!/(n+1)!!
}
return(System.Math.Exp(lfact + Gaussian.GetLogProb(y, 0, 1)
+ Gaussian.GetLogProb(diff, 0, 1)
+ 0.5 * n * System.Math.Log(omr2)) * MMath.NormalCdfMomentRatio(n, diff));
}
}
// this test shows that a Taylor expansion at zero is not reliable for x less than -0.5
internal void NormalCdfMomentRatioTaylorZeroTest()
{
double x = -1;
for (int n = 0; n < 20; n++)
{
Console.WriteLine("{0} {1} {2}", n, MMath.NormalCdfMomentRatio(n, x), NormalCdfMomentRatioTaylorZero(n, x));
}
}
public static void VarianceGammaTimesGaussianMoments2(double a, double m, double v, out double mu, out double vu)
{
// compute weights
Matrix laplacianMoments = new Matrix(nWeights, nWeights);
DenseVector exactMoments = DenseVector.Constant(laplacianMoments.Rows, 1.0);
// a=10: 7-1
// a=15: 8-1
// a=20: 10-1
// a=21: 10-1
// a=30: 12-1
// get best results if the lead term has flat moment ratio
int jMax = Math.Max(laplacianMoments.Cols, (int)Math.Round(a - 10)) - 1;
jMax = laplacianMoments.Cols - 1;
for (int i = 0; i < exactMoments.Count; i++)
{
//int ii = jMax-i;
int ii = i;
double logMoment = MMath.GammaLn(ii + a) - MMath.GammaLn(a) - MMath.GammaLn(ii + 1);
for (int j = 0; j < laplacianMoments.Cols; j++)
{
int jj = jMax - j;
laplacianMoments[i, j] = Math.Exp(MMath.GammaLn(2 * ii + jj + 1) - MMath.GammaLn(2 * ii + 1) - MMath.GammaLn(jj + 1) - logMoment);
}
}
//Console.WriteLine("exactMoments = {0}, laplacianMoments = ", exactMoments);
//Console.WriteLine(laplacianMoments);
(new LuDecomposition(laplacianMoments)).Solve(exactMoments);
DenseVector weights = exactMoments;
Console.WriteLine("weights = {0}", weights);
double Z0Plus = 0, Z1Plus = 0, Z2Plus = 0;
double Z0Minus = 0, Z1Minus = 0, Z2Minus = 0;
double sqrtV = Math.Sqrt(v);
double InvSqrtV = 1 / sqrtV;
double mPlus = (m - v) * InvSqrtV;
double mMinus = (-m - v) * InvSqrtV;
for (int j = 0; j < weights.Count; j++)
{
int jj = jMax - j;
Z0Plus += weights[j] * MMath.NormalCdfMomentRatio(0 + jj, mPlus) * Math.Pow(sqrtV, 0 + jj);
Z1Plus += weights[j] * MMath.NormalCdfMomentRatio(1 + jj, mPlus) * (1 + jj) * Math.Pow(sqrtV, 1 + jj);
Z2Plus += weights[j] * MMath.NormalCdfMomentRatio(2 + jj, mPlus) * (1 + jj) * (2 + jj) * Math.Pow(sqrtV, 2 + jj);
Z0Minus += weights[j] * MMath.NormalCdfMomentRatio(0 + jj, mMinus) * Math.Pow(sqrtV, 0 + jj);
Z1Minus += weights[j] * MMath.NormalCdfMomentRatio(1 + jj, mMinus) * (1 + jj) * Math.Pow(sqrtV, 1 + jj);
Z2Minus += weights[j] * MMath.NormalCdfMomentRatio(2 + jj, mMinus) * (1 + jj) * (2 + jj) * Math.Pow(sqrtV, 2 + jj);
}
double Z0 = Z0Plus + Z0Minus;
double Z1 = Z1Plus - Z1Minus;
double Z2 = Z2Plus + Z2Minus;
mu = Z1 / Z0;
vu = Z2 / Z0 - mu * mu;
}
/// <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);
}
public void NormalCdfMomentRatioSequenceTest()
{
int n = 0;
double x = -1.1;
var iter = MMath.NormalCdfMomentRatioSequence(n, x);
for (int i = 0; i < 20; i++)
{
iter.MoveNext();
double r1 = MMath.NormalCdfMomentRatio(n + i, x);
double r2 = iter.Current;
Console.WriteLine("{0}: {1} {2} {3}", i, r1, r2, Math.Abs(r1 - r2) / Math.Abs(r1));
}
}
public void NormalCdfMomentRatioTest()
{
/* Anaconda Python script to generate a true value (must not be indented):
* from mpmath import *
* mp.dps = 100; mp.pretty = True
* x = mpf('-2'); n=300;
* exp(x*x/4)*pcfu(0.5+n,-x)
*/
double[,] normcdfMomentRatio_pairs = ReadPairs(Path.Combine(TestUtils.DataFolderPath, "SpecialFunctionsValues", "NormalCdfMomentRatio.csv"));
CheckFunctionValues("NormalCdfMomentRatio", delegate(double n, double x)
{
return(MMath.NormalCdfMomentRatio((int)n, x));
}, normcdfMomentRatio_pairs);
}
public static double NormalCdfMomentDy(int n, double x, double y, double r)
{
double omr2 = 1 - r * r;
if (omr2 == 0)
{
return(System.Math.Pow(x - r * y, n) * System.Math.Exp(Gaussian.GetLogProb(y, 0, 1)));
}
else
{
double diff = (x - r * y) / System.Math.Sqrt(omr2);
return(System.Math.Exp(MMath.GammaLn(n + 1) + Gaussian.GetLogProb(y, 0, 1)
+ Gaussian.GetLogProb(diff, 0, 1)
+ 0.5 * n * System.Math.Log(omr2)) * MMath.NormalCdfMomentRatio(n, diff));
}
}
/// <summary>
/// Compute moments of 0.5*exp(-abs(x))*N(x;m,v)
/// </summary>
/// <param name="m"></param>
/// <param name="v"></param>
/// <param name="logZ"></param>
/// <param name="mu"></param>
/// <param name="vu"></param>
public static void LaplacianTimesGaussianMoments(double m, double v, out double logZ, out double mu, out double vu)
{
if (Double.IsPositiveInfinity(v))
{
// moments of Laplacian only
logZ = 0;
mu = 0;
vu = 2;
return;
}
double invV = 1 / v;
double invSqrtV = Math.Sqrt(invV);
double mPlus = (m - v) * invSqrtV;
double mMinus = (-m - v) * invSqrtV;
double Z0, Z1, Z2;
if (mPlus > 30)
{
double[] moments = NormalCdfMoments(2, m - v, v);
Z0 = moments[0];
Z1 = moments[1];
Z2 = moments[2];
logZ = Math.Log(0.5 * Z0) - m + 0.5 * v;
}
else if (mMinus > 30)
{
double[] moments = NormalCdfMoments(2, -m - v, v);
Z0 = moments[0];
Z1 = -moments[1];
Z2 = moments[2];
logZ = Math.Log(0.5 * Z0) + m + 0.5 * v;
}
else
{
Z0 = MMath.NormalCdfRatio(mPlus) + MMath.NormalCdfRatio(mMinus);
Z1 = Math.Sqrt(v) * (MMath.NormalCdfMomentRatio(1, mPlus) - MMath.NormalCdfMomentRatio(1, mMinus));
Z2 = 2 * v * (MMath.NormalCdfMomentRatio(2, mPlus) + MMath.NormalCdfMomentRatio(2, mMinus));
logZ = Math.Log(0.5 * Z0) - MMath.LnSqrt2PI - 0.5 * m * m * invV;
}
mu = Z1 / Z0;
double mu2 = Z2 / Z0;
vu = mu2 - mu * mu;
}
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);
}
private static IEnumerator <double> NormalCdfMomentRatioSequence(double x)
{
if (x > -1)
{
double rPrev = MMath.NormalCdfRatio(x);
yield return(rPrev);
double r = x * rPrev + 1;
yield return(r);
for (int i = 1; ; i++)
{
double rNew = (x * r + rPrev) / (i + 1);
rPrev = r;
r = rNew;
yield return(r);
}
}
else
{
int tableSize = 10;
// rtable[tableStart-i] = R_i
double[] rtable = new double[tableSize];
int tableStart = -1;
for (int i = 0; ; i++)
{
if (i > tableStart)
{
// build the table
tableStart = i + tableSize - 1;
rtable[0] = MMath.NormalCdfMomentRatio(tableStart, x);
rtable[1] = MMath.NormalCdfMomentRatio(tableStart - 1, x);
for (int j = 2; j < tableSize; j++)
{
int n = tableStart - j + 1;
rtable[j] = (n + 1) * rtable[j - 2] - x * rtable[j - 1];
}
}
yield return(rtable[tableStart - i]);
}
}
}
// 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));
}
}
// Returns NormalCdf divided by N(x;0,1) N((y-rx)/sqrt(1-r^2);0,1), multiplied by scale
// This version works best for small r^2
// We need x <= 0 and (y - r*x) <= 0
private static double NormalCdfRatioConFrac3b(double x, double y, double r, double scale)
{
if (scale == 0)
{
return(scale);
}
//if (r * (y - r * x) < 0)
// throw new ArgumentException("r*(y - r*x) < 0");
if (x - r * y > 0)
{
throw new ArgumentException("x - r*y > 0");
}
if (x > 0)
{
throw new ArgumentException("x > 0");
}
double omr2 = 1 - r * r;
double sqrtomr2 = System.Math.Sqrt(omr2);
double rxmy = r * x - y;
double ymrx = -rxmy / sqrtomr2;
if (ymrx > 0)
{
throw new ArgumentException("ymrx > 0");
}
double offset = MMath.NormalCdfRatio(x) * MMath.NormalCdfRatio(ymrx) * scale;
double omsomr2 = MMath.OneMinusSqrtOneMinus(r * r);
double delta = (r * y - x * omsomr2) / sqrtomr2;
double diff = (x - r * y) / sqrtomr2;
double Rdiff = MMath.NormalCdfRatio(diff);
//var RdiffIter = MMath.NormalCdfMomentRatioSequence(0, diff);
//RdiffIter.MoveNext();
//double Rdiff = RdiffIter.Current;
double scale2 = scale * omr2;
double numer;
if (System.Math.Abs(delta) > 0.5)
{
// for r =approx 0 this becomes inaccurate due to cancellation
numer = scale2 * (MMath.NormalCdfRatio(x) / sqrtomr2 - Rdiff);
}
else
{
numer = scale2 * (MMath.NormalCdfRatioDiff(diff, delta) + omsomr2 * Rdiff) / sqrtomr2;
}
double numerPrev = 0;
double denom = rxmy;
double denomPrev = 1;
double rOld = 0;
double result = 0;
double cEven = scale2;
double cIncr = r * sqrtomr2;
double cOdd = cEven * cIncr;
cIncr *= cIncr;
for (int i = 1; i < 10000; i++)
{
double numerNew, denomNew;
//RdiffIter.MoveNext();
double c = MMath.NormalCdfMomentRatio(i, diff);
//double c = RdiffIter.Current;
if (i % 2 == 1)
{
if (i > 1)
{
cOdd *= (i - 1) * cIncr;
}
c *= cOdd;
numerNew = rxmy * numer + omr2 * numerPrev - c;
denomNew = rxmy * denom + omr2 * denomPrev;
}
else
{
cEven *= i * cIncr;
c *= cEven;
numerNew = (rxmy * numer + omr2 * i * numerPrev - c) / (i + 1);
denomNew = (rxmy * denom + omr2 * i * denomPrev) / (i + 1);
}
numerPrev = numer;
numer = numerNew;
denomPrev = denom;
denom = denomNew;
if (i % 2 == 1)
{
result = -numer / denom;
Console.WriteLine($"iter {i.ToString().PadLeft(3)}: {result.ToString("r").PadRight(24)} {numer.ToString("r").PadRight(24)} {denom.ToString("r").PadRight(24)} {c}");
if (double.IsInfinity(result) || double.IsNaN(result))
{
throw new Exception(string.Format("NormalCdfRatioConFrac3 not converging for x={0} y={1} r={2} scale={3}", x, y, r, scale));
}
//if (result == rOld)
// return result + offset;
rOld = result;
}
}
throw new Exception(string.Format("NormalCdfRatioConFrac3 not converging for x={0} y={1} r={2} scale={3}", x, y, r, scale));
}
private double NormalCdfIntegralBasic2(double x, double y, double r)
{
double omr2 = 1 - r * r;
double sqrtomr2 = System.Math.Sqrt(omr2);
double ymrx = (y - r * x) / sqrtomr2;
double xmry = (x - r * y) / sqrtomr2;
double func(double t)
{
return((y - r * x + r * t) * System.Math.Exp(Gaussian.GetLogProb(t, x, 1) + MMath.NormalCdfLn(ymrx + r * t / sqrtomr2)));
}
func(0);
double func2(double t)
{
double ymrxt = ymrx + r * t / sqrtomr2;
return(sqrtomr2 * System.Math.Exp(Gaussian.GetLogProb(t, x, 1) + Gaussian.GetLogProb(ymrxt, 0, 1)) * (MMath.NormalCdfMomentRatio(1, ymrxt) - 1));
}
func2(0);
//return -MMath.NormalCdf(x, y, r) * (y / r - x) + Integrate(func2) / r;
double func3(double t)
{
double xmryt = xmry + r * t / sqrtomr2;
return(sqrtomr2 * System.Math.Exp(Gaussian.GetLogProb(t, y, 1) + Gaussian.GetLogProb(xmryt, 0, 1)) * MMath.NormalCdfMomentRatio(1, xmryt));
}
//double Z = MMath.NormalCdf(x, y, r, out double exponent);
double Z3 = Integrate(func3);
//return System.Math.Exp(exponent)*(-Z * (y / r - x) - omr2 / r * MMath.NormalCdfRatio(xmry)) + Z3/r;
return(Z3);
}
public static Gaussian XAverageConditional_Helper([SkipIfUniform] Bernoulli isPositive, [SkipIfUniform, Proper] Gaussian x, bool forceProper)
{
if (x.IsPointMass)
{
if (isPositive.IsPointMass && (isPositive.Point != (x.Point > 0)))
{
return(Gaussian.PointMass(0));
}
else
{
return(Gaussian.Uniform());
}
}
double tau = x.MeanTimesPrecision;
double prec = x.Precision;
if (prec == 0.0)
{
if (isPositive.IsPointMass)
{
if ((isPositive.Point && tau < 0) ||
(!isPositive.Point && tau > 0))
{
// posterior is proportional to I(x>0) exp(tau*x)
//double mp = -1 / tau;
//double vp = mp * mp;
return(Gaussian.FromNatural(-tau, tau * tau) / x);
}
}
return(Gaussian.Uniform());
}
else if (prec < 0)
{
throw new ImproperMessageException(x);
}
double sqrtPrec = Math.Sqrt(prec);
// m/sqrt(v) = (m/v)/sqrt(1/v)
double z = tau / sqrtPrec;
// epsilon = p(b=F)
// eq (51) in EP quickref
double alpha;
if (isPositive.IsPointMass)
{
if ((isPositive.Point && z < -10) || (!isPositive.Point && z > 10))
{
if (z > 10)
{
z = -z;
}
//double Y = MMath.NormalCdfRatio(z);
// dY = z*Y + 1
// d2Y = z*dY + Y
// posterior mean = m + sqrt(v)/Y = sqrt(v)*(z + 1/Y) = sqrt(v)*dY/Y
// = sqrt(v)*(d2Y/Y - 1)/z = (d2Y/Y - 1)/tau
// =approx sqrt(v)*(-1/z) = -1/tau
// posterior variance = v - v*dY/Y^2 =approx v/z^2
// posterior E[x^2] = v - v*dY/Y^2 + v*dY^2/Y^2 = v - v*dY/Y^2*(1 - dY) = v + v*z*dY/Y = v*d2Y/Y
// d3Y = z*d2Y + 2*dY
// = z*(z*dY + Y) + 2*dY
// = z^2*dY + z*Y + 2*dY
// = z^2*dY + 3*dY - 1
double d3Y = 6 * MMath.NormalCdfMomentRatio(3, z);
//double dY = MMath.NormalCdfMomentRatio(1, z);
double dY = (d3Y + 1) / (z * z + 3);
if (MMath.AreEqual(dY, 0))
{
double tau2 = tau * tau;
if (tau2 > double.MaxValue)
{
return(Gaussian.PointMass(-1 / tau));
}
else
{
return(Gaussian.FromNatural(-2 * tau, tau2));
}
}
// Y = (dY-1)/z
// alpha = sqrtPrec*z/(dY-1) = tau/(dY-1)
// alpha+tau = tau*(1 + 1/(dY-1)) = tau*dY/(dY-1) = alpha*dY
// beta = alpha * (alpha + tau)
// = alpha * tau * dY / (dY - 1)
// = prec * z^2 * dY/(dY-1)^2
// prec/beta = (dY-1)^2/(dY*z^2)
// prec/beta - 1 = ((dY-1)^2 - dY*z^2)/(dY*z^2)
// weight = beta/(prec - beta)
// = dY*z^2/((dY-1)^2 - dY*z^2)
// = dY*z^2/(dY^2 -2*dY + 1 - dY*z^2)
// = dY*z^2/(dY^2 + 1 + z*Y - d3Y)
// = dY*z^2/(dY^2 + dY - d3Y)
// = z^2/(dY + 1 - d3Y/dY)
double d3YidY = d3Y / dY;
double denom = dY - d3YidY + 1;
double msgPrec = tau * tau / denom;
// weight * (tau + alpha) + alpha
// = z^2/(dY + 1 - d3Y/dY) * alpha*dY + alpha
// = alpha*(z^2*dY/(dY + 1 - d3Y/dY) + 1)
// = alpha*(z^2*dY + dY + 1 - d3Y/dY)/(dY + 1 - d3Y/dY)
// = alpha*(z^2*dY + dY + 1 - d3Y/dY)/(dY + 1 - d3Y/dY)
// = alpha*(d3Y - 2*dY + 2 - d3Y/dY)/(dY + 1 - d3Y/dY)
// z^2*dY = d3Y - 3*dY + 1
double numer = (d3Y - 2 * dY - d3YidY + 2) / (dY - 1);
double msgMeanTimesPrec = tau * numer / denom;
if (msgPrec > double.MaxValue)
{
// In this case, the message should be the posterior.
// posterior mean = (msgMeanTimesPrec + tau)*denom/(tau*tau)
// = (numer + denom)/tau
return(Gaussian.PointMass((numer + denom) / tau));
}
else
{
return(Gaussian.FromNatural(msgMeanTimesPrec, msgPrec));
}
}
else if (isPositive.Point)
{
alpha = sqrtPrec / MMath.NormalCdfRatio(z);
}
else
{
alpha = -sqrtPrec / MMath.NormalCdfRatio(-z);
}
}
else
{
//double v = MMath.LogSumExp(isPositive.LogProbTrue + MMath.NormalCdfLn(z), isPositive.LogProbFalse + MMath.NormalCdfLn(-z));
double v = LogAverageFactor(isPositive, x);
alpha = sqrtPrec * Math.Exp(-z * z * 0.5 - MMath.LnSqrt2PI - v) * (2 * isPositive.GetProbTrue() - 1);
}
// eq (52) in EP quickref (where tau = mnoti/Vnoti)
double beta;
if (alpha == 0)
{
beta = 0; // avoid 0 * infinity
}
else
{
beta = alpha * (alpha + tau);
}
double weight = beta / (prec - beta);
if (forceProper && weight < 0)
{
weight = 0;
}
Gaussian result = new Gaussian();
if (weight == 0)
{
// avoid 0 * infinity
result.MeanTimesPrecision = alpha;
}
else
{
// eq (31) in EP quickref; same as inv(inv(beta)-inv(prec))
result.Precision = prec * weight;
// eq (30) in EP quickref times above and simplified
result.MeanTimesPrecision = weight * (tau + alpha) + alpha;
}
if (double.IsNaN(result.Precision) || double.IsNaN(result.MeanTimesPrecision))
{
throw new InferRuntimeException($"result is NaN. isPositive={isPositive}, x={x}, forceProper={forceProper}");
}
return(result);
}
public static Gaussian XAverageConditional_Helper([SkipIfUniform] Bernoulli isPositive, [SkipIfUniform, Proper] Gaussian x, bool forceProper)
{
if (x.IsPointMass)
{
if (isPositive.IsPointMass && (isPositive.Point != (x.Point > 0)))
{
return(Gaussian.PointMass(0));
}
else
{
return(Gaussian.Uniform());
}
}
double tau = x.MeanTimesPrecision;
double prec = x.Precision;
if (prec == 0.0)
{
if (isPositive.IsPointMass)
{
if ((isPositive.Point && tau < 0) ||
(!isPositive.Point && tau > 0))
{
// posterior is proportional to I(x>0) exp(tau*x)
double mp = -1 / tau;
double vp = mp * mp;
return((new Gaussian(mp, vp)) / x);
}
}
if (x.IsUniform())
{
return(Gaussian.Uniform());
}
throw new ImproperMessageException(x);
}
else if (prec < 0)
{
throw new ImproperMessageException(x);
}
double sqrtPrec = Math.Sqrt(prec);
// m/sqrt(v) = (m/v)/sqrt(1/v)
double z = tau / sqrtPrec;
// epsilon = p(b=F)
// eq (51) in EP quickref
double alpha;
if (isPositive.IsPointMass)
{
if (isPositive.Point)
{
if (z < -1e10)
{
double mp = -1 / (z * sqrtPrec);
double vp = 1 / (z * z * prec);
return((new Gaussian(mp, vp)) / x);
}
else if (z < -100)
{
double Y = MMath.NormalCdfRatio(z);
// double dY = MMath.NormalCdfMomentRatio(1, z);
// dY = z*Y + 1
// d2Y = z*dY + Y
// posterior mean = m + sqrt(v)/Y = sqrt(v)*(z + 1/Y) = sqrt(v)*dY/Y =approx sqrt(v)*(-1/z)
// posterior variance = v - v*dY/Y^2 =approx v/z^2
// posterior E[x^2] = v - v*dY/Y^2 + v*dY^2/Y^2 = v - v*dY/Y^2*(1 - dY) = v + v*z*dY/Y = v*d2Y/Y
double d2Y = 2 * MMath.NormalCdfMomentRatio(2, z);
// m2TimesPrec = z*dY/Y + 1
double m2TimesPrec = d2Y / Y;
Assert.IsTrue(tau != 0);
double mp = (m2TimesPrec - 1) / tau;
double vp = m2TimesPrec / prec - mp * mp;
return((new Gaussian(mp, vp)) / x);
}
alpha = sqrtPrec / MMath.NormalCdfRatio(z);
}
else
{
if (z > 1e10)
{
double mp = 1 / (z * sqrtPrec);
double vp = 1 / (z * z * prec);
return((new Gaussian(mp, vp)) / x);
}
else if (z > 100)
{
double Y = MMath.NormalCdfRatio(-z);
// dY = -(d2Y/prec - Y)/(-z)*sqrtPrec
// dY/Y/prec = -(d2Y/Y/prec/prec - 1/prec)/(-z)*sqrtPrec
// dY/Y/prec = -(d2Y/Y/prec - 1)/(-tau)
//double dY = -MMath.NormalCdfMomentRatio(1,-z)*sqrtPrec;
double d2Y = 2 * MMath.NormalCdfMomentRatio(2, -z);
double m2TimesPrec = d2Y / Y;
Assert.IsTrue(tau != 0);
double mp = (m2TimesPrec - 1) / tau;
double vp = m2TimesPrec / prec - mp * mp;
return((new Gaussian(mp, vp)) / x);
}
alpha = -sqrtPrec / MMath.NormalCdfRatio(-z);
}
}
else
{
//double v = MMath.LogSumExp(isPositive.LogProbTrue + MMath.NormalCdfLn(z), isPositive.LogProbFalse + MMath.NormalCdfLn(-z));
double v = LogAverageFactor(isPositive, x);
alpha = sqrtPrec * Math.Exp(-z * z * 0.5 - MMath.LnSqrt2PI - v) * (2 * isPositive.GetProbTrue() - 1);
}
// eq (52) in EP quickref (where tau = mnoti/Vnoti)
double beta = alpha * (alpha + tau);
double weight = beta / (prec - beta);
if (forceProper && weight < 0)
{
weight = 0;
}
Gaussian result = new Gaussian();
// eq (31) in EP quickref; same as inv(inv(beta)-inv(prec))
result.Precision = prec * weight;
// eq (30) in EP quickref times above and simplified
result.MeanTimesPrecision = weight * (tau + alpha) + alpha;
if (double.IsNaN(result.Precision) || double.IsNaN(result.MeanTimesPrecision))
{
throw new ApplicationException("result is nan");
}
return(result);
}
public static double NormalCdfMomentDyRatio(int n, double x, double y, double r)
{
double omr2 = 1 - r * r;
if (omr2 == 0)
{
throw new ArgumentException();
}
else
{
double diff = (x - r * y) / System.Math.Sqrt(omr2);
//return Math.Exp(MMath.GammaLn(n + 1) + 0.5 * n * Math.Log(omr2)) * MMath.NormalCdfMomentRatio(n, diff);
return(System.Math.Exp(MMath.GammaLn(n + 1) + 0.5 * n * System.Math.Log(omr2) + System.Math.Log(MMath.NormalCdfMomentRatio(n, diff))));
}
}
// returns phi_n(m,v)/n!
public static double NormalCdfMoment(int n, double m, double v)
{
double InvSqrtV = Math.Sqrt(1 / v);
return(MMath.NormalCdfMomentRatio(n, m * InvSqrtV) * MMath.InvSqrt2PI * Math.Exp(-0.5 * m * m / v) * Math.Pow(v, 0.5 * n));
}
/// <include file='FactorDocs.xml' path='factor_docs/message_op_class[@name="MaxGaussianOp"]/message_doc[@name="AAverageConditional(Gaussian, Gaussian, Gaussian)"]/*'/>
public static Gaussian AAverageConditional([SkipIfUniform] Gaussian max, [Proper] Gaussian a, [Proper] Gaussian b)
{
if (max.IsUniform())
{
return(Gaussian.Uniform());
}
if (!b.IsProper())
{
throw new ImproperMessageException(b);
}
double logw1, alpha1, vx1, mx1;
double logw2, alpha2, vx2, mx2;
double logz;
ComputeStats(max, a, b, out logz, out logw1, out alpha1, out vx1, out mx1,
out logw2, out alpha2, out vx2, out mx2);
double w1 = Math.Exp(logw1 - logz);
double w2 = Math.Exp(logw2 - logz);
bool checkDerivatives = false;
if (a.IsPointMass)
{
// f(a) = int p(x = max(a,b)) p(b) db
// f(a) = p(x = a) p(a >= b) + p(x = b) p(a < b | x = b)
// f(a) = N(a; mx, vx) phi((a - m2)/sqrt(v2)) + N(m2; mx, vx + v2) phi((mx2 - a)/sqrt(vx2))
// f'(a) = (mx-a)/vx N(a; mx, vx) phi((a - m2)/sqrt(v2)) + N(a; mx, vx) N(a; m2, v2) - N(m2; mx, vx + v2) N(a; mx2, vx2)
// = (mx-a)/vx N(a; mx, vx) phi((a - m2)/sqrt(v2))
// f''(a) = -1/vx N(a; mx, vx) phi((a - m2)/sqrt(v2)) + (mx-a)^2/vx^2 N(a; mx, vx) phi((a - m2)/sqrt(v2)) + (mx-a)/vx N(a; mx, vx) N(a; m2, v2)
// ddlogf = f''(a)/f(a) - (f'(a)/f(a))^2 = (f''(a) f(a) - f'(a)^2)/f(a)^2
double aPoint = a.Point;
if (max.IsPointMass)
{
return(max);
}
double z = max.MeanTimesPrecision - aPoint * max.Precision;
double alpha = z * w1;
double beta = (z * alpha1 - max.Precision + z * z) * w1 - alpha * alpha;
if (b.IsPointMass && b.Point != aPoint)
{
beta -= max.Precision * w2 * alpha2 * (b.Point - aPoint);
}
if (checkDerivatives)
{
double m1 = a.GetMean();
double delta = m1 * 1e-6;
double logzd, logw1d, alpha1d, vx1d, mx1d, logw2d, alpha2d, vx2d, mx2d;
ComputeStats(max, Gaussian.FromMeanAndPrecision(m1 + delta, a.Precision), b, out logzd, out logw1d, out alpha1d, out vx1d, out mx1d, out logw2d, out alpha2d, out vx2d, out mx2d);
double logzd2;
ComputeStats(max, Gaussian.FromMeanAndPrecision(m1 - delta, a.Precision), b, out logzd2, out logw1d, out alpha1d, out vx1d, out mx1d, out logw2d, out alpha2d, out vx2d, out mx2d);
double alphaCheck = (logzd - logzd2) / (2 * delta);
double alphaError = Math.Abs(alpha - alphaCheck);
double betaCheck = (logzd + logzd2 - 2 * logz) / (delta * delta);
double betaError = Math.Abs(beta - betaCheck);
Console.WriteLine($"alpha={alpha} check={alphaCheck} error={alphaError} beta={beta} check={betaCheck} error={betaError}");
}
return(Gaussian.FromDerivatives(aPoint, alpha, beta, ForceProper));
}
bool useMessage = (w1 > 0) || (logw2 > -100);
if (useMessage)
{
// vx1 = 1/(1/vx + 1/v1) = vx*v1/(vx + v1)
double z, alpha, beta;
if (max.IsPointMass)
{
if (w2 == 0)
{
return(max);
}
z = max.Point * a.Precision - a.MeanTimesPrecision;
alpha = z * w1 - w2 * alpha2;
beta = (z * z - a.Precision) * w1 - z * alpha2 * w2 - alpha * alpha;
}
else
{
//z = vx1 * (max.MeanTimesPrecision * a.Precision - a.MeanTimesPrecision * max.Precision);
z = (max.MeanTimesPrecision - a.GetMean() * max.Precision) / (max.Precision / a.Precision + 1);
alpha = z * w1 - vx1 * max.Precision * w2 * alpha2;
if (w2 == 0)
{
//beta = (z * alpha1 - max.Precision) / (max.Precision / a.Precision + 1);
//double resultPrecision = a.Precision * beta / (-a.Precision - beta);
//resultPrecision = beta / (-1 - beta/a.Precision);
//resultPrecision = 1 / (-1/beta - 1 / a.Precision);
//resultPrecision = a.Precision / (-a.Precision / beta - 1);
//resultPrecision = a.Precision / (-(a.Precision + max.Precision) / (z * alpha1 - max.Precision) - 1);
//resultPrecision = -a.Precision / ((a.Precision + z*alpha1) / (z * alpha1 - max.Precision));
double zalpha1 = z * alpha1;
double denom = (1 + zalpha1 / a.Precision);
double resultPrecision = (max.Precision - zalpha1) / denom;
//double weight = (max.Precision - z * alpha1) / (a.Precision + z * alpha1);
//double weightPlus1 = (max.Precision + a.Precision) / (a.Precision + z * alpha1);
//double resultMeanTimesPrecision = weight * (a.MeanTimesPrecision + alpha) + alpha;
//resultMeanTimesPrecision = weight * a.MeanTimesPrecision + weightPlus1 * alpha;
//double resultMeanTimesPrecision = (a.Precision * max.MeanTimesPrecision - z * alpha1 * a.MeanTimesPrecision) / (a.Precision + z * alpha1);
double resultMeanTimesPrecision = (max.MeanTimesPrecision - zalpha1 * a.GetMean()) / denom;
return(Gaussian.FromNatural(resultMeanTimesPrecision, resultPrecision));
}
else
{
beta = ((z * alpha1 - max.Precision) * vx1 * a.Precision + z * z) * w1
- max.Precision * vx1 * w2 * alpha2 * (mx2 * a.Precision - a.MeanTimesPrecision) / (vx2 * a.Precision + 1) - alpha * alpha;
}
}
//Console.WriteLine($"z={z} w1={w1:r} w2={w2:r} logw2={logw2} alpha1={alpha1} alpha2={alpha2} alpha={alpha:r} beta={beta:r}");
if (checkDerivatives)
{
double m1 = a.GetMean();
double delta = m1 * 1e-6;
double logzd, logw1d, alpha1d, vx1d, mx1d, logw2d, alpha2d, vx2d, mx2d;
ComputeStats(max, Gaussian.FromMeanAndPrecision(m1 + delta, a.Precision), b, out logzd, out logw1d, out alpha1d, out vx1d, out mx1d, out logw2d, out alpha2d, out vx2d, out mx2d);
double logzd2;
ComputeStats(max, Gaussian.FromMeanAndPrecision(m1 - delta, a.Precision), b, out logzd2, out logw1d, out alpha1d, out vx1d, out mx1d, out logw2d, out alpha2d, out vx2d, out mx2d);
double alphaCheck = (logzd - logzd2) / (2 * delta);
double alphaError = Math.Abs(alpha - alphaCheck);
double betaCheck = (logzd + logzd2 - 2 * logz) / (delta * delta);
double betaError = Math.Abs(beta - betaCheck);
Console.WriteLine($"alpha={alpha} check={alphaCheck} error={alphaError} beta={beta} check={betaCheck} error={betaError}");
}
return(GaussianOp.GaussianFromAlphaBeta(a, alpha, -beta, ForceProper));
}
else
{
double m1, v1, m2, v2;
a.GetMeanAndVariance(out m1, out v1);
b.GetMeanAndVariance(out m2, out v2);
// the posterior is a mixture model with weights exp(logw1-logz), exp(logw2-logz) and distributions
// N(a; mx1, vx1) phi((a - m2)/sqrt(v2)) / phi((mx1 - m2)/sqrt(vx1 + v2))
// N(a; m1, v1) phi((mx2 - a)/sqrt(vx2)) / phi((mx2 - m1)/sqrt(vx2 + v1))
// the moments of the posterior are computed via the moments of these two components.
if (vx1 == 0)
{
alpha1 = 0;
}
if (vx2 == 0)
{
alpha2 = 0;
}
double mc1 = mx1;
if (alpha1 != 0) // avoid 0*infinity
{
mc1 += alpha1 * vx1;
}
alpha2 = -alpha2;
double mc2 = m1;
if (alpha2 != 0) // avoid 0*infinity
{
mc2 += alpha2 * v1;
}
// z2 = (mx2 - m1) / Math.Sqrt(vx2 + v1)
// logw2 = MMath.NormalCdfLn(z2);
// alpha2 = -Math.Exp(Gaussian.GetLogProb(mx2, m1, vx2 + v1) - logw2);
// = -1/sqrt(vx2+v1)/NormalCdfRatio(z2)
// m1 + alpha2*v1 = sqrt(vx2+v1)*(m1/sqrt(vx2+v1) - v1/(vx2+v1)/NormalCdfRatio(z2))
// = sqrt(vx2+v1)*(mx2/sqrt(vx2+v1) - z2 - v1/(vx2+v1)/NormalCdfRatio(z2))
// = sqrt(vx2+v1)*(mx2/sqrt(vx2+v1) - dY/Y) if vx2=0
double z2 = 0, Y = 0, dY = 0;
const double z2small = 0;
if (vx2 == 0)
{
z2 = (mx2 - m1) / Math.Sqrt(vx2 + v1);
if (z2 < z2small)
{
Y = MMath.NormalCdfRatio(z2);
dY = MMath.NormalCdfMomentRatio(1, z2);
mc2 = mx2 - Math.Sqrt(v1) * dY / Y;
}
}
double m = w1 * mc1 + w2 * mc2;
double beta1;
if (alpha1 == 0)
{
beta1 = 0; // avoid 0*infinity
}
else
{
double r1 = (mx1 - m2) / (vx1 + v2);
beta1 = alpha1 * (alpha1 + r1);
}
double beta2;
if (alpha2 == 0)
{
beta2 = 0; // avoid 0*infinity
}
else
{
double r2 = (mx2 - m1) / (vx2 + v1);
beta2 = alpha2 * (alpha2 - r2);
}
double vc1 = vx1 * (1 - vx1 * beta1);
double vc2;
if (vx2 == 0 && z2 < z2small)
{
// beta2 = alpha2 * (alpha2 - z2/sqrt(v1))
// vc2 = v1 - v1^2 * alpha2 * (alpha2 - z2/sqrt(v1))
// = v1 - v1*dY/Y^2
// =approx v1/z2^2
// posterior E[x^2] = v - v*dY/Y^2 + v*dY^2/Y^2 = v - v*dY/Y^2*(1 - dY) = v + v*z*dY/Y = v*d2Y/Y
//vc2 = v1 * (1 - dY / (Y * Y));
double d2Y = 2 * MMath.NormalCdfMomentRatio(2, z2);
double dYiY = dY / Y;
vc2 = v1 * (d2Y / Y - dYiY * dYiY);
}
else if (beta2 == 0)
{
vc2 = v1;
}
else
{
vc2 = v1 * (1 - v1 * beta2);
}
double diff = mc1 - mc2;
double v = w1 * vc1 + w2 * vc2 + w1 * w2 * diff * diff;
Gaussian result = new Gaussian(m, v);
//Console.WriteLine($"z2={z2} m={m} v={v} vc2={vc2} diff={diff}");
result.SetToRatio(result, a, ForceProper);
if (Double.IsNaN(result.Precision) || Double.IsNaN(result.MeanTimesPrecision))
{
throw new InferRuntimeException($"result is NaN. max={max}, a={a}, b={b}");
}
return(result);
}
}
public static Gaussian XAverageConditional_Helper([SkipIfUniform] Bernoulli isPositive, [SkipIfUniform, Proper] Gaussian x, bool forceProper)
{
Gaussian result = new Gaussian();
if (x.IsPointMass)
{
result.SetToUniform();
return(result);
}
double prec = x.Precision;
if (prec == 0.0)
{
result.SetToUniform();
return(result);
}
else if (prec < 0)
{
throw new ImproperMessageException(x);
}
double sqrtPrec = Math.Sqrt(prec);
double tau = x.MeanTimesPrecision;
// m/sqrt(v) = (m/v)/sqrt(1/v)
double z = tau / sqrtPrec;
// epsilon = p(b=F)
// eq (51) in EP quickref
double alpha;
if (isPositive.IsPointMass)
{
if (isPositive.Point)
{
if (z < -100)
{
double Y = MMath.NormalCdfRatio(z);
double d2Y = 2 * MMath.NormalCdfMomentRatio(2, z);
double m2TimesPrec = d2Y / Y;
Assert.IsTrue(tau != 0);
double mp = (m2TimesPrec - 1) / tau;
double vp = m2TimesPrec / prec - mp * mp;
return((new Gaussian(mp, vp)) / x);
}
alpha = sqrtPrec / MMath.NormalCdfRatio(z);
}
else
{
if (z > 100)
{
double Y = MMath.NormalCdfRatio(-z);
// dY = -(d2Y/prec - Y)/(-z)*sqrtPrec
// dY/Y/prec = -(d2Y/Y/prec/prec - 1/prec)/(-z)*sqrtPrec
// dY/Y/prec = -(d2Y/Y/prec - 1)/(-tau)
//double dY = -MMath.NormalCdfMomentRatio(1,-z)*sqrtPrec;
double d2Y = 2 * MMath.NormalCdfMomentRatio(2, -z);
double m2TimesPrec = d2Y / Y;
Assert.IsTrue(tau != 0);
double mp = (m2TimesPrec - 1) / tau;
double vp = m2TimesPrec / prec - mp * mp;
return((new Gaussian(mp, vp)) / x);
}
alpha = -sqrtPrec / MMath.NormalCdfRatio(-z);
}
}
else
{
//double v = MMath.LogSumExp(isPositive.LogProbTrue + MMath.NormalCdfLn(z), isPositive.LogProbFalse + MMath.NormalCdfLn(-z));
double v = LogAverageFactor(isPositive, x);
alpha = sqrtPrec * Math.Exp(-z * z * 0.5 - MMath.LnSqrt2PI - v) * (2 * isPositive.GetProbTrue() - 1);
}
// eq (52) in EP quickref (where tau = mnoti/Vnoti)
double beta = alpha * (alpha + tau);
double weight = beta / (prec - beta);
if (forceProper && weight < 0)
{
weight = 0;
}
// eq (31) in EP quickref; same as inv(inv(beta)-inv(prec))
result.Precision = prec * weight;
// eq (30) in EP quickref times above and simplified
result.MeanTimesPrecision = weight * (tau + alpha) + alpha;
if (double.IsNaN(result.Precision) || double.IsNaN(result.MeanTimesPrecision))
{
throw new ApplicationException("result is nan");
}
return(result);
}
