/// <summary>
/// Constructs a Gamma distribution with the given mean and mean logarithm.
/// </summary>
/// <param name="mean">Desired expected value.</param>
/// <param name="meanLog">Desired expected logarithm.</param>
/// <returns>A new Gamma distribution.</returns>
/// <remarks>This function is equivalent to maximum-likelihood estimation of a Gamma distribution
/// from data given by sufficient statistics.
/// This function is significantly slower than the other constructors since it
/// involves nonlinear optimization. The algorithm is a generalized Newton iteration,
/// described in "Estimating a Gamma distribution" by T. Minka, 2002.
/// </remarks>
public static Gamma FromMeanAndMeanLog(double mean, double meanLog)
{
double delta = Math.Log(mean) - meanLog;
if (delta <= 2e-16)
{
return(Gamma.PointMass(mean));
}
double shape = 0.5 / delta;
for (int iter = 0; iter < 100; iter++)
{
double oldShape = shape;
double g = Math.Log(shape) - delta - MMath.Digamma(shape);
shape /= 1 + g / (1 - shape * MMath.Trigamma(shape));
if (Math.Abs(shape - oldShape) < 1e-8)
{
break;
}
}
if (Double.IsNaN(shape))
{
throw new Exception("shape is nan");
}
return(Gamma.FromShapeAndRate(shape, shape / mean));
}
/// <summary>
/// Constructs a Gamma distribution with the given log mean and mean logarithm.
/// </summary>
/// <param name="logMean">Log of desired expected value.</param>
/// <param name="meanLog">Desired expected logarithm.</param>
/// <returns>A new Gamma distribution.</returns>
/// <remarks>
/// This function is significantly slower than the other constructors since it
/// involves nonlinear optimization. The algorithm is a generalized Newton iteration,
/// described in "Estimating a Gamma distribution" by T. Minka, 2002.
/// </remarks>
public static Gamma FromLogMeanAndMeanLog(double logMean, double meanLog)
{
// logMean = log(shape)-log(rate)
// meanLog = Psi(shape)-log(rate)
// delta = log(shape)-Psi(shape)
double delta = logMean - meanLog;
if (delta <= 2e-16)
{
return(Gamma.PointMass(Math.Exp(logMean)));
}
double shape = 0.5 / delta;
for (int iter = 0; iter < 100; iter++)
{
double oldShape = shape;
double g = Math.Log(shape) - delta - MMath.Digamma(shape);
shape /= 1 + g / (1 - shape * MMath.Trigamma(shape));
if (Math.Abs(shape - oldShape) < 1e-8)
{
break;
}
}
if (Double.IsNaN(shape))
{
throw new Exception("shape is nan");
}
Gamma result = Gamma.FromShapeAndRate(shape, shape / Math.Exp(logMean));
return(result);
}
public void DigammaInvTest()
{
for (int i = 0; i < 1000; i++)
{
double y = -3 + i * 0.01;
double x = MMath.DigammaInv(y);
double y2 = MMath.Digamma(x);
double error = MMath.AbsDiff(y, y2, 1e-8);
Assert.True(error < 1e-8);
}
}
private static double LogMinusDigamma(double shape)
{
if (shape > largeShape)
{
// The next term in the series is -1/120/shape^4, which bounds the error.
return((0.5 - 1.0 / 12 / shape) / shape);
}
else
{
return(Math.Log(shape) - MMath.Digamma(shape));
}
}
static internal double ComputeMeanLogOneMinus(double trueCount, double falseCount)
{
if (double.IsPositiveInfinity(falseCount))
{
return(Math.Log(1 - trueCount));
}
if ((trueCount == 0.0) && (falseCount == 0.0))
{
throw new ImproperDistributionException(new Beta(trueCount, falseCount));
}
return(MMath.Digamma(falseCount) - MMath.Digamma(trueCount + falseCount));
}
/// <summary>
/// Used to compute log odds in the above operator
/// </summary>
/// <param name="trueCount"></param>
/// <param name="falseCount"></param>
/// <returns></returns>
internal static double ComputeLogOdds(double trueCount, double falseCount)
{
if (falseCount == Double.PositiveInfinity)
{
// compute log odds from prob true
return(MMath.Logit(trueCount));
}
else if ((trueCount == 0) || (falseCount == 0))
{
throw new ImproperMessageException(new Beta(trueCount, falseCount));
}
return(MMath.Digamma(trueCount) - MMath.Digamma(falseCount));
}
/// <summary>
/// Computes E[log(x)]
/// </summary>
/// <returns></returns>
public double GetMeanLog()
{
if (IsPointMass)
{
return(Math.Log(Point));
}
else if (!IsProper())
{
throw new ImproperDistributionException(this);
}
else
{
return(Power * (MMath.Digamma(Shape) - Math.Log(Rate)));
}
}
/// <summary>
/// The expected logarithm E[log(p)].
/// </summary>
/// <returns></returns>
public double GetMeanLog()
{
if (IsPointMass)
{
return(System.Math.Log(Point));
}
else if (!IsProper())
{
throw new ImproperDistributionException(this);
}
else
{
return(MMath.Digamma(TrueCount) - MMath.Digamma(TotalCount));
}
}
public void RandWishart()
{
// multivariate Gamma
double a = 2.7;
int d = 3;
PositiveDefiniteMatrix mTrue = new PositiveDefiniteMatrix(d, d);
mTrue.SetToIdentity();
mTrue.SetToProduct(mTrue, a);
LowerTriangularMatrix L = new LowerTriangularMatrix(d, d);
PositiveDefiniteMatrix X = new PositiveDefiniteMatrix(d, d);
PositiveDefiniteMatrix m = new PositiveDefiniteMatrix(d, d);
m.SetAllElementsTo(0);
double s = 0;
for (int i = 0; i < nsamples; i++)
{
Rand.Wishart(a, L);
X.SetToProduct(L, L.Transpose());
m = m + X;
s = s + X.LogDeterminant();
}
double sTrue = 0;
for (int i = 0; i < d; i++)
{
sTrue += MMath.Digamma(a - i * 0.5);
}
m.Scale(1.0 / nsamples);
s = s / nsamples;
Console.WriteLine("");
Console.WriteLine("Multivariate Gamma");
Console.WriteLine("-------------------");
Console.WriteLine("m = \n{0}", m);
double dError = m.MaxDiff(mTrue);
if (dError > TOLERANCE)
{
Assert.True(false, String.Format("Wishart({0}) mean: (should be {0}*I), error = {1}", a, dError));
}
if (System.Math.Abs(s - sTrue) > TOLERANCE)
{
Assert.True(false, string.Format("E[logdet]: {0} (should be {1})", s, sTrue));
}
}
/// <summary>
/// Gets the mean log determinant
/// </summary>
/// <returns>The mean log determinant</returns>
public double GetMeanLogDeterminant()
{
if (IsPointMass)
{
return(Point.LogDeterminant());
}
// E[logdet(X)] = -logdet(B) + sum_{i=0..d-1} digamma(a)
double s = 0;
int d = Dimension;
for (int i = 0; i < d; i++)
{
s += MMath.Digamma(Shape - i * 0.5);
}
s -= rate.LogDeterminant();
return(s);
}
/// <summary>VMP message to <c>sample</c>.</summary>
/// <param name="probTrue">Incoming message from <c>probTrue</c>. Must be a proper distribution. If uniform, the result will be uniform.</param>
/// <returns>The outgoing VMP message to the <c>sample</c> argument.</returns>
/// <remarks>
/// <para>The outgoing message is the exponential of the average log-factor value, where the average is over all arguments except <c>sample</c>. The formula is <c>exp(sum_(probTrue) p(probTrue) log(factor(sample,probTrue)))</c>.</para>
/// </remarks>
/// <exception cref="ImproperMessageException">
/// <paramref name="probTrue" /> is not a proper distribution.</exception>
public static Bernoulli SampleAverageLogarithm([SkipIfUniform] Beta probTrue)
{
if (probTrue.IsPointMass)
{
return(new Bernoulli(probTrue.Point));
}
else if (!probTrue.IsProper())
{
throw new ImproperMessageException(probTrue);
}
else
{
// E[x*log(p) + (1-x)*log(1-p)] = x*E[log(p)] + (1-x)*E[log(1-p)]
// p(x=true) = exp(E[log(p)])/(exp(E[log(p)]) + exp(E[log(1-p)]))
// log(p(x=true)/p(x=false)) = E[log(p)] - E[log(1-p)] = digamma(trueCount) - digamma(falseCount)
return(Bernoulli.FromLogOdds(MMath.Digamma(probTrue.TrueCount) - MMath.Digamma(probTrue.FalseCount)));
}
}
/// <summary>
/// The expected logarithms E[log(p)] and E[log(1-p)].
/// </summary>
/// <param name="eLogP"></param>
/// <param name="eLogOneMinusP"></param>
public void GetMeanLogs(out double eLogP, out double eLogOneMinusP)
{
if (IsPointMass)
{
eLogP = System.Math.Log(Point);
eLogOneMinusP = System.Math.Log(1 - Point);
}
else if (!IsProper())
{
throw new ImproperDistributionException(this);
}
else
{
double d = MMath.Digamma(TotalCount);
eLogP = MMath.Digamma(TrueCount) - d;
eLogOneMinusP = MMath.Digamma(FalseCount) - d;
}
}
public static void GetDerivLogZ(GammaPower sum, GammaPower toSum, double ds, double dds, double dr, double ddr, out double dlogZ, out double ddlogZ)
{
if (sum.Power != toSum.Power)
{
throw new ArgumentException($"sum.Power ({sum.Power}) != toSum.Power ({toSum.Power})");
}
if (toSum.IsPointMass)
{
throw new NotSupportedException();
}
if (toSum.IsUniform())
{
dlogZ = 0;
ddlogZ = 0;
return;
}
if (sum.IsPointMass)
{
// Z = toSum.GetLogProb(sum.Point)
// log(Z) = (toSum.Shape/toSum.Power - 1)*log(sum.Point) - toSum.Rate*sum.Point^(1/toSum.Power) + toSum.Shape*log(toSum.Rate) - GammaLn(toSum.Shape)
if (sum.Point == 0)
{
throw new NotSupportedException();
}
double logSumOverPower = Math.Log(sum.Point) / toSum.Power;
double powSum = Math.Exp(logSumOverPower);
double logRate = Math.Log(toSum.Rate);
double digammaShape = MMath.Digamma(toSum.Shape);
double shapeOverRate = toSum.Shape / toSum.Rate;
dlogZ = ds * logSumOverPower - dr * powSum + ds * logRate + shapeOverRate * dr - digammaShape * ds;
ddlogZ = dds * logSumOverPower - ddr * powSum + dds * logRate + 2 * ds * dr / toSum.Rate + shapeOverRate * ddr - MMath.Trigamma(toSum.Shape) * ds - digammaShape * dds;
}
else
{
GammaPower product = sum * toSum;
double cs = (MMath.Digamma(product.Shape) - Math.Log(product.Shape)) - (MMath.Digamma(toSum.Shape) - Math.Log(toSum.Shape));
double cr = toSum.Shape / toSum.Rate - product.Shape / product.Rate;
double css = MMath.Trigamma(product.Shape) - MMath.Trigamma(toSum.Shape);
double csr = 1 / toSum.Rate - 1 / product.Rate;
double crr = product.Shape / (product.Rate * product.Rate) - toSum.Shape / (toSum.Rate * toSum.Rate);
dlogZ = cs * ds + cr * dr;
ddlogZ = cs * dds + cr * ddr + css * ds * ds + 2 * csr * ds * dr + crr * dr * dr;
}
}
public static Gaussian DAverageConditional([SkipIfUniform] Gamma exp, [Proper] Gaussian d)
{
// as a function of d, the factor is Ga(exp(d); shape, rate) = exp(d*(shape-1) -rate*exp(d))
if (exp.IsUniform())
{
return(Gaussian.Uniform());
}
if (exp.IsPointMass)
{
return(ExpOp.DAverageConditional(exp.Point));
}
if (exp.Rate < 0)
{
throw new ImproperMessageException(exp);
}
if (exp.Rate == 0)
{
return(Gaussian.FromNatural(exp.Shape - 1, 0));
}
if (d.IsUniform())
{
if (exp.Shape <= 1)
{
throw new ArgumentException("The posterior has infinite variance due to input of Exp distributed as " + d + " and output of Exp distributed as " + exp +
" (shape <= 1)");
}
// posterior for d is a shifted log-Gamma distribution:
// exp((a-1)*d - b*exp(d)) =propto exp(a*(d+log(b)) - exp(d+log(b)))
// we find the Gaussian with same moments.
// u = d+log(b)
// E[u] = digamma(a-1)
// E[d] = E[u]-log(b) = digamma(a-1)-log(b)
// var(d) = var(u) = trigamma(a-1)
double lnRate = Math.Log(exp.Rate);
return(new Gaussian(MMath.Digamma(exp.Shape - 1) - lnRate, MMath.Trigamma(exp.Shape - 1)));
}
double aMinus1 = exp.Shape - 1;
double b = exp.Rate;
if (d.IsPointMass)
{
double x = d.Point;
double expx = Math.Exp(x);
double dlogf = aMinus1 - b * expx;
double ddlogf = -b * expx;
return(Gaussian.FromDerivatives(x, dlogf, ddlogf, true));
}
double dmode, dmin, dmax;
GetIntegrationBounds(exp, d, out dmode, out dmin, out dmax);
double expmode = Math.Exp(dmode);
int n = QuadratureNodeCount;
double inc = (dmax - dmin) / (n - 1);
MeanVarianceAccumulator mva = new MeanVarianceAccumulator();
for (int i = 0; i < n; i++)
{
double x = dmin + i * inc;
double xMinusMode = x - dmode;
double diff = aMinus1 * xMinusMode - b * (Math.Exp(x) - expmode)
- 0.5 * ((x * x - dmode * dmode) * d.Precision - 2 * xMinusMode * d.MeanTimesPrecision);
double p = Math.Exp(diff);
mva.Add(x, p);
if (double.IsNaN(mva.Variance))
{
throw new Exception();
}
}
double dMean = mva.Mean;
double dVariance = mva.Variance;
Gaussian result = Gaussian.FromMeanAndVariance(dMean, dVariance);
result.SetToRatio(result, d, true);
return(result);
}
/// <summary>
/// EP message to 'exp'
/// </summary>
/// <param name="exp">Incoming message from 'exp'.</param>
/// <param name="d">Incoming message from 'd'. Must be a proper distribution. If uniform, the result will be uniform.</param>
/// <param name="to_d">Previous outgoing message to 'd'.</param>
/// <returns>The outgoing EP message to the 'exp' argument</returns>
/// <remarks><para>
/// The outgoing message is a distribution matching the moments of 'exp' as the random arguments are varied.
/// The formula is <c>proj[p(exp) sum_(d) p(d) factor(exp,d)]/p(exp)</c>.
/// </para></remarks>
/// <exception cref="ImproperMessageException"><paramref name="d"/> is not a proper distribution</exception>
public static Gamma ExpAverageConditional(Gamma exp, [Proper] Gaussian d, Gaussian to_d)
{
if (d.IsPointMass)
{
return(Gamma.PointMass(Math.Exp(d.Point)));
}
if (d.IsUniform())
{
return(Gamma.FromShapeAndRate(0, 0));
}
if (exp.IsPointMass)
{
// Z = int_y delta(x - exp(y)) N(y; my, vy) dy
// = int_u delta(x - u) N(log(u); my, vy)/u du
// = N(log(x); my, vy)/x
// logZ = -log(x) -0.5/vy*(log(x)-my)^2
// dlogZ/dx = -1/x -1/vy*(log(x)-my)/x
// d2logZ/dx2 = -dlogZ/dx/x -1/vy/x^2
// log Ga(x;a,b) = (a-1)*log(x) - bx
// dlogGa/dx = (a-1)/x - b
// d2logGa/dx2 = -(a-1)/x^2
// match derivatives and solve for (a,b)
double shape = (1 + d.GetMean() - Math.Log(exp.Point)) * d.Precision;
double rate = d.Precision / exp.Point;
return(Gamma.FromShapeAndRate(shape, rate));
}
if (exp.IsUniform())
{
return(ExpAverageLogarithm(d));
}
if (to_d.IsUniform() && exp.Shape > 1)
{
to_d = new Gaussian(MMath.Digamma(exp.Shape - 1) - Math.Log(exp.Rate), MMath.Trigamma(exp.Shape - 1));
}
double mD, vD;
Gaussian dMarginal = d * to_d;
dMarginal.GetMeanAndVariance(out mD, out vD);
double Z = 0;
double sumy = 0;
double sumexpy = 0;
if (vD < 1e-6)
{
double m, v;
d.GetMeanAndVariance(out m, out v);
return(Gamma.FromLogMeanAndMeanLog(m + v / 2.0, m));
}
//if (vD < 10)
if (true)
{
// Use Gauss-Hermite quadrature
double[] nodes = new double[QuadratureNodeCount];
double[] weights = new double[QuadratureNodeCount];
Quadrature.GaussianNodesAndWeights(mD, vD, nodes, weights);
for (int i = 0; i < weights.Length; i++)
{
weights[i] = Math.Log(weights[i]);
}
if (!to_d.IsUniform())
{
// modify the weights to include q(y_k)/N(y_k;m,v)
for (int i = 0; i < weights.Length; i++)
{
weights[i] += d.GetLogProb(nodes[i]) - dMarginal.GetLogProb(nodes[i]);
}
}
double maxLogF = Double.NegativeInfinity;
// f(x,y) = Ga(exp(y); shape, rate) = exp(y*(shape-1) -rate*exp(y))
// Z E[x] = int_y int_x x Ga(x;a,b) delta(x - exp(y)) N(y;my,vy) dx dy
// = int_y exp(y) Ga(exp(y);a,b) N(y;my,vy) dy
// Z E[log(x)] = int_y y Ga(exp(y);a,b) N(y;my,vy) dy
for (int i = 0; i < weights.Length; i++)
{
double y = nodes[i];
double logf = weights[i] + (exp.Shape - 1) * y - exp.Rate * Math.Exp(y);
if (logf > maxLogF)
{
maxLogF = logf;
}
weights[i] = logf;
}
for (int i = 0; i < weights.Length; i++)
{
double y = nodes[i];
double f = Math.Exp(weights[i] - maxLogF);
double f_y = f * y;
double fexpy = f * Math.Exp(y);
Z += f;
sumy += f_y;
sumexpy += fexpy;
}
}
else
{
Converter <double, double> p = delegate(double y) {
return(d.GetLogProb(y) + (exp.Shape - 1) * y - exp.Rate * Math.Exp(y));
};
double sc = Math.Sqrt(vD);
double offset = p(mD);
Z = Quadrature.AdaptiveClenshawCurtis(z => Math.Exp(p(sc * z + mD) - offset), 1, 16, 1e-6);
sumy = Quadrature.AdaptiveClenshawCurtis(z => (sc * z + mD) * Math.Exp(p(sc * z + mD) - offset), 1, 16, 1e-6);
sumexpy = Quadrature.AdaptiveClenshawCurtis(z => Math.Exp(sc * z + mD + p(sc * z + mD) - offset), 1, 16, 1e-6);
}
if (Z == 0)
{
throw new ApplicationException("Z==0");
}
double s = 1.0 / Z;
if (Double.IsPositiveInfinity(s))
{
throw new ApplicationException("s is -inf");
}
double meanLog = sumy * s;
double mean = sumexpy * s;
Gamma result = Gamma.FromMeanAndMeanLog(mean, meanLog);
if (ForceProper)
{
result.SetToRatioProper(result, exp);
}
else
{
result.SetToRatio(result, exp);
}
if (Double.IsNaN(result.Shape) || Double.IsNaN(result.Rate))
{
throw new ApplicationException("result is nan");
}
return(result);
}
/// <summary>
/// EP message to 'd'
/// </summary>
/// <param name="exp">Incoming message from 'exp'. Must be a proper distribution. If uniform, the result will be uniform.</param>
/// <param name="d">Incoming message from 'd'. Must be a proper distribution. If uniform, the result will be uniform.</param>
/// <param name="result">Modified to contain the outgoing message</param>
/// <returns><paramref name="result"/></returns>
/// <remarks><para>
/// The outgoing message is a distribution matching the moments of 'd' as the random arguments are varied.
/// The formula is <c>proj[p(d) sum_(exp) p(exp) factor(exp,d)]/p(d)</c>.
/// </para></remarks>
/// <exception cref="ImproperMessageException"><paramref name="exp"/> is not a proper distribution</exception>
/// <exception cref="ImproperMessageException"><paramref name="d"/> is not a proper distribution</exception>
//internal static Gaussian DAverageConditional_slow([SkipIfUniform] Gamma exp, [Proper] Gaussian d)
//{
// Gaussian to_d = exp.Shape<=1 || exp.Rate==0 ?
// Gaussian.Uniform()
// : new Gaussian(MMath.Digamma(exp.Shape-1) - Math.Log(exp.Rate), MMath.Trigamma(exp.Shape));
// //var to_d = Gaussian.Uniform();
// for (int i = 0; i < QuadratureIterations; i++) {
// to_d = DAverageConditional(exp, d, to_d);
// }
// return to_d;
//}
// to_d does not need to be Fresh. it is only used for quadrature proposal.
public static Gaussian DAverageConditional([SkipIfUniform] Gamma exp, [Proper] Gaussian d, Gaussian result)
{
if (exp.IsUniform() || d.IsPointMass)
{
return(Gaussian.Uniform());
}
if (exp.IsPointMass)
{
return(DAverageConditional(exp.Point));
}
if (exp.Rate < 0)
{
throw new ImproperMessageException(exp);
}
if (d.IsUniform())
{
// posterior for d is a shifted log-Gamma distribution:
// exp((a-1)*d - b*exp(d)) =propto exp(a*(d+log(b)) - exp(d+log(b)))
// we find the Gaussian with same moments.
// u = d+log(b)
// E[u] = digamma(a-1)
// E[d] = E[u]-log(b) = digamma(a-1)-log(b)
// var(d) = var(u) = trigamma(a-1)
double lnRate = Math.Log(exp.Rate);
return(new Gaussian(MMath.Digamma(exp.Shape - 1) - lnRate, MMath.Trigamma(exp.Shape - 1)));
}
// We use moment matching to find the best Gaussian message.
// The moments are computed via quadrature.
// Z = int_y f(x,y) q(y) dy =approx sum_k w_k f(x,y_k) q(y_k)/N(y_k;m,v)
// f(x,y) = Ga(exp(y); shape, rate) = exp(y*(shape-1) -rate*exp(y))
double[] nodes = new double[QuadratureNodeCount];
double[] weights = new double[QuadratureNodeCount];
double moD, voD;
d.GetMeanAndVariance(out moD, out voD);
double mD, vD;
if (result.IsUniform() && exp.Shape > 1)
{
result = new Gaussian(MMath.Digamma(exp.Shape - 1) - Math.Log(exp.Rate), MMath.Trigamma(exp.Shape - 1));
}
Gaussian dMarginal = d * result;
dMarginal.GetMeanAndVariance(out mD, out vD);
Quadrature.GaussianNodesAndWeights(mD, vD, nodes, weights);
if (!result.IsUniform())
{
// modify the weights to include q(y_k)/N(y_k;m,v)
for (int i = 0; i < weights.Length; i++)
{
weights[i] *= Math.Exp(d.GetLogProb(nodes[i]) - Gaussian.GetLogProb(nodes[i], mD, vD));
}
}
double Z = 0;
double sumy = 0;
double sumy2 = 0;
double maxLogF = Double.NegativeInfinity;
for (int i = 0; i < weights.Length; i++)
{
double y = nodes[i];
double logf = Math.Log(weights[i]) + (exp.Shape - 1) * y - exp.Rate * Math.Exp(y);
if (logf > maxLogF)
{
maxLogF = logf;
}
weights[i] = logf;
}
for (int i = 0; i < weights.Length; i++)
{
double y = nodes[i];
double f = Math.Exp(weights[i] - maxLogF);
double f_y = f * y;
double fyy = f_y * y;
Z += f;
sumy += f_y;
sumy2 += fyy;
}
if (Z == 0)
{
return(Gaussian.Uniform());
}
double s = 1.0 / Z;
double mean = sumy * s;
double var = sumy2 * s - mean * mean;
if (var <= 0.0)
{
double quadratureGap = 0.1;
var = 2 * vD * quadratureGap * quadratureGap;
}
result = new Gaussian(mean, var);
if (ForceProper)
{
result.SetToRatioProper(result, d);
}
else
{
result.SetToRatio(result, d);
}
if (result.Precision < -1e10)
{
throw new ApplicationException("result has negative precision");
}
if (Double.IsPositiveInfinity(result.Precision))
{
throw new ApplicationException("result is point mass");
}
if (Double.IsNaN(result.Precision) || Double.IsNaN(result.MeanTimesPrecision))
{
throw new ApplicationException("result is nan");
}
return(result);
}
//internal static Gaussian DAverageConditional_slow([SkipIfUniform] Gamma exp, [Proper] Gaussian d)
//{
// Gaussian to_d = exp.Shape<=1 || exp.Rate==0 ?
// Gaussian.Uniform()
// : new Gaussian(MMath.Digamma(exp.Shape-1) - Math.Log(exp.Rate), MMath.Trigamma(exp.Shape));
// //var to_d = Gaussian.Uniform();
// for (int i = 0; i < QuadratureIterations; i++) {
// to_d = DAverageConditional(exp, d, to_d);
// }
// return to_d;
//}
// to_d does not need to be Fresh. it is only used for quadrature proposal.
/// <include file='FactorDocs.xml' path='factor_docs/message_op_class[@name="ExpOp"]/message_doc[@name="DAverageConditional(Gamma, Gaussian, Gaussian)"]/*'/>
public static Gaussian DAverageConditional([SkipIfUniform] Gamma exp, [Proper] Gaussian d, Gaussian result)
{
if (exp.IsUniform() || d.IsUniform() || d.IsPointMass || exp.IsPointMass || exp.Rate <= 0)
{
return(ExpOp_Slow.DAverageConditional(exp, d));
}
// We use moment matching to find the best Gaussian message.
// The moments are computed via quadrature.
// Z = int_y f(x,y) q(y) dy =approx sum_k w_k f(x,y_k) q(y_k)/N(y_k;m,v)
// f(x,y) = Ga(exp(y); shape, rate) = exp(y*(shape-1) -rate*exp(y))
double[] nodes = new double[QuadratureNodeCount];
double[] weights = new double[QuadratureNodeCount];
double moD, voD;
d.GetMeanAndVariance(out moD, out voD);
double mD, vD;
if (result.IsUniform() && exp.Shape > 1)
{
result = new Gaussian(MMath.Digamma(exp.Shape - 1) - Math.Log(exp.Rate), MMath.Trigamma(exp.Shape - 1));
}
Gaussian dMarginal = d * result;
dMarginal.GetMeanAndVariance(out mD, out vD);
if (vD == 0)
{
return(ExpOp_Slow.DAverageConditional(exp, d));
}
Quadrature.GaussianNodesAndWeights(mD, vD, nodes, weights);
if (!result.IsUniform())
{
// modify the weights to include q(y_k)/N(y_k;m,v)
for (int i = 0; i < weights.Length; i++)
{
weights[i] *= Math.Exp(d.GetLogProb(nodes[i]) - Gaussian.GetLogProb(nodes[i], mD, vD));
}
}
double Z = 0;
double sumy = 0;
double sumy2 = 0;
double maxLogF = Double.NegativeInfinity;
for (int i = 0; i < weights.Length; i++)
{
double y = nodes[i];
double logf = Math.Log(weights[i]) + (exp.Shape - 1) * y - exp.Rate * Math.Exp(y);
if (logf > maxLogF)
{
maxLogF = logf;
}
weights[i] = logf;
}
for (int i = 0; i < weights.Length; i++)
{
double y = nodes[i];
double f = Math.Exp(weights[i] - maxLogF);
double f_y = f * y;
double fyy = f_y * y;
Z += f;
sumy += f_y;
sumy2 += fyy;
}
if (Z == 0)
{
return(Gaussian.Uniform());
}
double s = 1.0 / Z;
double mean = sumy * s;
double var = sumy2 * s - mean * mean;
// TODO: explain this
if (var <= 0.0)
{
double quadratureGap = 0.1;
var = 2 * vD * quadratureGap * quadratureGap;
}
result = new Gaussian(mean, var);
result.SetToRatio(result, d, ForceProper);
if (result.Precision < -1e10)
{
throw new InferRuntimeException("result has negative precision");
}
if (Double.IsPositiveInfinity(result.Precision))
{
throw new InferRuntimeException("result is point mass");
}
if (Double.IsNaN(result.Precision) || Double.IsNaN(result.MeanTimesPrecision))
{
return(ExpOp_Slow.DAverageConditional(exp, d));
}
return(result);
}
#pragma warning disable 162
#endif
/// <summary>
/// Find a Beta distribution with given integral and mean times a Beta weight function.
/// </summary>
/// <param name="mean">The desired value of the mean</param>
/// <param name="logZ">The desired value of the integral</param>
/// <param name="a">trueCount-1 of the weight function</param>
/// <param name="b">falseCount-1 of the weight function</param>
/// <returns></returns>
private static Beta BetaFromMeanAndIntegral(double mean, double logZ, double a, double b)
{
// The constraints are:
// 1. int_p to_p(p) p^a (1-p)^b dp = exp(logZ)
// 2. int_p to_p(p) p p^a (1-p)^b dp = mean*exp(logZ)
// Let to_p(p) = Beta(p; af, bf)
// The LHS of (1) is gamma(af+bf)/gamma(af+bf+a+b) gamma(af+a)/gamma(af) gamma(bf+b)/gamma(bf)
// The LHS of (2) is gamma(af+bf)/gamma(af+bf+a+b+1) gamma(af+a+1)/gamma(af) gamma(bf+b)/gamma(bf)
// The ratio of (2)/(1) is gamma(af+a+1)/gamma(af+a) gamma(af+bf+a+b)/gamma(af+bf+a+b+1) = (af+a)/(af+bf+a+b) = mean
// Solving for bf gives bf = (af+a)/mean - (af+a+b).
// To solve for af, we apply a generalized Newton algorithm to solve equation (1) with bf substituted.
// af0 is the smallest value of af that ensures (af >= 0, bf >= 0).
if (mean <= 0)
{
throw new ArgumentException("mean <= 0");
}
if (mean >= 1)
{
throw new ArgumentException("mean >= 1");
}
if (double.IsNaN(mean))
{
throw new ArgumentException("mean is NaN");
}
// If exp(logZ) exceeds the largest possible value of (1), then we return a point mass.
// gammaln(x) =approx (x-0.5)*log(x) - x + 0.5*log(2pi)
// (af+x)*log(af+x) =approx (af+x)*log(af) + x + 0.5*x*x/af
// For large af, logZ = (af+bf-0.5)*log(af+bf) - (af+bf+a+b-0.5)*log(af+bf+a+b) +
// (af+a-0.5)*log(af+a) - (af-0.5)*log(af) +
// (bf+b-0.5)*log(bf+b) - (bf-0.5)*log(bf)
// =approx (af+bf-0.5)*log(af+bf) - ((af+bf+a+b-0.5)*log(af+bf) + (a+b) + 0.5*(a+b)*(a+b)/(af+bf) -0.5*(a+b)/(af+bf)) +
// ((af+a-0.5)*log(af) + a + 0.5*a*a/af - 0.5*a/af) - (af-0.5)*log(af) +
// ((bf+b-0.5)*log(bf) + b + 0.5*b*b/bf - 0.5*b/bf) - (bf-0.5)*log(bf)
// = -(a+b)*log(af+bf) - 0.5*(a+b)*(a+b-1)/(af+bf) + a*log(af) + 0.5*a*(a-1)/af + b*log(bf) + 0.5*b*(b-1)/bf
// =approx (a+b)*log(m) + b*log((1-m)/m) + 0.5*(a+b)*(a+b-1)*m/af - 0.5*a*(a+1)/af - 0.5*b*(b+1)*m/(1-m)/af
// =approx (a+b)*log(mean) + b*log((1-mean)/mean)
double maxLogZ = (a + b) * Math.Log(mean) + b * Math.Log((1 - mean) / mean);
// slope determines whether maxLogZ is the maximum or minimum possible value of logZ
double slope = (a + b) * (a + b - 1) * mean - a * (a + 1) - b * (b + 1) * mean / (1 - mean);
if ((slope <= 0 && logZ >= maxLogZ) || (slope > 0 && logZ <= maxLogZ))
{
// optimal af is infinite
return(Beta.PointMass(mean));
}
// bf = (af+bx)*(1-m)/m
double bx = -(mean * (a + b) - a) / (1 - mean);
// af0 is the lower bound for af
// we need both af>0 and bf>0
double af0 = Math.Max(0, -bx);
double x = Math.Max(0, bx);
double af = af0 + 1; // initial guess for af
double invMean = 1 / mean;
double bf = (af + a) * invMean - (af + a + b);
int numIters = 20;
for (int iter = 0; iter < numIters; iter++)
{
double old_af = af;
double f = (MMath.GammaLn(af + bf) - MMath.GammaLn(af + bf + a + b)) + (MMath.GammaLn(af + a) - MMath.GammaLn(af)) +
(MMath.GammaLn(bf + b) - MMath.GammaLn(bf));
double g = (MMath.Digamma(af + bf) - MMath.Digamma(af + bf + a + b)) * invMean + (MMath.Digamma(af + a) - MMath.Digamma(af)) +
(MMath.Digamma(bf + b) - MMath.Digamma(bf)) * (invMean - 1);
// fit a fcn of the form: s*log((af-af0)/(af+x)) + c
// whose deriv is s/(af-af0) - s/(af+x)
double s = g / (1 / (af - af0) - 1 / (af + x));
double c = f - s * Math.Log((af - af0) / (af + x));
bool isIncreasing = (x > -af0);
if ((!isIncreasing && c >= logZ) || (isIncreasing && c <= logZ))
{
// the approximation doesn't fit; use Gauss-Newton instead
af += (logZ - f) / g;
}
else
{
// now solve s*log((af-af0)/(af+x))+c = logz
// af-af0 = exp((logz-c)/s) (af+x)
af = af0 + (x + af0) / MMath.ExpMinus1((c - logZ) / s);
//if (af == af0)
// throw new ArgumentException("logZ is out of range");
}
if (double.IsNaN(af))
{
throw new InferRuntimeException("af is nan");
}
bf = (af + a) / mean - (af + a + b);
if (Math.Abs(af - old_af) < 1e-8 || af == af0)
{
break;
}
//if (iter == numIters-1)
// throw new Exception("not converging");
}
if (false)
{
// check that integrals are correct
double f = (MMath.GammaLn(af + bf) - MMath.GammaLn(af + bf + a + b)) + (MMath.GammaLn(af + a) - MMath.GammaLn(af)) +
(MMath.GammaLn(bf + b) - MMath.GammaLn(bf));
if (Math.Abs(f - logZ) > 1e-6)
{
throw new InferRuntimeException("wrong f");
}
double f2 = (MMath.GammaLn(af + bf) - MMath.GammaLn(af + bf + a + b + 1)) + (MMath.GammaLn(af + a + 1) - MMath.GammaLn(af)) +
(MMath.GammaLn(bf + b) - MMath.GammaLn(bf));
if (Math.Abs(f2 - (Math.Log(mean) + logZ)) > 1e-6)
{
throw new InferRuntimeException("wrong f2");
}
}
return(new Beta(af, bf));
}
/// <summary>
/// Find a Beta distribution with given integral and mean times a Beta weight function.
/// </summary>
/// <param name="mean">The desired value of the mean</param>
/// <param name="logZ">The desired value of the integral</param>
/// <param name="a">trueCount-1 of the weight function</param>
/// <param name="b">falseCount-1 of the weight function</param>
/// <returns></returns>
private static Beta BetaFromMeanAndIntegral(double mean, double logZ, double a, double b)
{
// The constraints are:
// 1. int_p to_p(p) p^a (1-p)^b dp = exp(logZ)
// 2. int_p to_p(p) p p^a (1-p)^b dp = mean*exp(logZ)
// Let to_p(p) = Beta(p; af, bf)
// The LHS of (1) is gamma(af+bf)/gamma(af+bf+a+b) gamma(af+a)/gamma(af) gamma(bf+b)/gamma(bf)
// The LHS of (2) is gamma(af+bf)/gamma(af+bf+a+b+1) gamma(af+a+1)/gamma(af) gamma(bf+b)/gamma(bf)
// The ratio of (2)/(1) is gamma(af+a+1)/gamma(af+a) gamma(af+bf+a+b)/gamma(af+bf+a+b+1) = (af+a)/(af+bf+a+b) = mean
// Solving for bf gives bf = (af+a)/mean - (af+a+b).
// To solve for af, we apply a generalized Newton algorithm to solve equation (1) with bf substituted.
// af0 is the smallest value of af that ensures (af >= 0, bf >= 0).
if (mean <= 0)
{
throw new ArgumentException("mean <= 0");
}
if (mean >= 1)
{
throw new ArgumentException("mean >= 1");
}
if (double.IsNaN(mean))
{
throw new ArgumentException("mean is NaN");
}
// bf = (af+bx)*(1-m)/m
double bx = -(mean * (a + b) - a) / (1 - mean);
// af0 is the lower bound for af
// we need both af>0 and bf>0
double af0 = Math.Max(0, -bx);
double x = Math.Max(0, bx);
double af = af0 + 1; // initial guess for af
double invMean = 1 / mean;
double bf = (af + a) * invMean - (af + a + b);
for (int iter = 0; iter < 20; iter++)
{
double old_af = af;
double f = (MMath.GammaLn(af + bf) - MMath.GammaLn(af + bf + a + b)) + (MMath.GammaLn(af + a) - MMath.GammaLn(af)) + (MMath.GammaLn(bf + b) - MMath.GammaLn(bf));
double g = (MMath.Digamma(af + bf) - MMath.Digamma(af + bf + a + b)) * invMean + (MMath.Digamma(af + a) - MMath.Digamma(af)) + (MMath.Digamma(bf + b) - MMath.Digamma(bf)) * (invMean - 1);
// fit a fcn of the form: s*log((af-af0)/(af+x)) + c
// whose deriv is s/(af-af0) - s/(af+x)
double s = g / (1 / (af - af0) - 1 / (af + x));
double c = f - s * Math.Log((af - af0) / (af + x));
bool isIncreasing = (x > -af0);
if ((!isIncreasing && c >= logZ) || (isIncreasing && c <= logZ))
{
// the approximation doesn't fit; use Gauss-Newton instead
af += (logZ - f) / g;
}
else
{
// now solve s*log((af-af0)/(af+x))+c = logz
// af-af0 = exp((logz-c)/s) (af+x)
af = af0 + (x + af0) / MMath.ExpMinus1((c - logZ) / s);
if (af == af0)
{
throw new ArgumentException("logZ is out of range");
}
}
if (double.IsNaN(af))
{
throw new ApplicationException("af is nan");
}
bf = (af + a) / mean - (af + a + b);
if (Math.Abs(af - old_af) < 1e-8)
{
break;
}
}
if (false)
{
// check that integrals are correct
double f = (MMath.GammaLn(af + bf) - MMath.GammaLn(af + bf + a + b)) + (MMath.GammaLn(af + a) - MMath.GammaLn(af)) + (MMath.GammaLn(bf + b) - MMath.GammaLn(bf));
if (Math.Abs(f - logZ) > 1e-6)
{
throw new ApplicationException("wrong f");
}
double f2 = (MMath.GammaLn(af + bf) - MMath.GammaLn(af + bf + a + b + 1)) + (MMath.GammaLn(af + a + 1) - MMath.GammaLn(af)) + (MMath.GammaLn(bf + b) - MMath.GammaLn(bf));
if (Math.Abs(f2 - (Math.Log(mean) + logZ)) > 1e-6)
{
throw new ApplicationException("wrong f2");
}
}
return(new Beta(af, bf));
}