/// <include file='FactorDocs.xml' path='factor_docs/message_op_class[@name="PowerOp"]/message_doc[@name="XAverageConditional(Pareto, Gamma, double)"]/*'/>
public static Gamma XAverageConditional(Pareto pow, Gamma x, double y)
{
// factor is delta(pow - x^y)
// marginal for x is Pareto(x^y; s,L) Ga(x; a,b)
// =propto x^(a-1-y(s+1)) exp(-bx) for x >= L^(1/y)
// we can compute moments via the incomplete Gamma function.
// change variables to: z=bx, dz=b dx
// int_LL^inf x^(c-1) exp(-bx) dx = int_(bLL)^inf z^(c-1)/b^c exp(-z) dz = gammainc(c,bLL,inf)/b^c
double lowerBound = System.Math.Pow(pow.LowerBound, 1 / y);
double b = x.Rate;
double c = x.Shape - y * (pow.Shape + 1);
double bL = b * lowerBound;
double m, m2;
if (y > 0)
{
// note these ratios can be simplified
double z = MMath.GammaUpper(c, bL);
m = MMath.GammaUpper(c + 1, bL) * c / z / b;
m2 = MMath.GammaUpper(c + 2, bL) * c * (c + 1) / z / (b * b);
}
else
{
double z = MMath.GammaLower(c, bL);
m = MMath.GammaLower(c + 1, bL) * c / z / b;
m2 = MMath.GammaLower(c + 2, bL) * c * (c + 1) / z / (b * b);
}
double v = m2 - m * m;
Gamma xPost = Gamma.FromMeanAndVariance(m, v);
Gamma result = new Gamma();
result.SetToRatio(xPost, x, true);
return(result);
}
/// <summary>
/// Get the mean and variance after truncation.
/// </summary>
/// <param name="mean"></param>
/// <param name="variance"></param>
public void GetMeanAndVariance(out double mean, out double variance)
{
if (this.Gamma.IsPointMass)
{
mean = this.Gamma.Point;
variance = 0.0;
}
else if (!IsProper())
{
throw new ImproperDistributionException(this);
}
else
{
double Z = GetNormalizer();
if (Z == 0)
{
mean = Math.Min(UpperBound, Math.Max(LowerBound, this.Gamma.GetMean()));
variance = 0.0;
return;
}
double m = this.Gamma.Shape / this.Gamma.Rate;
double sum = m * (MMath.GammaLower(this.Gamma.Shape + 1, this.Gamma.Rate * UpperBound) - MMath.GammaLower(this.Gamma.Shape + 1, this.Gamma.Rate * LowerBound));
mean = sum / Z;
double sum2 = m * (this.Gamma.Shape + 1) / this.Gamma.Rate * (MMath.GammaLower(this.Gamma.Shape + 2, this.Gamma.Rate * UpperBound) - MMath.GammaLower(this.Gamma.Shape + 2, this.Gamma.Rate * LowerBound));
variance = sum2 / Z - mean * mean;
}
}
/// <include file='FactorDocs.xml' path='factor_docs/message_op_class[@name="IsGreaterThanOp"]/message_doc[@name="IsGreaterThanAverageConditional(Poisson, int)"]/*'/>
public static Bernoulli IsGreaterThanAverageConditional([Proper] Poisson a, int b)
{
if (a.IsPointMass)
{
return(Bernoulli.PointMass(a.Point > b));
}
if (a.Precision == 1)
{
if (b < 0)
{
return(Bernoulli.PointMass(true));
}
else
{
return(new Bernoulli(MMath.GammaLower(b + 1, a.Rate)));
}
}
else
{
double sum = 0;
for (int i = 0; i <= b; i++)
{
sum += Math.Exp(a.GetLogProb(i));
}
if (sum > 1)
{
sum = 1; // this can happen due to round-off errors
}
return(new Bernoulli(1 - sum));
}
}
/// <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 E[x^power]
/// </summary>
/// <returns></returns>
public double GetMeanPower(double power)
{
if (power == 0.0)
{
return(1.0);
}
else if (IsPointMass)
{
return(Math.Pow(Point, power));
}
//else if (Rate == 0.0) return (power > 0) ? Double.PositiveInfinity : 0.0;
else if (!IsProper())
{
throw new ImproperDistributionException(this);
}
else if (this.Gamma.Shape <= -power && LowerBound == 0)
{
throw new ArgumentException("Cannot compute E[x^" + power + "] for " + this + " (shape <= " + (-power) + ")");
}
else
{
double Z = GetNormalizer();
double shapePlusPower = this.Gamma.Shape + power;
double Z1;
bool regularized = shapePlusPower >= 1;
if (regularized)
{
Z1 = Math.Exp(MMath.GammaLn(shapePlusPower) - MMath.GammaLn(this.Gamma.Shape)) *
(MMath.GammaLower(shapePlusPower, this.Gamma.Rate * UpperBound) - MMath.GammaLower(shapePlusPower, this.Gamma.Rate * LowerBound));
}
else
{
Z1 = Math.Exp(-MMath.GammaLn(this.Gamma.Shape)) *
(MMath.GammaUpper(shapePlusPower, this.Gamma.Rate * LowerBound, regularized) - MMath.GammaUpper(shapePlusPower, this.Gamma.Rate * UpperBound, regularized));
}
return(Math.Pow(this.Gamma.Rate, -power) * Z1 / Z);
}
}
/// <summary>
/// Compute the probability that a sample from this distribution is less than x.
/// </summary>
/// <param name="x">Any real number.</param>
/// <returns>The cumulative gamma distribution at <paramref name="x"/></returns>
public double GetProbLessThan(double x)
{
if (x < 0.0)
{
return(0.0);
}
else if (double.IsPositiveInfinity(x))
{
return(1.0);
}
else if (this.IsPointMass)
{
return((this.Point < x) ? 1.0 : 0.0);
}
else if (this.IsUniform())
{
throw new ImproperDistributionException(this);
}
else
{
return(MMath.GammaLower(Shape, x * Rate));
}
}
/// <summary>
/// Returns the mean (first moment) of the distribution
/// </summary>
/// <returns></returns>
public double GetMean()
{
if (this.Gamma.IsPointMass)
{
return(this.Gamma.Point);
}
else if (!IsProper())
{
throw new ImproperDistributionException(this);
}
else
{
double Z = GetNormalizer();
if (Z == 0)
{
double mean = this.Gamma.GetMean();
return(Math.Min(UpperBound, Math.Max(LowerBound, mean)));
}
// if Z is not zero, then Z1 cannot be zero.
double Z1 = MMath.GammaLower(this.Gamma.Shape + 1, this.Gamma.Rate * UpperBound) - MMath.GammaLower(this.Gamma.Shape + 1, this.Gamma.Rate * LowerBound);
double sum = this.Gamma.Shape / this.Gamma.Rate * Z1;
return(sum / Z);
}
}
