/// <include file='FactorDocs.xml' path='factor_docs/message_op_class[@name="ExpOp_Laplace"]/message_doc[@name="DAverageConditional(Gamma, Gaussian, Gaussian)"]/*'/>
public static Gaussian DAverageConditional([SkipIfUniform] Gamma exp, [Proper] Gaussian d, Gaussian to_d)
{
if (exp.IsPointMass)
{
return(ExpOp.DAverageConditional(exp.Point));
}
Gaussian dPost = d * to_d;
double dhat = dPost.GetMean();
double ehat = Math.Exp(dhat);
double a = exp.Shape;
double b = exp.Rate;
double dlogf = (a - 1) - b * ehat;
double ddlogf = -b * ehat;
double r = -ddlogf;
if (ForceProper && r < 0)
{
r = 0;
}
return(Gaussian.FromNatural(r * dhat + dlogf, r));
}
/// <include file='FactorDocs.xml' path='factor_docs/message_op_class[@name="ExpOp_Laplace2"]/message_doc[@name="DAverageConditional(Gamma, Gaussian, double)"]/*'/>
public static Gaussian DAverageConditional([SkipIfUniform] Gamma exp, [Proper] Gaussian d, double x)
{
if (exp.IsPointMass)
{
return(ExpOp.DAverageConditional(exp.Point));
}
double expx = Math.Exp(x);
double a = exp.Shape;
double b = exp.Rate;
double ddlogf = -b * expx;
double dddlogf = ddlogf;
double d4logf = ddlogf;
double dlogf = (a - 1) + ddlogf;
double v = 1 / (d.Precision - ddlogf);
// this is Laplace's estimate of the posterior mean and variance
double mpost = x + 0.5 * dddlogf * v * v;
double vpost = v + 0.5 * d4logf * v * v * v + dddlogf * dddlogf * v * v * v * v;
//double vpost = v + 0.25 * dddlogf * dddlogf * v * v * v * v;
Gaussian result = Gaussian.FromMeanAndVariance(mpost, vpost);
result.SetToRatio(result, d, ExpOp.ForceProper);
return(result);
}
/// <include file='FactorDocs.xml' path='factor_docs/message_op_class[@name="LogOp_EP"]/message_doc[@name="LogAverageConditional(Gaussian, Gamma, Gaussian)"]/*'/>
public static Gaussian LogAverageConditional([Proper] Gaussian log, [SkipIfUniform] Gamma d, Gaussian result)
{
var g = Gamma.FromShapeAndRate(d.Shape + 1, d.Rate);
return(ExpOp.DAverageConditional(g, log, result));
}
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);
}
