/// <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));
}
/// <include file='FactorDocs.xml' path='factor_docs/message_op_class[@name="LogOp_EP"]/message_doc[@name="DAverageConditional(Gaussian, Gamma, Gaussian)"]/*'/>
public static Gamma DAverageConditional([Proper] Gaussian log, Gamma d, Gaussian to_log)
{
var g = Gamma.FromShapeAndRate(d.Shape + 1, d.Rate);
return(ExpOp.ExpAverageConditional(g, log, to_log));
}
/// <include file='FactorDocs.xml' path='factor_docs/message_op_class[@name="LogOp_EP"]/message_doc[@name="LogAverageFactor(Gaussian, Gamma, Gaussian)"]/*'/>
public static double LogAverageFactor(Gaussian log, Gamma d, [Fresh] Gaussian to_log)
{
Gamma g = Gamma.FromShapeAndRate(d.Shape + 1, d.Rate);
return(d.Shape / d.Rate * ExpOp.LogAverageFactor(g, log, to_log));
}
/// <include file='FactorDocs.xml' path='factor_docs/message_op_class[@name="ExpOp_BFGS"]/message_doc[@name="DAverageLogarithm(Gamma, Gaussian, Gaussian)"]/*'/>
public static Gaussian DAverageLogarithm([Proper] Gamma exp, [Proper, Stochastic] Gaussian d, Gaussian to_d)
{
if (exp.IsPointMass)
{
return(ExpOp.DAverageLogarithm(exp.Point));
}
double m, v;
d.GetMeanAndVariance(out m, out v);
double mu, s2;
var prior = d / to_d;
prior.GetMeanAndVariance(out mu, out s2);
var z = Vector.Zero(2);
z[0] = m;
z[1] = Math.Log(v);
double startingValue = GradientAndValueAtPoint(mu, s2, z, exp.Shape, exp.Rate, null);
var s = new BFGS();
int evalCounter = 0;
s.MaximumStep = 1e3;
s.MaximumIterations = 100;
s.Epsilon = 1e-5;
s.convergenceCriteria = BFGS.ConvergenceCriteria.Objective;
z = s.Run(z, 1.0, delegate(Vector y, ref Vector grad)
{
evalCounter++;
return(GradientAndValueAtPoint(mu, s2, y, exp.Shape, exp.Rate, grad));
});
m = z[0];
v = Math.Exp(z[1]);
to_d.SetMeanAndVariance(m, v);
to_d.SetToRatio(to_d, prior);
double endValue = GradientAndValueAtPoint(mu, s2, z, exp.Shape, exp.Rate, null);
//Console.WriteLine("Went from {0} to {1} in {2} steps, {3} evals", startingValue, endValue, s.IterationsPerformed, evalCounter);
if (startingValue < endValue)
{
Console.WriteLine("Warning: BFGS resulted in an increased objective function");
}
return(to_d);
/* ---------------- NEWTON ITERATION VERSION 1 -------------------
* double meanTimesPrec, prec;
* d.GetNatural(out meanTimesPrec, out prec);
* Matrix K = new Matrix(2, 2);
* K[0, 0]=1/prec; // d2K by dmu^2
* K[1, 0]=K[0, 1]=-meanTimesPrec/(prec*prec);
* K[1, 1]=meanTimesPrec*meanTimesPrec/Math.Pow(prec, 3)+1/(2*prec*prec);
* double[,,] Kprime = new double[2, 2, 2];
* Kprime[0, 0, 0]=0;
* Kprime[0, 0, 1]=Kprime[0, 1, 0]=Kprime[1, 0, 0]=-1/(prec*prec);
* Kprime[0, 1, 1]=Kprime[1, 1, 0]=Kprime[1, 0, 1]=2*meanTimesPrec/Math.Pow(prec, 3);
* Kprime[1, 1, 1]=-3*meanTimesPrec*meanTimesPrec/Math.Pow(prec, 4)-1/Math.Pow(prec, 3);
* Vector gradS = new Vector(2);
* gradS[0]=(exp.Shape-1)/prec-exp.Rate/prec*Math.Exp((meanTimesPrec+.5)/prec);
* gradS[1]=-(exp.Shape-1)*meanTimesPrec/(prec*prec)+exp.Rate*(meanTimesPrec+.5)/(prec*prec)*Math.Exp((meanTimesPrec+.5)/prec);
* Matrix grad2S = new Matrix(2, 2);
* grad2S[0, 0]=-exp.Rate/(prec*prec)*Math.Exp((meanTimesPrec+.5)/prec);
* grad2S[0, 1]=grad2S[1, 0]=-(exp.Shape-1)/(prec*prec)+exp.Rate*(1/(prec*prec)+(meanTimesPrec+.5)/Math.Pow(prec, 3))*Math.Exp((meanTimesPrec+.5)/prec);
* grad2S[1, 1]=2*(exp.Shape-1)*meanTimesPrec/Math.Pow(prec, 3)-exp.Rate*(meanTimesPrec+.5)/(prec*prec)*(2/prec+(meanTimesPrec+.5)/(prec*prec))*Math.Exp((meanTimesPrec+.5)/prec);
* Vector phi = new Vector(new double[] { result.MeanTimesPrecision, result.Precision });
* Vector gradKL = K*phi-gradS;
* Matrix hessianKL = K - grad2S;
* for (int i=0; i<2; i++)
* for (int j=0; j<2; j++)
* for (int k=0; k<2; k++)
* hessianKL[i, j]+=Kprime[i, j, k]*phi[k];
* double step = 1;
* Vector direction = GammaFromShapeAndRate.twoByTwoInverse(hessianKL)*gradKL;
* Vector newPhi = phi - step * direction;
* result.SetNatural(newPhi[0], newPhi[1]);
* return result;
*
* ---------------- NEWTON ITERATION VERSION 2 -------------------
* double mean, variance;
* d.GetMeanAndVariance(out mean, out variance);
* Gaussian prior = d / result;
* double mean1, variance1;
* prior.GetMeanAndVariance(out mean1, out variance1);
* Vector gradKL = new Vector(2);
* gradKL[0]=-(exp.Shape-1)+exp.Rate*Math.Exp(mean+variance/2)+mean/variance1-mean1/variance1;
* gradKL[1]=-1/(2*variance)+exp.Rate*Math.Exp(mean+variance/2)+1/(2*variance1);
* Matrix hessianKL = new Matrix(2, 2);
* hessianKL[0, 0]=exp.Rate*Math.Exp(mean+variance/2)+1/variance1;
* hessianKL[0, 1]=hessianKL[1, 0]=.5*exp.Rate*Math.Exp(mean+variance/2);
* hessianKL[1, 1]=1/(2*variance*variance)+exp.Rate*Math.Exp(mean+variance/2)/4;
* result.GetMeanAndVariance(out mean, out variance);
* if (double.IsInfinity(variance))
* variance=1000;
* Vector theta = new Vector(new double[] { mean, variance });
* theta -= GammaFromShapeAndRate.twoByTwoInverse(hessianKL)*gradKL;
* result.SetMeanAndVariance(theta[0], theta[1]);
* return 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);
}
