/// <summary>EP message to <c>mean</c>.</summary>
/// <param name="sample">Incoming message from <c>sample</c>. Must be a proper distribution. If uniform, the result will be uniform.</param>
/// <param name="mean">Incoming message from <c>mean</c>. Must be a proper distribution. If uniform, the result will be uniform.</param>
/// <returns>The outgoing EP message to the <c>mean</c> argument.</returns>
/// <remarks>
/// <para>The outgoing message is a distribution matching the moments of <c>mean</c> as the random arguments are varied. The formula is <c>proj[p(mean) sum_(sample) p(sample) factor(sample,mean)]/p(mean)</c>.</para>
/// </remarks>
/// <exception cref="ImproperMessageException">
/// <paramref name="sample" /> is not a proper distribution.</exception>
/// <exception cref="ImproperMessageException">
/// <paramref name="mean" /> is not a proper distribution.</exception>
public static Gamma MeanAverageConditional([SkipIfUniform] Poisson sample, [Proper] Gamma mean)
{
if (sample.IsUniform())
{
return(Gamma.Uniform());
}
if (sample.IsPointMass)
{
return(MeanAverageConditional(sample.Point));
}
if (sample.Precision != 0)
{
throw new NotImplementedException("sample.Precision != 0 is not implemented");
}
// Z = int_m sum_x r^x m^x exp(-m)/x! q(m)
// = int_m exp(rm -m) q(m)
// = (1 + (1-r)/b)^(-a)
// logZ = -a log(1 + (1-r)/b)
// alpha = -b dlogZ/db
// = -a(1-r)/(b + (1-r))
// beta = -b dalpha/db
// = -ba(1-r)/(b + (1-r))^2
double omr = 1 - sample.Rate;
double denom = 1 / (mean.Rate + omr);
double alpha = -mean.Shape * omr * denom;
double beta = mean.Rate * alpha * denom;
return(GaussianOp.GammaFromAlphaBeta(mean, alpha, beta, ForceProper));
}
public static GammaPower GammaPowerFromDerivLogZ(GammaPower a, double dlogZ, double ddlogZ)
{
bool method1 = false;
if (method1)
{
GetPosteriorMeanAndVariance(Gamma.FromShapeAndRate(a.Shape, a.Rate), dlogZ, ddlogZ, out double iaMean, out double iaVariance);
Gamma ia = Gamma.FromMeanAndVariance(iaMean, iaVariance);
return(GammaPower.FromShapeAndRate(ia.Shape, ia.Rate, a.Power) / a);
}
else
{
double alpha = -a.Rate * dlogZ;
// dalpha/dr = -dlogZ - r*ddlogZ
// beta = -r * dalpha/dr
double beta = a.Rate * dlogZ + a.Rate * a.Rate * ddlogZ;
Gamma prior = Gamma.FromShapeAndRate(a.Shape, a.Rate);
// ia is the marginal of a^(1/a.Power)
Gamma ia = GaussianOp.GammaFromAlphaBeta(prior, alpha, beta, true) * prior;
return(GammaPower.FromShapeAndRate(ia.Shape, ia.Rate, a.Power) / a);
}
}