/// <include file='FactorDocs.xml' path='factor_docs/message_op_class[@name="PlusGammaVmpOp"]/message_doc[@name="AAverageLogarithm(GammaPower, GammaPower, GammaPower)"]/*'/>
public static GammaPower AAverageLogarithm([SkipIfUniform] GammaPower sum, [Proper] GammaPower a, [Proper] GammaPower b, GammaPower to_a, GammaPower to_b)
{
// f = int_sum sum^(ss/c -1)*exp(-sr*sum^(1/c))*delta(sum = a+b) dsum
// = (a+b)^(ss/c -1)*exp(-sr*(a+b)^(1/c))
// log(f) = (ss/c -1)*log(a+b) - sr*(a+b)^(1/c)
// apply a lower bound:
// log(a+b) >= p*log(a) + (1-p)*log(b) - p*log(p) - (1-p)*log(1-p)
// optimal p = exp(a)/(exp(a)+exp(b)) if (a,b) are fixed
// optimal p = exp(E[log(a)])/(exp(E[log(a)])+exp(E[log(b)])) if (a,b) are random
// This generalizes the bound used by Cemgil (2008).
// (a+b)^(1/c) = (a*q/q + b*(1-q)/(1-q))^(1/c)
// <= q*(a/q)^(1/c) + (1-q)*(b/(1-q))^(1/c) if c < 0
// = q^(1-1/c)*a^(1/c) + (1-q)^(1-1/c)*b^(1/c)
// d/dq = (1-1/c)*(q^(-1/c)*a^(1/c) - (1-q)^(-1/c)*b^(1/c))
// optimal q = a/(a + b) if (a,b) are fixed
// optimal q = E[a^(1/c)]^c/(E[a^(1/c)]^c + E[b^(1/c)]^c) if (a,b) are random
// The message to A has shape (ss-c)*p + c and rate sr*q^(1-1/c).
// If sum is a point mass, then the message to A is pointmass(s*p^c*q^(1-c))
if (a.Power != sum.Power)
{
throw new NotSupportedException($"a.Power ({a.Power}) != sum.Power ({sum.Power})");
}
double x = sum.Shape - sum.Power;
GammaPower aPost = a * to_a;
if (aPost.IsUniform())
{
return(sum);
}
GammaPower bPost = b * to_b;
if (bPost.IsUniform())
{
if (sum.IsPointMass)
{
return(sum);
}
return(GammaPower.FromShapeAndRate(sum.Power, sum.Rate, sum.Power));
}
double ma = Math.Exp(aPost.GetMeanLog());
double mb = Math.Exp(bPost.GetMeanLog());
double denom = ma + mb;
double p = (denom == 0) ? 0.5 : ma / (ma + mb);
double m = x * p;
double mac = Math.Exp(a.Power * aPost.GetLogMeanPower(1 / a.Power));
double mbc = Math.Exp(b.Power * bPost.GetLogMeanPower(1 / b.Power));
double denom2 = mac + mbc;
double q = (denom2 == 0) ? 0.5 : mac / (mac + mbc);
if (sum.IsPointMass)
{
return(GammaPower.PointMass(sum.Point * Math.Pow(p / q, sum.Power) * q, sum.Power));
}
return(GammaPower.FromShapeAndRate(m + sum.Power, sum.Rate * Math.Pow(q, 1 - 1 / sum.Power), sum.Power));
}
