/// <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));
}
/// <summary>
/// EP message to 'mean'
/// </summary>
/// <param name="sample">Incoming message from 'sample'. Must be a proper distribution. If uniform, the result will be uniform.</param>
/// <returns>The outgoing EP message to the 'mean' argument</returns>
/// <remarks><para>
/// The outgoing message is a distribution matching the moments of 'mean' 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>
public static Gamma MeanAverageConditional([SkipIfUniform] Poisson sample)
{
// Z = sum_sample sum_mean (mean*rate)^sample exp(-mean)/Gamma(sample+1)^(nu+1)/Z(rate,nu)
if (sample.IsUniform())
{
return(Gamma.Uniform());
}
throw new NotSupportedException(NotSupportedMessage);
}
/// <summary>Evidence message for EP.</summary>
/// <param name="sample">Incoming message from <c>sample</c>.</param>
/// <param name="mean">Incoming message from <c>mean</c>.</param>
/// <returns>Logarithm of the factor's average value across the given argument distributions.</returns>
/// <remarks>
/// <para>The formula for the result is <c>log(sum_(sample,mean) p(sample,mean) factor(sample,mean))</c>.</para>
/// </remarks>
public static double LogAverageFactor(Poisson sample, Gamma mean)
{
if (sample.IsUniform())
{
return(0);
}
if (sample.IsPointMass)
{
return(LogAverageFactor(sample.Point, mean, MeanAverageConditional(sample.Point)));
}
if (sample.Precision != 0)
{
throw new NotImplementedException("sample.Precision != 0 is not implemented");
}
return(-mean.Shape * Math.Log(1 + (1 - sample.Rate) / mean.Rate) - sample.GetLogNormalizer());
}
