/// <summary>
/// VMP message to 'logistic'
/// </summary>
/// <param name="x">Incoming message from 'x'. Must be a proper distribution. If uniform, the result will be uniform.</param>
/// <returns>The outgoing VMP message to the 'logistic' argument</returns>
/// <remarks><para>
/// The outgoing message is a distribution matching the moments of 'logistic' as the random arguments are varied.
/// The formula is <c>proj[sum_(x) p(x) factor(logistic,x)]</c>.
/// </para></remarks>
/// <exception cref="ImproperMessageException"><paramref name="x"/> is not a proper distribution</exception>
public static Beta LogisticAverageLogarithm([Proper] Gaussian x)
{
double m, v;
x.GetMeanAndVariance(out m, out v);
#if true
// for consistency with XAverageLogarithm
double eLogOneMinusP = BernoulliFromLogOddsOp.AverageLogFactor(false, x);
#else
// E[log (1-sigma(x))] = E[log sigma(-x)] = -E[log(1+exp(x))]
double eLogOneMinusP = -MMath.Log1PlusExpGaussian(m, v);
#endif
// E[log sigma(x)] = -E[log(1+exp(-x))] = -E[log(1+exp(x))-x] = -E[log(1+exp(x))] + E[x]
double eLogP = eLogOneMinusP + m;
return(Beta.FromMeanLogs(eLogP, eLogOneMinusP));
}
/// <summary>
/// Evidence message for VMP
/// </summary>
/// <param name="sample">Constant value for 'sample'.</param>
/// <param name="logProbs">Incoming message from 'logProbs'. Must be a proper distribution. If any element is uniform, the result will be uniform.</param>
/// <returns>Average of the factor's log-value across the given argument distributions</returns>
/// <remarks><para>
/// The formula for the result is <c>sum_(logProbs) p(logProbs) log(factor(sample,logProbs))</c>.
/// Adding up these values across all factors and variables gives the log-evidence estimate for VMP.
/// </para></remarks>
/// <exception cref="ImproperMessageException"><paramref name="logProbs"/> is not a proper distribution</exception>
public static double AverageLogFactor(int sample, [Proper] IList <Gaussian> logProbs)
{
double result = 0;
double ms, vs;
logProbs[sample].GetMeanAndVariance(out ms, out vs);
for (int k = 0; k < logProbs.Count; k++)
{
if (k == sample)
{
continue;
}
double m, v;
logProbs[k].GetMeanAndVariance(out m, out v);
Gaussian logProb = new Gaussian(ms - m, vs + v);
result += BernoulliFromLogOddsOp.AverageLogFactor(true, logProb);
}
return(result);
}