/// <summary>
/// Update the buffer 'falseMsg'
/// </summary>
/// <param name="logistic">Incoming message from 'logistic'. Must be a proper distribution. If uniform, the result will be uniform.</param>
/// <param name="x">Incoming message from 'x'. Must be a proper distribution. If uniform, the result will be uniform.</param>
/// <param name="falseMsg">Buffer 'falseMsg'.</param>
/// <returns>New value of buffer 'falseMsg'</returns>
/// <remarks><para>
///
/// </para></remarks>
/// <exception cref="ImproperMessageException"><paramref name="logistic"/> is not a proper distribution</exception>
/// <exception cref="ImproperMessageException"><paramref name="x"/> is not a proper distribution</exception>
public static Gaussian FalseMsg([SkipIfUniform] Beta logistic, [Proper] Gaussian x, Gaussian falseMsg)
{
// falseMsg approximates sigma(-x)
// logistic(sigma(x)) N(x;m,v)
// = sigma(x)^(a-1) sigma(-x)^(b-1) N(x;m,v)
// = e^((a-1)x) sigma(-x)^(a+b-2) N(x;m,v)
// = sigma(-x)^(a+b-2) N(x;m+(a-1)v,v) exp((a-1)m + (a-1)^2 v/2)
// = sigma(-x) (prior)
// where prior = sigma(-x)^(a+b-3) N(x;m+(a-1)v,v)
double tc1 = logistic.TrueCount - 1;
double fc1 = logistic.FalseCount - 1;
double m, v;
x.GetMeanAndVariance(out m, out v);
if (tc1 + fc1 == 0)
{
falseMsg.SetToUniform();
return(falseMsg);
}
else if (tc1 + fc1 < 0)
{
// power EP update, using 1/sigma(-x) as the factor
Gaussian prior = new Gaussian(m + tc1 * v, v) * (falseMsg ^ (tc1 + fc1 + 1));
double mprior, vprior;
prior.GetMeanAndVariance(out mprior, out vprior);
// posterior moments can be computed exactly
double w = MMath.Logistic(mprior + 0.5 * vprior);
Gaussian post = new Gaussian(mprior + w * vprior, vprior * (1 + w * (1 - w) * vprior));
return(prior / post);
}
else
{
// power EP update
Gaussian prior = new Gaussian(m + tc1 * v, v) * (falseMsg ^ (tc1 + fc1 - 1));
Gaussian newMsg = BernoulliFromLogOddsOp.LogOddsAverageConditional(false, prior);
//Console.WriteLine("prior = {0}, falseMsg = {1}, newMsg = {2}", prior, falseMsg, newMsg);
if (true)
{
// adaptive damping scheme
Gaussian ratio = newMsg / falseMsg;
if ((ratio.MeanTimesPrecision < 0 && prior.MeanTimesPrecision > 0) ||
(ratio.MeanTimesPrecision > 0 && prior.MeanTimesPrecision < 0))
{
// if the update would change the sign of the mean, take a fractional step so that the new prior has exactly zero mean
// newMsg = falseMsg * (ratio^step)
// newPrior = prior * (ratio^step)^(tc1+fc1-1)
// 0 = prior.mp + ratio.mp*step*(tc1+fc1-1)
double step = -prior.MeanTimesPrecision / (ratio.MeanTimesPrecision * (tc1 + fc1 - 1));
if (step > 0 && step < 1)
{
newMsg = falseMsg * (ratio ^ step);
// check that newPrior has zero mean
//Gaussian newPrior = prior * ((ratio^step)^(tc1+fc1-1));
//Console.WriteLine(newPrior);
}
}
}
return(newMsg);
}
}
/// <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);
}
/// <summary>
/// VMP message to 'x'
/// </summary>
/// <param name="logistic">Incoming message from 'logistic'. Must be a proper distribution. If uniform, the result will be uniform.</param>
/// <param name="x">Incoming message from 'x'. Must be a proper distribution. If uniform, the result will be uniform.</param>
/// <param name="to_X">Previous outgoing message to 'X'.</param>
/// <returns>The outgoing VMP message to the 'x' argument</returns>
/// <remarks><para>
/// The outgoing message is the factor viewed as a function of 'x' with 'logistic' integrated out.
/// The formula is <c>sum_logistic p(logistic) factor(logistic,x)</c>.
/// </para></remarks>
/// <exception cref="ImproperMessageException"><paramref name="logistic"/> is not a proper distribution</exception>
/// <exception cref="ImproperMessageException"><paramref name="x"/> is not a proper distribution</exception>
public static Gaussian XAverageLogarithm([SkipIfUniform] Beta logistic, [Proper, SkipIfUniform] Gaussian x, Gaussian to_X)
{
if (logistic.IsPointMass)
{
return(XAverageLogarithm(logistic.Point));
}
// f(x) = sigma(x)^(a-1) sigma(-x)^(b-1)
// = sigma(x)^(a+b-2) exp(-x(b-1))
// since sigma(-x) = sigma(x) exp(-x)
double a = logistic.TrueCount;
double b = logistic.FalseCount;
double scale = a + b - 2;
if (scale == 0.0)
{
return(Gaussian.Uniform());
}
double shift = -(b - 1);
Gaussian toLogOddsPrev = Gaussian.FromNatural((to_X.MeanTimesPrecision - shift) / scale, to_X.Precision / scale);
Gaussian toLogOdds = BernoulliFromLogOddsOp.LogOddsAverageLogarithm(true, x, toLogOddsPrev);
return(Gaussian.FromNatural(scale * toLogOdds.MeanTimesPrecision + shift, scale * toLogOdds.Precision));
}