public static Gaussian BAverageConditional([SkipIfUniform] VectorGaussian product, [SkipIfUniform] VectorGaussian a, Gaussian b)
{
if (!b.IsPointMass)
{
throw new ArgumentException("b is not a point mass", nameof(b));
}
double bPoint = b.Point;
Vector productMean = Vector.Zero(product.Dimension);
PositiveDefiniteMatrix productVariance = new PositiveDefiniteMatrix(product.Dimension, product.Dimension);
product.GetMeanAndVariance(productMean, productVariance);
Vector aMean = Vector.Zero(product.Dimension);
PositiveDefiniteMatrix aVariance = new PositiveDefiniteMatrix(product.Dimension, product.Dimension);
a.GetMeanAndVariance(aMean, aVariance);
PositiveDefiniteMatrix variance = new PositiveDefiniteMatrix(product.Dimension, product.Dimension);
variance.SetToSum(1, productVariance, bPoint * bPoint, aVariance);
// variance = productVariance + aVariance * b^2
PositiveDefiniteMatrix precision = variance.Inverse();
Vector diff = aMean * bPoint;
diff.SetToDifference(productMean, diff);
// diff = productMean - aMean * b
var precisionDiff = precision * diff;
double dlogf = aMean.Inner(precisionDiff) + bPoint * (precisionDiff.Inner(aVariance * precisionDiff) - Matrix.TraceOfProduct(precision, aVariance));
double ddlogf = -1;
return(Gaussian.FromDerivatives(bPoint, dlogf, ddlogf, GaussianProductOp.ForceProper));
}
/// <include file='FactorDocs.xml' path='factor_docs/message_op_class[@name="IntegralOp"]/message_doc[@name="UpperBoundAverageConditional(Gaussian, double, Func{double,double}, ITruncatableDistribution{double})"]/*'/>
public static Gaussian UpperBoundAverageConditional(Gaussian integral, double upperBound, Func <double, double> func, ITruncatableDistribution <double> distribution)
{
CanGetLogProb <double> canGetLogProb = (CanGetLogProb <double>)distribution;
double dlogf = integral.MeanTimesPrecision * func(upperBound) * Math.Exp(canGetLogProb.GetLogProb(upperBound));
double ddlogf = 0; // approximation ignoring integral.Precision
return(Gaussian.FromDerivatives(upperBound, dlogf, ddlogf, false));
}
/// <include file='FactorDocs.xml' path='factor_docs/message_op_class[@name="RatioGaussianOp"]/message_doc[@name="ProbabilityAverageConditional(Gaussian, CanGetQuantile{double}, double)"]/*'/>
public static Gaussian ProbabilityAverageConditional(Gaussian quantile, CanGetQuantile <double> canGetQuantile, double probability)
{
// The quantile function is the inverse function of the cdf.
// The derivative of the quantile function is the reciprocal of the pdf.
double x = canGetQuantile.GetQuantile(probability);
CanGetLogProb <double> canGetLogProb = (CanGetLogProb <double>)canGetQuantile;
double dlogp = quantile.MeanTimesPrecision / Math.Exp(canGetLogProb.GetLogProb(x));
double ddlogp = 0; // approximation ignoring quantile.Precision
return(Gaussian.FromDerivatives(probability, dlogp, ddlogp, false));
}
/// <include file='FactorDocs.xml' path='factor_docs/message_op_class[@name="ProbBetweenOp"]/message_doc[@name="LowerBoundAverageConditional(Gaussian, CanGetProbLessThan{double}, double)"]/*'/>
public static Gaussian LowerBoundAverageConditional(Gaussian probBetween, CanGetProbLessThan <double> canGetProbLessThan, double lowerBound)
{
CanGetLogProb <double> canGetLogProb = (CanGetLogProb <double>)canGetProbLessThan;
// factor is N(ProbBetween(left, right); m, v)
// logp = -(m - ProbBetween(left, right))^2/(2v)
// dlogp = (m - ProbBetween(left, right))/v * dProbBetween(left, right)
// dProbBetween(left, right)/dleft = -p(left)
// ddlogp = (m - ProbBetween(left, right))/v * ddProbBetween(left, right) - dProbBetween(left, right)/v * dProbBetween(left, right)
// (m - ProbBetween(left, right))/v -> m/v
double dlogp = -Math.Exp(canGetLogProb.GetLogProb(lowerBound)) * probBetween.MeanTimesPrecision;
double ddlogp = 0; // approximation ignoring probBetween.Precision
return(Gaussian.FromDerivatives(lowerBound, dlogp, ddlogp, false));
}
/// <include file='FactorDocs.xml' path='factor_docs/message_op_class[@name="RatioGaussianOp"]/message_doc[@name="AAverageConditional(Gaussian, Gaussian, Gaussian)"]/*'/>
public static Gaussian BAverageConditional([SkipIfUniform] Gaussian ratio, [SkipIfUniform] Gaussian a, Gaussian b)
{
//if (string.Empty.Length == 0) return GaussianProductOp.BAverageConditional(Gaussian.FromNatural(a.MeanTimesPrecision, Math.Min(a.Precision, 1e16)), Gaussian.FromNatural(ratio.MeanTimesPrecision, ratio.Precision + 1e-16), b);
if (b.IsPointMass && ratio.Precision == 0)
{
// f(b) = int_a N(m; a/b, v) p(a) da
// = N(m; ma/b, v + va/b^2)
// logf = -0.5*log(v + va/b^2) -0.5*(m - ma/b)^2/(v + va/b^2)
// dlogf = va/b^3/(v + va/b^2) -(m - ma/b)/(v + va/b^2)*(ma/b^2) + (m - ma/b)^2/(v + va/b^2)^2*(va/b^3)
// -> -(m - ma/b)/v * (ma/b^2)
double dlogp = -a.GetMean() / (b.Point * b.Point) * ratio.MeanTimesPrecision;
double ddlogp = 0;
return(Gaussian.FromDerivatives(b.Point, dlogp, ddlogp, false));
}
else
{
throw new NotSupportedException();
}
}
/// <include file='FactorDocs.xml' path='factor_docs/message_op_class[@name="ProbBetweenOp"]/message_doc[@name="UpperBoundAverageConditional(Gaussian, CanGetProbLessThan{double}, Gaussian)"]/*'/>
public static Gaussian UpperBoundAverageConditional(Gaussian probBetween, CanGetProbLessThan <double> canGetProbLessThan, [RequiredArgument] Gaussian upperBound)
{
CanGetLogProb <double> canGetLogProb = (CanGetLogProb <double>)canGetProbLessThan;
// factor is N(ProbBetween(left, right); m, v)
if (upperBound.IsPointMass)
{
// logp = -(m - ProbBetween(left, right))^2/(2v)
// dlogp = (m - ProbBetween(left, right))/v * dProbBetween(left, right)
// dProbBetween(left, right)/dright = p(right)
// ddlogp = (m - ProbBetween(left, right))/v * ddProbBetween(left, right) - dProbBetween(left, right)/v * dProbBetween(left, right)
double dlogp = Math.Exp(canGetLogProb.GetLogProb(upperBound.Point)) * probBetween.MeanTimesPrecision;
double ddlogp = 0; // approximation ignoring probBetween.Precision
return(Gaussian.FromDerivatives(upperBound.Point, dlogp, ddlogp, false));
}
else
{
throw new NotSupportedException();
}
}
/// <include file='FactorDocs.xml' path='factor_docs/message_op_class[@name="MaxGaussianOp"]/message_doc[@name="AAverageConditional(Gaussian, Gaussian, Gaussian)"]/*'/>
public static Gaussian AAverageConditional([SkipIfUniform] Gaussian max, [Proper] Gaussian a, [Proper] Gaussian b)
{
if (max.IsUniform())
{
return(Gaussian.Uniform());
}
if (!b.IsProper())
{
throw new ImproperMessageException(b);
}
double logw1, alpha1, vx1, mx1;
double logw2, alpha2, vx2, mx2;
double logz;
ComputeStats(max, a, b, out logz, out logw1, out alpha1, out vx1, out mx1,
out logw2, out alpha2, out vx2, out mx2);
double w1 = Math.Exp(logw1 - logz);
double w2 = Math.Exp(logw2 - logz);
bool checkDerivatives = false;
if (a.IsPointMass)
{
// f(a) = int p(x = max(a,b)) p(b) db
// f(a) = p(x = a) p(a >= b) + p(x = b) p(a < b | x = b)
// f(a) = N(a; mx, vx) phi((a - m2)/sqrt(v2)) + N(m2; mx, vx + v2) phi((mx2 - a)/sqrt(vx2))
// f'(a) = (mx-a)/vx N(a; mx, vx) phi((a - m2)/sqrt(v2)) + N(a; mx, vx) N(a; m2, v2) - N(m2; mx, vx + v2) N(a; mx2, vx2)
// = (mx-a)/vx N(a; mx, vx) phi((a - m2)/sqrt(v2))
// f''(a) = -1/vx N(a; mx, vx) phi((a - m2)/sqrt(v2)) + (mx-a)^2/vx^2 N(a; mx, vx) phi((a - m2)/sqrt(v2)) + (mx-a)/vx N(a; mx, vx) N(a; m2, v2)
// ddlogf = f''(a)/f(a) - (f'(a)/f(a))^2 = (f''(a) f(a) - f'(a)^2)/f(a)^2
double aPoint = a.Point;
if (max.IsPointMass)
{
return(max);
}
double z = max.MeanTimesPrecision - aPoint * max.Precision;
double alpha = z * w1;
double beta = (z * alpha1 - max.Precision + z * z) * w1 - alpha * alpha;
if (b.IsPointMass && b.Point != aPoint)
{
beta -= max.Precision * w2 * alpha2 * (b.Point - aPoint);
}
if (checkDerivatives)
{
double m1 = a.GetMean();
double delta = m1 * 1e-6;
double logzd, logw1d, alpha1d, vx1d, mx1d, logw2d, alpha2d, vx2d, mx2d;
ComputeStats(max, Gaussian.FromMeanAndPrecision(m1 + delta, a.Precision), b, out logzd, out logw1d, out alpha1d, out vx1d, out mx1d, out logw2d, out alpha2d, out vx2d, out mx2d);
double logzd2;
ComputeStats(max, Gaussian.FromMeanAndPrecision(m1 - delta, a.Precision), b, out logzd2, out logw1d, out alpha1d, out vx1d, out mx1d, out logw2d, out alpha2d, out vx2d, out mx2d);
double alphaCheck = (logzd - logzd2) / (2 * delta);
double alphaError = Math.Abs(alpha - alphaCheck);
double betaCheck = (logzd + logzd2 - 2 * logz) / (delta * delta);
double betaError = Math.Abs(beta - betaCheck);
Console.WriteLine($"alpha={alpha} check={alphaCheck} error={alphaError} beta={beta} check={betaCheck} error={betaError}");
}
return(Gaussian.FromDerivatives(aPoint, alpha, beta, ForceProper));
}
bool useMessage = (w1 > 0) || (logw2 > -100);
if (useMessage)
{
// vx1 = 1/(1/vx + 1/v1) = vx*v1/(vx + v1)
double z, alpha, beta;
if (max.IsPointMass)
{
if (w2 == 0)
{
return(max);
}
z = max.Point * a.Precision - a.MeanTimesPrecision;
alpha = z * w1 - w2 * alpha2;
beta = (z * z - a.Precision) * w1 - z * alpha2 * w2 - alpha * alpha;
}
else
{
//z = vx1 * (max.MeanTimesPrecision * a.Precision - a.MeanTimesPrecision * max.Precision);
z = (max.MeanTimesPrecision - a.GetMean() * max.Precision) / (max.Precision / a.Precision + 1);
alpha = z * w1 - vx1 * max.Precision * w2 * alpha2;
if (w2 == 0)
{
//beta = (z * alpha1 - max.Precision) / (max.Precision / a.Precision + 1);
//double resultPrecision = a.Precision * beta / (-a.Precision - beta);
//resultPrecision = beta / (-1 - beta/a.Precision);
//resultPrecision = 1 / (-1/beta - 1 / a.Precision);
//resultPrecision = a.Precision / (-a.Precision / beta - 1);
//resultPrecision = a.Precision / (-(a.Precision + max.Precision) / (z * alpha1 - max.Precision) - 1);
//resultPrecision = -a.Precision / ((a.Precision + z*alpha1) / (z * alpha1 - max.Precision));
double zalpha1 = z * alpha1;
double denom = (1 + zalpha1 / a.Precision);
double resultPrecision = (max.Precision - zalpha1) / denom;
//double weight = (max.Precision - z * alpha1) / (a.Precision + z * alpha1);
//double weightPlus1 = (max.Precision + a.Precision) / (a.Precision + z * alpha1);
//double resultMeanTimesPrecision = weight * (a.MeanTimesPrecision + alpha) + alpha;
//resultMeanTimesPrecision = weight * a.MeanTimesPrecision + weightPlus1 * alpha;
//double resultMeanTimesPrecision = (a.Precision * max.MeanTimesPrecision - z * alpha1 * a.MeanTimesPrecision) / (a.Precision + z * alpha1);
double resultMeanTimesPrecision = (max.MeanTimesPrecision - zalpha1 * a.GetMean()) / denom;
return(Gaussian.FromNatural(resultMeanTimesPrecision, resultPrecision));
}
else
{
beta = ((z * alpha1 - max.Precision) * vx1 * a.Precision + z * z) * w1
- max.Precision * vx1 * w2 * alpha2 * (mx2 * a.Precision - a.MeanTimesPrecision) / (vx2 * a.Precision + 1) - alpha * alpha;
}
}
//Console.WriteLine($"z={z} w1={w1:r} w2={w2:r} logw2={logw2} alpha1={alpha1} alpha2={alpha2} alpha={alpha:r} beta={beta:r}");
if (checkDerivatives)
{
double m1 = a.GetMean();
double delta = m1 * 1e-6;
double logzd, logw1d, alpha1d, vx1d, mx1d, logw2d, alpha2d, vx2d, mx2d;
ComputeStats(max, Gaussian.FromMeanAndPrecision(m1 + delta, a.Precision), b, out logzd, out logw1d, out alpha1d, out vx1d, out mx1d, out logw2d, out alpha2d, out vx2d, out mx2d);
double logzd2;
ComputeStats(max, Gaussian.FromMeanAndPrecision(m1 - delta, a.Precision), b, out logzd2, out logw1d, out alpha1d, out vx1d, out mx1d, out logw2d, out alpha2d, out vx2d, out mx2d);
double alphaCheck = (logzd - logzd2) / (2 * delta);
double alphaError = Math.Abs(alpha - alphaCheck);
double betaCheck = (logzd + logzd2 - 2 * logz) / (delta * delta);
double betaError = Math.Abs(beta - betaCheck);
Console.WriteLine($"alpha={alpha} check={alphaCheck} error={alphaError} beta={beta} check={betaCheck} error={betaError}");
}
return(GaussianOp.GaussianFromAlphaBeta(a, alpha, -beta, ForceProper));
}
else
{
double m1, v1, m2, v2;
a.GetMeanAndVariance(out m1, out v1);
b.GetMeanAndVariance(out m2, out v2);
// the posterior is a mixture model with weights exp(logw1-logz), exp(logw2-logz) and distributions
// N(a; mx1, vx1) phi((a - m2)/sqrt(v2)) / phi((mx1 - m2)/sqrt(vx1 + v2))
// N(a; m1, v1) phi((mx2 - a)/sqrt(vx2)) / phi((mx2 - m1)/sqrt(vx2 + v1))
// the moments of the posterior are computed via the moments of these two components.
if (vx1 == 0)
{
alpha1 = 0;
}
if (vx2 == 0)
{
alpha2 = 0;
}
double mc1 = mx1;
if (alpha1 != 0) // avoid 0*infinity
{
mc1 += alpha1 * vx1;
}
alpha2 = -alpha2;
double mc2 = m1;
if (alpha2 != 0) // avoid 0*infinity
{
mc2 += alpha2 * v1;
}
// z2 = (mx2 - m1) / Math.Sqrt(vx2 + v1)
// logw2 = MMath.NormalCdfLn(z2);
// alpha2 = -Math.Exp(Gaussian.GetLogProb(mx2, m1, vx2 + v1) - logw2);
// = -1/sqrt(vx2+v1)/NormalCdfRatio(z2)
// m1 + alpha2*v1 = sqrt(vx2+v1)*(m1/sqrt(vx2+v1) - v1/(vx2+v1)/NormalCdfRatio(z2))
// = sqrt(vx2+v1)*(mx2/sqrt(vx2+v1) - z2 - v1/(vx2+v1)/NormalCdfRatio(z2))
// = sqrt(vx2+v1)*(mx2/sqrt(vx2+v1) - dY/Y) if vx2=0
double z2 = 0, Y = 0, dY = 0;
const double z2small = 0;
if (vx2 == 0)
{
z2 = (mx2 - m1) / Math.Sqrt(vx2 + v1);
if (z2 < z2small)
{
Y = MMath.NormalCdfRatio(z2);
dY = MMath.NormalCdfMomentRatio(1, z2);
mc2 = mx2 - Math.Sqrt(v1) * dY / Y;
}
}
double m = w1 * mc1 + w2 * mc2;
double beta1;
if (alpha1 == 0)
{
beta1 = 0; // avoid 0*infinity
}
else
{
double r1 = (mx1 - m2) / (vx1 + v2);
beta1 = alpha1 * (alpha1 + r1);
}
double beta2;
if (alpha2 == 0)
{
beta2 = 0; // avoid 0*infinity
}
else
{
double r2 = (mx2 - m1) / (vx2 + v1);
beta2 = alpha2 * (alpha2 - r2);
}
double vc1 = vx1 * (1 - vx1 * beta1);
double vc2;
if (vx2 == 0 && z2 < z2small)
{
// beta2 = alpha2 * (alpha2 - z2/sqrt(v1))
// vc2 = v1 - v1^2 * alpha2 * (alpha2 - z2/sqrt(v1))
// = v1 - v1*dY/Y^2
// =approx v1/z2^2
// posterior E[x^2] = v - v*dY/Y^2 + v*dY^2/Y^2 = v - v*dY/Y^2*(1 - dY) = v + v*z*dY/Y = v*d2Y/Y
//vc2 = v1 * (1 - dY / (Y * Y));
double d2Y = 2 * MMath.NormalCdfMomentRatio(2, z2);
double dYiY = dY / Y;
vc2 = v1 * (d2Y / Y - dYiY * dYiY);
}
else if (beta2 == 0)
{
vc2 = v1;
}
else
{
vc2 = v1 * (1 - v1 * beta2);
}
double diff = mc1 - mc2;
double v = w1 * vc1 + w2 * vc2 + w1 * w2 * diff * diff;
Gaussian result = new Gaussian(m, v);
//Console.WriteLine($"z2={z2} m={m} v={v} vc2={vc2} diff={diff}");
result.SetToRatio(result, a, ForceProper);
if (Double.IsNaN(result.Precision) || Double.IsNaN(result.MeanTimesPrecision))
{
throw new InferRuntimeException($"result is NaN. max={max}, a={a}, b={b}");
}
return(result);
}
}
/// <include file='FactorDocs.xml' path='factor_docs/message_op_class[@name="ArrayFromVectorOp"]/message_doc[@name="ArrayAverageConditional{GaussianList}(IList{Gaussian}, VectorGaussian, GaussianList)"]/*'/>
/// <typeparam name="GaussianList">The type of the resulting array.</typeparam>
public static GaussianList ArrayAverageConditional <GaussianList>(
[NoInit] IList <Gaussian> array, [SkipIfUniform] VectorGaussian vector, GaussianList result)
where GaussianList : IList <Gaussian>
{
if (result.Count != vector.Dimension)
{
throw new ArgumentException("vector.Dimension (" + vector.Dimension + ") != result.Count (" + result.Count + ")");
}
int length = result.Count;
bool allPointMass = array.All(g => g.IsPointMass);
if (allPointMass)
{
// efficient special case
for (int i = 0; i < length; i++)
{
double x = array[i].Point;
// -prec*(x-m) = -prec*x + prec*m
double dlogp = vector.MeanTimesPrecision[i];
for (int j = 0; j < length; j++)
{
dlogp -= vector.Precision[i, j] * array[j].Point;
}
double ddlogp = -vector.Precision[i, i];
result[i] = Gaussian.FromDerivatives(x, dlogp, ddlogp, false);
}
}
else if (vector.IsPointMass)
{
// efficient special case
Vector mean = vector.Point;
for (int i = 0; i < length; i++)
{
result[i] = Gaussian.PointMass(mean[i]);
}
}
else if (vector.IsUniform())
{
for (int i = 0; i < length; i++)
{
result[i] = Gaussian.Uniform();
}
}
else if (array.Any(g => g.IsPointMass))
{
// Z = N(m1; m2, V1+V2)
// logZ = -0.5 (m1-m2)'inv(V1+V2)(m1-m2)
// dlogZ = (m1-m2)'inv(V1+V2) dm2
// ddlogZ = -dm2'inv(V1+V2) dm2
Vector mean = Vector.Zero(length);
PositiveDefiniteMatrix variance = new PositiveDefiniteMatrix(length, length);
vector.GetMeanAndVariance(mean, variance);
for (int i = 0; i < length; i++)
{
if (array[i].IsUniform())
{
continue;
}
double m, v;
array[i].GetMeanAndVariance(out m, out v);
variance[i, i] += v;
mean[i] -= m;
}
PositiveDefiniteMatrix precision = variance.Inverse();
Vector meanTimesPrecision = precision * mean;
for (int i = 0; i < length; i++)
{
if (array[i].IsUniform())
{
result[i] = Gaussian.FromMeanAndVariance(mean[i], variance[i, i]);
}
else
{
double alpha = meanTimesPrecision[i];
double beta = precision[i, i];
result[i] = GaussianOp.GaussianFromAlphaBeta(array[i], alpha, beta, false);
}
}
}
else
{
// Compute inv(V1+V2)*(m1-m2) as inv(V2)*inv(inv(V1) + inv(V2))*(inv(V1)*m1 + inv(V2)*m2) - inv(V2)*m2 = inv(V2)*(m - m2)
// Compute inv(V1+V2) as inv(V2)*inv(inv(V1) + inv(V2))*inv(V2) - inv(V2)
PositiveDefiniteMatrix precision = (PositiveDefiniteMatrix)vector.Precision.Clone();
Vector meanTimesPrecision = vector.MeanTimesPrecision.Clone();
for (int i = 0; i < length; i++)
{
Gaussian g = array[i];
precision[i, i] += g.Precision;
meanTimesPrecision[i] += g.MeanTimesPrecision;
}
bool fastMethod = true;
if (fastMethod)
{
bool isPosDef;
// this destroys precision
LowerTriangularMatrix precisionChol = precision.CholeskyInPlace(out isPosDef);
if (!isPosDef)
{
throw new PositiveDefiniteMatrixException();
}
// variance = inv(precisionChol*precisionChol') = inv(precisionChol)'*inv(precisionChol) = varianceChol*varianceChol'
// this destroys meanTimesPrecision
var mean = meanTimesPrecision.PredivideBy(precisionChol);
mean = mean.PredivideByTranspose(precisionChol);
var varianceCholTranspose = precisionChol;
// this destroys precisionChol
varianceCholTranspose.SetToInverse(precisionChol);
for (int i = 0; i < length; i++)
{
Gaussian g = array[i];
double variance_ii = GetSquaredLengthOfColumn(varianceCholTranspose, i);
// works when g is uniform, but not when g is point mass
result[i] = Gaussian.FromMeanAndVariance(mean[i], variance_ii) / g;
}
}
else
{
// equivalent to above, but slower
PositiveDefiniteMatrix variance = precision.Inverse();
var mean = variance * meanTimesPrecision;
for (int i = 0; i < length; i++)
{
Gaussian g = array[i];
// works when g is uniform, but not when g is point mass
result[i] = Gaussian.FromMeanAndVariance(mean[i], variance[i, i]) / g;
}
}
}
return(result);
}
/// <include file='FactorDocs.xml' path='factor_docs/message_op_class[@name="MatrixVectorProductOp"]/message_doc[@name="AAverageConditional(VectorGaussian, DistributionArray2D{Gaussian, double}, Vector, PositiveDefiniteMatrix, DistributionStructArray2D{Gaussian, double})"]/*'/>
public static DistributionStructArray2D <Gaussian, double> AAverageConditional([SkipIfUniform] VectorGaussian product, DistributionArray2D <Gaussian, double> A, Vector BMean, PositiveDefiniteMatrix BVariance, DistributionStructArray2D <Gaussian, double> result)
{
if (product.IsUniform())
{
result.SetToUniform();
return(result);
}
if (!A.IsPointMass)
{
throw new ArgumentException("A is not a point mass");
}
// logZ = log N(mProduct; A*BMean, vProduct + A*BVariance*A')
// = -0.5 (mProduct - A*BMean)' inv(vProduct + A*BVariance*A') (mProduct - A*BMean) - 0.5 logdet(vProduct + A*BVariance*A')
// = -0.5 (mProduct - A*BMean)' pPrec inv(pProduct + pProduct*A*BVariance*A'*pProduct) pProduct (mProduct - A*BMean)
// - 0.5 logdet(pProduct + pProduct*A*BVariance*A'*pProduct) + logdet(pProduct)
// dlogZ = 0.5 (dA*BMean)' pProduct inv(pProduct + pProduct*A*BVariance*A'*pProduct) pProduct (mProduct - A*BMean)
// +0.5 (mProduct - A*BMean)' pProduct inv(pProduct + pProduct*A*BVariance*A'*pProduct) pProduct (dA*BMean)
// +0.5 (mProduct - A*BMean)' pProduct inv(pProduct + pProduct*A*BVariance*A'*pProduct) (pProduct*dA*BVariance*A'*pProduct + pProduct*A*BVariance*dA'*pProduct) inv(pProduct + pProduct*A*BVariance*A'*pProduct) pProduct (mProduct - A*BMean)
// - 0.5 tr(inv(pProduct + pProduct*A*BVariance*A'*pProduct) (pProduct*dA*BVariance*A'*pProduct + pProduct*A*BVariance*dA'*pProduct))
// dlogZ/dA = pProduct inv(pProduct + pProduct*A*BVariance*A'*pProduct) pProduct (mProduct - A*BMean) BMean'
// + pProduct inv(pProduct + pProduct*A*BVariance*A'*pProduct) pProduct (mProduct - A*BMean) (mProduct - A*BMean)' pProduct inv(pProduct + pProduct*A*BVariance*A'*pProduct) pProduct*A*BVariance
// - pProduct inv(pProduct + pProduct*A*BVariance*A'*pProduct) pProduct A*BVariance
var Amatrix = new Matrix(A.Point);
var pProductA = product.Precision * Amatrix;
var pProductABV = pProductA * BVariance;
PositiveDefiniteMatrix prec = new PositiveDefiniteMatrix(product.Dimension, product.Dimension);
prec.SetToSum(product.Precision, pProductABV * pProductA.Transpose());
// pProductA is now free
for (int i = 0; i < prec.Rows; i++)
{
if (prec[i, i] == 0)
{
prec[i, i] = 1;
}
}
var v = prec.Inverse();
var ABM = Amatrix * BMean;
var pProductABM = product.Precision * ABM;
var diff = pProductABM;
diff.SetToDifference(product.MeanTimesPrecision, pProductABM);
// ABM is now free
var pProductV = product.Precision * v;
var pProductVdiff = ABM;
pProductVdiff.SetToProduct(pProductV, diff);
var Vdiff = v * diff;
pProductV.Scale(-1);
pProductV.SetToSumWithOuter(pProductV, 1, pProductVdiff, Vdiff);
Matrix dlogZ = pProductA;
dlogZ.SetToProduct(pProductV, pProductABV);
dlogZ.SetToSumWithOuter(dlogZ, 1, pProductVdiff, BMean);
int rows = A.GetLength(0);
int cols = A.GetLength(1);
for (int i = 0; i < rows; i++)
{
for (int j = 0; j < cols; j++)
{
double dlogp = dlogZ[i, j];
// for now, we don't compute the second derivative.
double ddlogp = -1;
result[i, j] = Gaussian.FromDerivatives(A[i, j].Point, dlogp, ddlogp, false);
}
}
return(result);
}
public static Gaussian DAverageConditional([SkipIfUniform] Gamma exp, [Proper] Gaussian d)
{
// as a function of d, the factor is Ga(exp(d); shape, rate) = exp(d*(shape-1) -rate*exp(d))
if (exp.IsUniform())
{
return(Gaussian.Uniform());
}
if (exp.IsPointMass)
{
return(ExpOp.DAverageConditional(exp.Point));
}
if (exp.Rate < 0)
{
throw new ImproperMessageException(exp);
}
if (exp.Rate == 0)
{
return(Gaussian.FromNatural(exp.Shape - 1, 0));
}
if (d.IsUniform())
{
if (exp.Shape <= 1)
{
throw new ArgumentException("The posterior has infinite variance due to input of Exp distributed as " + d + " and output of Exp distributed as " + exp +
" (shape <= 1)");
}
// posterior for d is a shifted log-Gamma distribution:
// exp((a-1)*d - b*exp(d)) =propto exp(a*(d+log(b)) - exp(d+log(b)))
// we find the Gaussian with same moments.
// u = d+log(b)
// E[u] = digamma(a-1)
// E[d] = E[u]-log(b) = digamma(a-1)-log(b)
// var(d) = var(u) = trigamma(a-1)
double lnRate = Math.Log(exp.Rate);
return(new Gaussian(MMath.Digamma(exp.Shape - 1) - lnRate, MMath.Trigamma(exp.Shape - 1)));
}
double aMinus1 = exp.Shape - 1;
double b = exp.Rate;
if (d.IsPointMass)
{
double x = d.Point;
double expx = Math.Exp(x);
double dlogf = aMinus1 - b * expx;
double ddlogf = -b * expx;
return(Gaussian.FromDerivatives(x, dlogf, ddlogf, true));
}
double dmode, dmin, dmax;
GetIntegrationBounds(exp, d, out dmode, out dmin, out dmax);
double expmode = Math.Exp(dmode);
int n = QuadratureNodeCount;
double inc = (dmax - dmin) / (n - 1);
MeanVarianceAccumulator mva = new MeanVarianceAccumulator();
for (int i = 0; i < n; i++)
{
double x = dmin + i * inc;
double xMinusMode = x - dmode;
double diff = aMinus1 * xMinusMode - b * (Math.Exp(x) - expmode)
- 0.5 * ((x * x - dmode * dmode) * d.Precision - 2 * xMinusMode * d.MeanTimesPrecision);
double p = Math.Exp(diff);
mva.Add(x, p);
if (double.IsNaN(mva.Variance))
{
throw new Exception();
}
}
double dMean = mva.Mean;
double dVariance = mva.Variance;
Gaussian result = Gaussian.FromMeanAndVariance(dMean, dVariance);
result.SetToRatio(result, d, true);
return(result);
}