// /// <include file='FactorDocs.xml' path='factor_docs/message_op_class[@name="LogisticOp"]/message_doc[@name="LogisticAverageConditional(Beta, Gaussian, Gaussian)"]/*'/>
public static Beta LogisticAverageConditional(Beta logistic, [Proper] Gaussian x, Gaussian falseMsg, Gaussian to_x)
{
if (x.IsPointMass)
{
return(Beta.PointMass(MMath.Logistic(x.Point)));
}
if (logistic.IsPointMass || x.IsUniform())
{
return(Beta.Uniform());
}
Gaussian post = to_x * x;
double m, v;
post.GetMeanAndVariance(out m, out v);
double mean = MMath.LogisticGaussian(m, v);
bool useVariance = logistic.IsUniform(); // useVariance gives lower accuracy on tests, but is required for the uniform case
if (useVariance)
{
// meanTF = E[p] - E[p^2]
double meanTF = MMath.LogisticGaussianDerivative(m, v);
double meanSquare = mean - meanTF;
Beta result = Beta.FromMeanAndVariance(mean, meanSquare - mean * mean);
result.SetToRatio(result, logistic, true);
return(result);
}
else
{
double logZ = LogAverageFactor(logistic, x, falseMsg) + logistic.GetLogNormalizer(); // log int_x logistic(sigma(x)) N(x;m,v) dx
double tc1 = logistic.TrueCount - 1;
double fc1 = logistic.FalseCount - 1;
return(BetaFromMeanAndIntegral(mean, logZ, tc1, fc1));
}
}
/// <summary>
/// VMP message to LogOdds
/// </summary>
/// <param name="sample">Incoming message from sample</param>
/// <param name="logOdds">Incoming message from logOdds</param>
/// <returns><c>sum_x marginal(x)*log(factor(x))</c></returns>
/// <remarks><para>
/// The outgoing message is the exponential of the integral of the log-factor times incoming messages, over all arguments except 'logOdds'.
/// The formula is <c>int log(f(logOdds,x)) q(x) dx</c> where <c>x = (sample)</c>.
/// </para></remarks>
public static Gaussian LogOddsAverageLogarithm(bool sample, Gaussian logOdds)
{
// This is the non-conjugate VMP update using the Saul and Jordan (1999) bound.
double m, v;
logOdds.GetMeanAndVariance(out m, out v);
double a = 0.5;
// TODO: use a buffer to store the value of 'a', so it doesn't need to be re-optimised each time.
for (int iter = 0; iter < 10; iter++)
{
double aOld = a;
a = MMath.Logistic(m + (1 - 2 * a) * v * 0.5);
if (Math.Abs(a - aOld) < 1e-8)
{
break;
}
}
double sa = MMath.Logistic(m + (1 - 2 * a) * v * 0.5);
double vf = 1 / (a * a + (1 - 2 * a) * sa);
double mf = m + vf * (sample ? 1 - sa : sa);
return(Gaussian.FromMeanAndVariance(mf, vf));
}
public static bool BernoulliFromLogOdds(double logOdds)
{
return(Bernoulli(MMath.Logistic(logOdds)));
}
/// <summary>
/// Gets the probability of the binary variable being false
/// </summary>
/// <returns>p(x=false)</returns>
public double GetProbFalse()
{
return(MMath.Logistic(-LogOdds));
}
/// <summary>
/// Gets the probability of the binary variable being true
/// </summary>
/// <returns>p(x=true)</returns>
public double GetProbTrue()
{
return(MMath.Logistic(LogOdds));
}
/// <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>
/// <param name="a">Buffer 'a'.</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, double a)
{
if (logistic.IsPointMass)
{
return(LogisticOp.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 scale = logistic.TrueCount + logistic.FalseCount - 2;
if (scale == 0.0)
{
return(Gaussian.Uniform());
}
double shift = -(logistic.FalseCount - 1);
double m, v;
x.GetMeanAndVariance(out m, out v);
double sa;
if (double.IsPositiveInfinity(v))
{
a = 0.5;
sa = MMath.Logistic(m);
}
else
{
sa = MMath.Logistic(m + (1 - 2 * a) * v * 0.5);
}
double precision = a * a + (1 - 2 * a) * sa;
// meanTimesPrecision = m*a*a + 1-2*a*sa;
double meanTimesPrecision = m * precision + 1 - sa;
//double vf = 1/(a*a + (1-2*a)*sa);
//double mf = m + vf*(true ? 1-sa : sa);
//double precision = 1/vf;
//double meanTimesPrecision = mf*precision;
Gaussian result = Gaussian.FromNatural(scale * meanTimesPrecision + shift, scale * precision);
double step = (LogisticOp_SJ99.global_step == 0.0) ? 1.0 : (Rand.Double() * LogisticOp_SJ99.global_step); // random damping helps convergence, especially with parallel updates
if (false && !x.IsPointMass)
{
// if the update would change the sign of 1-2*sa, send a message to make sa=0.5
double newPrec = x.Precision - to_x.Precision + result.Precision;
double newv = 1 / newPrec;
double newm = newv * (x.MeanTimesPrecision - to_x.MeanTimesPrecision + result.MeanTimesPrecision);
double newarg = newm + (1 - 2 * a) * newv * 0.5;
if ((sa < 0.5 && newarg > 0) || (sa > 0.5 && newarg < 0))
{
// send a message to make newarg=0
// it is sufficient to make (x.MeanTimesPrecision + step*(result.MeanTimesPrecision - to_x.MeanTimesPrecision) + 0.5-a) = 0
double mpOffset = x.MeanTimesPrecision + 0.5 - a;
double precOffset = x.Precision;
double mpScale = result.MeanTimesPrecision - to_x.MeanTimesPrecision;
double precScale = result.Precision - to_x.Precision;
double arg = m + (1 - 2 * a) * v * 0.5;
//arg = 0;
step = (arg * precOffset - mpOffset) / (mpScale - arg * precScale);
//step = (a-0.5-x.MeanTimesPrecision)/(result.MeanTimesPrecision - to_x.MeanTimesPrecision);
//Console.WriteLine(step);
}
}
if (step != 1.0)
{
result.Precision = step * result.Precision + (1 - step) * to_x.Precision;
result.MeanTimesPrecision = step * result.MeanTimesPrecision + (1 - step) * to_x.MeanTimesPrecision;
}
return(result);
}
/// <summary>
/// Evidence message for EP
/// </summary>
/// <param name="logistic">Incoming message from 'logistic'.</param>
/// <param name="x">Constant value for 'x'.</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_(logistic) p(logistic) factor(logistic,x))</c>.
/// </para></remarks>
public static double LogAverageFactor(Beta logistic, double x)
{
return(logistic.GetLogProb(MMath.Logistic(x)));
}
/// <summary>
/// Evidence message for EP
/// </summary>
/// <param name="logistic">Constant value for 'logistic'.</param>
/// <param name="x">Constant value for 'x'.</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(factor(logistic,x))</c>.
/// </para></remarks>
public static double LogAverageFactor(double logistic, double x)
{
return((logistic == MMath.Logistic(x)) ? 0.0 : Double.NegativeInfinity);
}
/// <include file='FactorDocs.xml' path='factor_docs/message_op_class[@name="LogisticOp"]/message_doc[@name="FalseMsg(Beta, Gaussian, Gaussian)"]/*'/>
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 (string.Empty.Length == 0)
{
// 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);
}
}
}
else
{
for (int iter = 0; iter < 10; iter++)
{
newMsg = falseMsg * ((newMsg / falseMsg) ^ 0.5);
falseMsg = newMsg;
//Console.WriteLine("prior = {0}, falseMsg = {1}, newMsg = {2}", prior, falseMsg, newMsg);
prior = new Gaussian(m + tc1 * v, v) * (falseMsg ^ (tc1 + fc1 - 1));
newMsg = BernoulliFromLogOddsOp.LogOddsAverageConditional(false, prior);
}
}
return(newMsg);
}
}
/// <summary>
/// Gets the probability of the binary variable being false
/// </summary>
/// <returns>p(x=false)</returns>
public double GetProbFalse(int index)
{
return(MMath.Logistic(-LogOddsVector[index]));
}
/// <summary>
/// The expected logarithm of that distribution under this distribution.
/// </summary>
/// <param name="that">The distribution to take the logarithm of.</param>
/// <returns><c>sum_x this.Evaluate(x)*Math.Log(that.Evaluate(x))</c></returns>
/// <remarks>This is also known as the cross entropy.</remarks>
public double GetAverageLog(BernoulliIntegerSubset that)
{
var res = SparseVector.Zero(LogOddsVector.Count);
var res2 = SparseVector.Zero(LogOddsVector.Count);
res.SetToFunction(LogOddsVector, that.GetLogProbTrueVector(), (logpr, logProbTrue) => double.IsNegativeInfinity(logpr) ?0 :MMath.Logistic(logpr) * logProbTrue);
res2.SetToFunction(LogOddsVector, that.GetLogProbFalseVector(), (logpr, logProbFalse) => double.IsPositiveInfinity(logpr) ?0 :MMath.Logistic(-logpr) * logProbFalse);
return(res.Sum() + res2.Sum());
}
public void Sample(Options options, Matrix data)
{
if (options.numParams > 2)
{
throw new Exception("numParams > 2");
}
int numStudents = data.Rows;
int numQuestions = data.Cols;
// initialize the sampler at the mean of the priors (not sampling from the priors)
double abilityMean = abilityMeanPrior.GetMean();
double abilityPrec = abilityPrecPrior.GetMean();
double difficultyMean = difficultyMeanPrior.GetMean();
double difficultyPrec = difficultyPrecPrior.GetMean();
double discriminationMean = discriminationMeanPrior.GetMean();
double discriminationPrec = discriminationPrecPrior.GetMean();
double[] ability = new double[numStudents];
double[] difficulty = new double[numQuestions];
List <double>[] difficultySamples = new List <double> [numQuestions];
GaussianEstimator[] difficultyEstimator = new GaussianEstimator[numQuestions];
for (int question = 0; question < numQuestions; question++)
{
difficultyEstimator[question] = new GaussianEstimator();
difficultySamples[question] = new List <double>();
if (difficultyObserved != null)
{
difficulty[question] = difficultyObserved[question];
difficultyEstimator[question].Add(difficultyObserved[question]);
difficultySamples[question].Add(difficultyObserved[question]);
}
}
List <double>[] abilitySamples = new List <double> [numStudents];
GaussianEstimator[] abilityEstimator = new GaussianEstimator[ability.Length];
for (int student = 0; student < abilityEstimator.Length; student++)
{
abilityEstimator[student] = new GaussianEstimator();
abilitySamples[student] = new List <double>();
if (abilityObserved != null)
{
ability[student] = abilityObserved[student];
abilityEstimator[student].Add(abilityObserved[student]);
abilitySamples[student].Add(abilityObserved[student]);
}
}
double[] discrimination = new double[numQuestions];
List <double>[] discriminationSamples = new List <double> [numQuestions];
GammaEstimator[] discriminationEstimator = new GammaEstimator[numQuestions];
for (int question = 0; question < numQuestions; question++)
{
discriminationEstimator[question] = new GammaEstimator();
discriminationSamples[question] = new List <double>();
discrimination[question] = 1;
if (discriminationObserved != null)
{
discrimination[question] = discriminationObserved[question];
discriminationEstimator[question].Add(discriminationObserved[question]);
discriminationSamples[question].Add(discriminationObserved[question]);
}
}
responseProbMean = new Matrix(numStudents, numQuestions);
int niters = options.numberOfSamples;
int burnin = options.burnIn;
double logisticVariance = Math.PI * Math.PI / 3;
double shape = 4.5;
Gamma precPrior = Gamma.FromShapeAndRate(shape, (shape - 1) * logisticVariance);
precPrior = Gamma.PointMass(1);
double[,] prec = new double[numStudents, numQuestions];
double[,] x = new double[numStudents, numQuestions];
int numRejected = 0, numAttempts = 0;
for (int iter = 0; iter < niters; iter++)
{
for (int student = 0; student < numStudents; student++)
{
for (int question = 0; question < numQuestions; question++)
{
// sample prec given ability, difficulty, x
// N(x; ability-difficulty, 1/prec) = Gamma(prec; 1.5, (x-ability+difficulty)^2/2)
Gamma precPost = precPrior;
double xMean = (ability[student] - difficulty[question]) * discrimination[question];
double delta = x[student, question] - xMean;
Gamma like = Gamma.FromShapeAndRate(1.5, 0.5 * delta * delta);
precPost.SetToProduct(precPost, like);
prec[student, question] = precPost.Sample();
// sample x given ability, difficulty, prec, data
// using an independence chain MH
bool y = (data[student, question] > 0);
double sign = y ? 1.0 : -1.0;
Gaussian xPrior = Gaussian.FromMeanAndPrecision(xMean, prec[student, question]);
// we want to sample from xPrior*I(x>0)
// instead we sample from xPost
Gaussian xPost = xPrior * IsPositiveOp.XAverageConditional(y, xPrior);
double oldx = x[student, question];
double newx = xPost.Sample();
numAttempts++;
if (newx * sign < 0)
{
newx = oldx; // rejected
numRejected++;
}
else
{
// importance weights
double oldw = xPrior.GetLogProb(oldx) - xPost.GetLogProb(oldx);
double neww = xPrior.GetLogProb(newx) - xPost.GetLogProb(newx);
// acceptance ratio
double paccept = Math.Exp(neww - oldw);
if (paccept < 1 && Rand.Double() > paccept)
{
newx = oldx; // rejected
numRejected++;
}
}
x[student, question] = newx;
if (iter >= burnin)
{
double responseProb = MMath.Logistic(xMean);
responseProbMean[student, question] += responseProb;
}
}
}
if (abilityObserved == null)
{
// sample ability given difficulty, prec, x
for (int student = 0; student < numStudents; student++)
{
Gaussian post = Gaussian.FromMeanAndPrecision(abilityMean, abilityPrec);
for (int question = 0; question < numQuestions; question++)
{
// N(x; disc*(ability-difficulty), 1/prec) =propto N(x/disc; ability-difficulty, 1/disc^2/prec) = N(ability; x/disc+difficulty, 1/disc^2/prec)
Gaussian abilityLike = Gaussian.FromMeanAndPrecision(x[student, question] / discrimination[question] + difficulty[question], prec[student, question] * discrimination[question] * discrimination[question]);
post.SetToProduct(post, abilityLike);
}
ability[student] = post.Sample();
if (iter >= burnin)
{
abilityEstimator[student].Add(post);
abilitySamples[student].Add(ability[student]);
}
}
}
// sample difficulty given ability, prec, x
for (int question = 0; question < numQuestions; question++)
{
Gaussian post = Gaussian.FromMeanAndPrecision(difficultyMean, difficultyPrec);
for (int student = 0; student < numStudents; student++)
{
// N(x; disc*(ability-difficulty), 1/prec) =propto N(x/disc; ability-difficulty, 1/disc^2/prec) = N(difficulty; ability-x/disc, 1/disc^2/prec)
if (discrimination[question] > 0)
{
Gaussian like = Gaussian.FromMeanAndPrecision(ability[student] - x[student, question] / discrimination[question], prec[student, question] * discrimination[question] * discrimination[question]);
post.SetToProduct(post, like);
}
}
difficulty[question] = post.Sample();
if (iter >= burnin)
{
//if (difficulty[question] > 100)
// Console.WriteLine("difficulty[{0}] = {1}", question, difficulty[question]);
difficultyEstimator[question].Add(post);
difficultySamples[question].Add(difficulty[question]);
}
}
if (options.numParams > 1 && discriminationObserved == null)
{
// sample discrimination given ability, difficulty, prec, x
for (int question = 0; question < numQuestions; question++)
{
// moment-matching on the prior
Gaussian approxPrior = Gaussian.FromMeanAndVariance(Math.Exp(discriminationMean + 0.5 / discriminationPrec), Math.Exp(2 * discriminationMean + 1 / discriminationPrec) * (Math.Exp(1 / discriminationPrec) - 1));
Gaussian post = approxPrior;
for (int student = 0; student < numStudents; student++)
{
// N(x; disc*delta, 1/prec) =propto N(x/delta; disc, 1/prec/delta^2)
double delta = ability[student] - difficulty[question];
if (delta > 0)
{
Gaussian like = Gaussian.FromMeanAndPrecision(x[student, question] / delta, prec[student, question] * delta * delta);
post.SetToProduct(post, like);
}
}
TruncatedGaussian postTrunc = new TruncatedGaussian(post, 0, double.PositiveInfinity);
double olddisc = discrimination[question];
double newdisc = postTrunc.Sample();
// importance weights
Func <double, double> priorLogProb = delegate(double d)
{
double logd = Math.Log(d);
return(Gaussian.GetLogProb(logd, discriminationMean, 1 / discriminationPrec) - logd);
};
double oldw = priorLogProb(olddisc) - approxPrior.GetLogProb(olddisc);
double neww = priorLogProb(newdisc) - approxPrior.GetLogProb(newdisc);
// acceptance ratio
double paccept = Math.Exp(neww - oldw);
if (paccept < 1 && Rand.Double() > paccept)
{
// rejected
}
else
{
discrimination[question] = newdisc;
}
if (iter >= burnin)
{
discriminationEstimator[question].Add(discrimination[question]);
discriminationSamples[question].Add(discrimination[question]);
}
}
}
// sample abilityMean given ability, abilityPrec
Gaussian abilityMeanPost = abilityMeanPrior;
for (int student = 0; student < numStudents; student++)
{
Gaussian like = GaussianOp.MeanAverageConditional(ability[student], abilityPrec);
abilityMeanPost *= like;
}
abilityMean = abilityMeanPost.Sample();
// sample abilityPrec given ability, abilityMean
Gamma abilityPrecPost = abilityPrecPrior;
for (int student = 0; student < numStudents; student++)
{
Gamma like = GaussianOp.PrecisionAverageConditional(ability[student], abilityMean);
abilityPrecPost *= like;
}
abilityPrec = abilityPrecPost.Sample();
// sample difficultyMean given difficulty, difficultyPrec
Gaussian difficultyMeanPost = difficultyMeanPrior;
for (int question = 0; question < numQuestions; question++)
{
Gaussian like = GaussianOp.MeanAverageConditional(difficulty[question], difficultyPrec);
difficultyMeanPost *= like;
}
difficultyMean = difficultyMeanPost.Sample();
// sample difficultyPrec given difficulty, difficultyMean
Gamma difficultyPrecPost = difficultyPrecPrior;
for (int question = 0; question < numQuestions; question++)
{
Gamma like = GaussianOp.PrecisionAverageConditional(difficulty[question], difficultyMean);
difficultyPrecPost *= like;
}
difficultyPrec = difficultyPrecPost.Sample();
// sample discriminationMean given discrimination, discriminationPrec
Gaussian discriminationMeanPost = discriminationMeanPrior;
for (int question = 0; question < numQuestions; question++)
{
Gaussian like = GaussianOp.MeanAverageConditional(Math.Log(discrimination[question]), discriminationPrec);
discriminationMeanPost *= like;
}
discriminationMean = discriminationMeanPost.Sample();
// sample discriminationPrec given discrimination, discriminationMean
Gamma discriminationPrecPost = discriminationPrecPrior;
for (int question = 0; question < numQuestions; question++)
{
Gamma like = GaussianOp.PrecisionAverageConditional(Math.Log(discrimination[question]), discriminationMean);
discriminationPrecPost *= like;
}
discriminationPrec = discriminationPrecPost.Sample();
//if (iter % 1 == 0)
// Console.WriteLine("iter = {0}", iter);
}
//Console.WriteLine("abilityMean = {0}, abilityPrec = {1}", abilityMean, abilityPrec);
//Console.WriteLine("difficultyMean = {0}, difficultyPrec = {1}", difficultyMean, difficultyPrec);
int numSamplesUsed = niters - burnin;
responseProbMean.Scale(1.0 / numSamplesUsed);
//Console.WriteLine("acceptance rate = {0}", ((double)numAttempts - numRejected)/numAttempts);
difficultyPost = Array.ConvertAll(difficultyEstimator, est => est.GetDistribution(Gaussian.Uniform()));
abilityPost = Array.ConvertAll(abilityEstimator, est => est.GetDistribution(Gaussian.Uniform()));
if (options.numParams > 1)
{
discriminationPost = Array.ConvertAll(discriminationEstimator, est => est.GetDistribution(new Gamma()));
}
abilityCred = GetCredibleIntervals(options.credibleIntervalProbability, abilitySamples);
difficultyCred = GetCredibleIntervals(options.credibleIntervalProbability, difficultySamples);
bool saveSamples = false;
if (saveSamples)
{
using (MatlabWriter writer = new MatlabWriter(@"..\..\samples.mat"))
{
int q = 11;
writer.Write("difficulty", difficultySamples[q]);
writer.Write("discrimination", discriminationSamples[q]);
}
}
}
