/// <summary>
/// Bayesian estimation of the difference in means between two samples.
/// </summary>
public static void TestTwoSampleBayesianEstimation()
{
List <double> data1 = new List <double> {
1.0, 0.9, 1.1, 1.0, 0.9, 1.1, 1.0, 0.9, 1.1
};
List <double> data2 = new List <double> {
1.0, 0.9, 1.1, 1.0, 0.9, 1.1, 1.0, 0.9, 1.1
};
double pooledMean = data1.Concat(data2).Mean();
double pooledSd = data1.Concat(data2).StandardDeviation();
// construct a t-distribution model fitter for each dataset
var sampler1 = new AdaptiveMetropolisWithinGibbs(
data1.ToArray(),
new StudentTDistributionModel(
priorMuMean: pooledMean, priorMuSd: pooledSd * 1000000, muInitialGuess: pooledMean,
priorSdStart: pooledSd / 1000, priorSdEnd: pooledSd * 1000, sdInitialGuess: pooledSd,
priorNuExponent: 1.0 / 29.0, nuInitialGuess: 5),
seed: 0);
var sampler2 = new AdaptiveMetropolisWithinGibbs(
data2.ToArray(),
new StudentTDistributionModel(
priorMuMean: pooledMean, priorMuSd: pooledSd * 1000000, muInitialGuess: pooledMean,
priorSdStart: pooledSd / 1000, priorSdEnd: pooledSd * 1000, sdInitialGuess: pooledSd,
priorNuExponent: 1.0 / 29.0, nuInitialGuess: 5),
seed: 1);
// burn in and then sample the MCMC chains
sampler1.Run(1000, 1000);
sampler2.Run(1000, 1000);
var chain1 = sampler1.MarkovChain;
var chain2 = sampler2.MarkovChain;
// calculate difference in means
List <double> meanDiffs = new List <double>();
for (int i = 0; i < chain1.Count; i++)
{
meanDiffs.Add(chain1[i][0] - chain2[i][0]);
}
double avgMeanDiff = meanDiffs.Average(); // point estimate of mean diff
Assert.That(Math.Round(avgMeanDiff, 3) == -0.011);
var highestDensityInterval = BayesianEstimation.Util.GetHighestDensityInterval(meanDiffs.ToArray());
Assert.That(Math.Round(highestDensityInterval.hdi_start, 3) == -0.118);
Assert.That(Math.Round(highestDensityInterval.hdi_end, 3) == 0.095);
}
/// <summary>
/// This test demonstrates the utility of estimating the mean and standard deviation of a set of datapoints using a Bayesian
/// method (which uses Monte-Carlo sampling) rather than using the standard equations for the mean and standard deviations. The example data
/// contains an outlier, which skews the estimate of mean and standard deviation if you compute these using mean/std dev equations.
/// The Bayesian method fits a series of t-distributions to the data, which is more tolerant of the outlier, providing a better
/// estimate of the mean and standard deviation.
/// </summary>
public static void TestOneSampleBayesianEstimation()
{
List <double> data = new List <double> {
1.0, 1.0, 1.1, 0.9, 0.8, 0.9, 1.0, 100.0
};
// the std dev and mean of the data are very high because the data contain an outlier (100)
double stdev = data.StandardDeviation(); // sd of the data including outlier is 35.0
double mean = data.Mean(); // mean of the data including outlier is 13.3
// let's see what the mean and std dev are without the outlier
List <double> dataWithoutTheOutlier = new List <double> {
1.0, 1.0, 1.1, 0.9, 0.8, 0.9, 1.0
};
double stdevWithoutTheOutlier = dataWithoutTheOutlier.StandardDeviation(); // sd of the data excluding the outlier is 0.098
double meanWithoutTheOutlier = dataWithoutTheOutlier.Mean(); // mean of the data excluding the outlier is 0.96
// let's do a Bayesian estimate of the mean and standard deviation of the data set that includes the outlier...
AdaptiveMetropolisWithinGibbs sampler = new AdaptiveMetropolisWithinGibbs(
data.ToArray(),
new StudentTDistributionModel(
priorMuMean: data.Average(), priorMuSd: data.StandardDeviation() * 1000000, muInitialGuess: data.Average(),
priorSdStart: data.StandardDeviation() / 1000, priorSdEnd: data.StandardDeviation() * 1000, sdInitialGuess: data.StandardDeviation(),
priorNuExponent: 1.0 / 29.0, nuInitialGuess: 5),
seed: 0);
// burn in and then sample the MCMC chain
sampler.Run(1000, 1000);
var chain = sampler.MarkovChain;
// each iteration of the MCMC chain produces one t-distribution fit result
// in this example, 1000 different t-distributions are fit to the data
// each of these distributions has a mean, sd, and degree of normality
var mus = chain.Select(p => p[0]).ToList();
var sigmas = chain.Select(p => p[1]).ToList();
var nus = chain.Select(p => p[2]).ToList();
Assert.That(mus.Count == 1000);
// here are the mean/sd/normality results of the Bayesian estimation:
// (the point estimate is the average of each parameter over the 1000 iterations)
double muPointEstimate = mus.Average(); // point estimate of mu (mean)
double sigmaPointEstimate = sigmas.Average(); // point estimate of sigma (std dev)
double nuPointEstimate = nus.Average(); // point estimate of nu (degree of normality)
// the mean is estimated at 0.96
Assert.That(Math.Round(muPointEstimate, 3) == 0.962);
// std dev is estimated at 0.10
Assert.That(Math.Round(sigmaPointEstimate, 3) == 0.104);
// nu (degree of normality) is estimated at 0.65
Assert.That(Math.Round(nuPointEstimate, 3) == 0.654);
// instead of only a point estimate of the mean, the Bayesian method also gives a range of credible values.
// sort of like a 95% confidence interval, we can construct a 95% highest density interval where 95% of the
// probability density for a parameter is contained
var highestDensityInterval = BayesianEstimation.Util.GetHighestDensityInterval(mus.ToArray());
Assert.That(Math.Round(highestDensityInterval.hdi_start, 3) == 0.862);
Assert.That(Math.Round(highestDensityInterval.hdi_end, 3) == 1.062);
}
