source: branches/HeuristicLab.Problems.GrammaticalOptimization-gkr/HeuristicLab.Distributions/GaussianModel.cs @ 12438

using System;
using HeuristicLab.Common;

namespace HeuristicLab.Algorithms.Bandits.Models {
  // bayesian estimation of a Gaussian with 
  // 1) unknown mean and known variance
  // 2) unknown mean and unknown variance
  public class GaussianModel : IModel {

    private OnlineMeanAndVarianceEstimator estimator = new OnlineMeanAndVarianceEstimator();

    // parameters of Gaussian prior for mean
    private readonly double meanPriorMu;
    private readonly double meanPriorVariance;

    private readonly bool knownVariance;
    private readonly double rewardVariance = 0.1; // assumed know reward variance

    // parameters of Gamma prior for precision (= inverse variance)
    private readonly int precisionPriorAlpha;
    private readonly double precisionPriorBeta;

    // non-informative prior
    private const double priorK = 1.0;

    // this constructor assumes the variance is known
    public GaussianModel(double meanPriorMu, double meanPriorVariance, double rewardVariance = 0.1) {
      this.meanPriorMu = meanPriorMu;
      this.meanPriorVariance = meanPriorVariance;

      this.knownVariance = true;
      this.rewardVariance = rewardVariance;
    }

    // this constructor assumes the variance is also unknown 
    // uses Murphy 2007: Conjugate Bayesian analysis of the Gaussian distribution equation 85 - 89
    public GaussianModel(double meanPriorMu, double meanPriorVariance, int precisionPriorAlpha, double precisionPriorBeta) {
      this.meanPriorMu = meanPriorMu;
      this.meanPriorVariance = meanPriorVariance;
      
      this.knownVariance = false;
      this.precisionPriorAlpha = precisionPriorAlpha;
      this.precisionPriorBeta = precisionPriorBeta;
    }


    public double Sample(Random random) {
      if (knownVariance) {
        return SampleExpectedRewardKnownVariance(random);
      } else {
        return SampleExpectedRewardUnknownVariance(random);
      }
    }

    private double SampleExpectedRewardKnownVariance(Random random) {
      // expected values for reward
      // calculate posterior mean and variance (for mean reward)

      // see Murphy 2007: Conjugate Bayesian analysis of the Gaussian distribution (http://www.cs.ubc.ca/~murphyk/Papers/bayesGauss.pdf)
      var posteriorMeanVariance = 1.0 / (estimator.N / rewardVariance + 1.0 / meanPriorVariance);
      var posteriorMeanMean = posteriorMeanVariance * (meanPriorMu / meanPriorVariance + estimator.Sum / rewardVariance);

      // sample a mean from the posterior 
      var posteriorMeanSample = Rand.RandNormal(random) * Math.Sqrt(posteriorMeanVariance) + posteriorMeanMean;
      // theta already represents the expected reward value => nothing else to do
      return posteriorMeanSample;

      // return 0.99-quantile value
      //return alglib.invnormaldistribution(0.99) * Math.Sqrt(rewardVariance + posteriorMeanVariance) + posteriorMeanMean;
    }

    // see Murphy 2007: Conjugate Bayesian analysis of the Gaussian distribution page 6 onwards (http://www.cs.ubc.ca/~murphyk/Papers/bayesGauss.pdf)
    private double SampleExpectedRewardUnknownVariance(Random random) {

      var posteriorMean = (priorK * meanPriorMu + estimator.Sum) / (priorK + estimator.N);
      var posteriorK = priorK + estimator.N;
      var posteriorAlpha = precisionPriorAlpha + estimator.N / 2.0;
      double posteriorBeta;
      if (estimator.N > 0) {
        posteriorBeta = precisionPriorBeta + 0.5 * estimator.N * estimator.Variance + priorK * estimator.N * Math.Pow(estimator.Avg - meanPriorMu, 2) / (2.0 * (priorK + estimator.N));
      } else {
        posteriorBeta = precisionPriorBeta;
      }


      // sample from the posterior marginal for mu (expected value) equ. 91
      // p(µ|D) = T2αn (µ| µn, βn/(αnκn)) 

      var t2alpha = alglib.invstudenttdistribution((int)(2 * posteriorAlpha), Math.Max(random.NextDouble(), 0.00001));

      var theta = t2alpha * posteriorBeta / (posteriorAlpha * posteriorK) + posteriorMean;
      return theta;


      /*
       * value function : 0.99-quantile
      // sample posterior mean and posterior variance independently
      var sampledPrec = Rand.GammaRand(random, posteriorAlpha) * posteriorBeta;
      var t2alpha = alglib.invstudenttdistribution((int)(2 * posteriorAlpha), random.NextDouble());
      var sampledMean = t2alpha * posteriorBeta / (posteriorAlpha * posteriorK) + posteriorMean;

      return alglib.invnormaldistribution(0.99) / Math.Sqrt(sampledPrec) + sampledMean;
       */
    }


    public void Update(double reward) {
      estimator.UpdateReward(reward);
    }

    public void Reset() {
      estimator.Reset();
    }

    public object Clone() {
      if (knownVariance)
        return new GaussianModel(meanPriorMu, meanPriorVariance, rewardVariance);
      else
        return new GaussianModel(meanPriorMu, meanPriorVariance, precisionPriorAlpha, precisionPriorBeta);
    }

    public override string ToString() {
      if (knownVariance) {
        var posteriorMeanVariance = 1.0 / (estimator.N / rewardVariance + 1.0 / meanPriorVariance);
        var posteriorMeanMean = posteriorMeanVariance * (meanPriorMu / meanPriorVariance + estimator.Sum / rewardVariance);
        return string.Format("Gaussian(mu, var=0.1), mu ~ Gaussian(mu'={0:F3}, var'={1:F3})", posteriorMeanMean, posteriorMeanVariance);
      } else {
        var posteriorMean = (priorK * meanPriorMu + estimator.Sum) / (priorK + estimator.N);
        var posteriorK = priorK + estimator.N;
        var posteriorAlpha = precisionPriorAlpha + estimator.N / 2.0;
        double posteriorBeta;
        if (estimator.N > 0) {
          posteriorBeta = precisionPriorBeta + 0.5 * estimator.N * estimator.Variance + priorK * estimator.N * Math.Pow(estimator.Avg - meanPriorMu, 2) / (2.0 * (priorK + estimator.N));
        } else {
          posteriorBeta = precisionPriorBeta;
        }
        var nu = (int)(2 * posteriorAlpha);
        var meanVariance = posteriorBeta / (posteriorAlpha * posteriorK) * (nu / (double)(nu - 2));
        return string.Format("Gaussian(mu, var), mu ~ T{0}(mu'={1:F3}, var'={2:F3}), 1.0/var ~ Gamma(mu={3:F3}, var={4:F3})",
          nu, posteriorMean, meanVariance,
          posteriorAlpha / posteriorBeta, posteriorAlpha / (posteriorBeta * posteriorBeta));
      }
    }
  }
}