• 均值和方差未知的多元正态分布的后验Multivariate normal with unknown mean and variance
  • 从后验分布中采样均值mu和方差Sigma


1. 均值和方差未知的多元正态分布的后验（Multivariate normal with unknown mean and variance）

假设有N个观测值{xi|i=1,2,...,N}，且服从均值为μ方差为Σ的多元正态分布，即: 
 
xi∼N(μ,Σ)

 均值和方差都未知的情况下，多元正态分布的共轭先验是正态逆威沙特分布（Normal-Inverse-Wishart），即有： 

 
(μ,Σ)Σμ|Σ∼NIW(μ0,κ0;ν0,Λ0)∼Inv−Wishart(ν0,Λ0)∼N(μ0,Σ/κ0)

 其中逆威沙特分布的概率密度函数为如下形式： 

 
p(Σ|Λ0,ν0)=|Λ0|ν0/2|Σ|−(ν0+k+1)/2exp(−tr(Λ0Σ−1)/2)2ν0k/2Γk(ν0/2)

 
Γk(⋅)是多变量Gamma分布， 
 
ν0和 
 
Λ0分别是逆威沙特分布的自由度和尺度矩阵， 
 
k是数据的维度。 
     依据文献[1]，在观测到数据{xi|i=1,2,...,N}后，均值 
 
μ和方差 
 
Σ的后验分布依然服从正态逆威沙特分布，如下所示： 

 
(μ,Σ)∼NIW(μ′,κ′;ν′,Λ′)

 其中： 

 
μ′κ′ν′Λ′=κ0κ0+nμ0+Nκ0+Nx¯=κ0+Nν0+N=Λ0+∑n=1N(xi−x¯)(xi−x¯)T+κ0Nκ0+N(x¯−μ0)(x¯−μ0)T

 得到了后验分布的概率密度函数，我们就可以通过其采样多元正态分布的均值 
 
μ和方差 
 
Σ了。


2. 从后验分布中采样均值μ和方差Σ

均值μ的采样需要依赖于Σ，因此采样顺序一般为：首先采样Σ∼Inv−Wishart(ν′,Λ′)，然后采样μ|Σ,x∼N(μ′,Σ/κ′)。关于均值的采样，可以看这篇博客。下面介绍一下如何从逆威沙特分布中采样方差Σ。首先介绍一下Odell&Feiveson于1966年提出的基本采样思路[2]，然后给出Java代码。

一、 假设Vi(1≤i≤k)是独立的随机变量，并且采样自自由度为ν−i+1的卡方分布，所有有ν−k+1≤ν−i+1≤ν.
二、假设Nij是独立的采样自均值为0方差为1的正态分布中的随机变量，且有1≤i≤j≤k，Nij独立于Vi.
三、定义随机变量bij，且1≤i,j≤k，当1≤i≤j≤k时，有bji=bij，我们通过如下公式构造bij。

biibij=Vi+∑r=1i−1N2ri,1≤i≤k=NijVi−−√+∑r=1i−1NriNrj,i<j≤k

四、对方阵 Λ进行Cholesky分解，即 LLT=Λ−1
五、构造矩阵 R=LBLT
六、则 Σ′=R−1为该逆威沙特分布的样本。 至于为什么这么做，大家可参考文献[3]或者[2]。上面的过程已经很清晰了，下面我们给出具体的Java代码，，并且做了一点的修改（Note，下面的代码使用的依然是commons.math的3.0版本，事实上，现在已经更新到4.0版本的。） 


import java.util.Arrays;
import java.util.logging.Level;
import java.util.logging.Logger;
import org.apache.commons.math3.distribution.GammaDistribution;
import org.apache.commons.math3.linear.Array2DRowRealMatrix;
import org.apache.commons.math3.linear.CholeskyDecomposition;
import org.apache.commons.math3.linear.LUDecomposition;
import org.apache.commons.math3.linear.RealMatrix;
import org.apache.commons.math3.linear.SingularMatrixException;
import org.apache.commons.math3.random.RandomGenerator;
import org.apache.commons.math3.random.Well19937c;

/**
 * Inverse Wishart distribution implementation, to sample random covariances matrices for
 * multivariate gaussian distributions.
 * <p/>
 * The sampling method follows the procedure described by Odell & Feiveson, 1966 to get samples
 * from a Wishart distribution, and then computes the inverse of the obtained samples.
 *
 * @author Marc Pujol <mpujol@iiia.csic.es>
 */
public class InverseWishartDistribution {
    private static final Logger LOG = Logger.getLogger(InverseWishartDistribution.class.getName());

    private GammaDistribution[] gammas;
    private double df;
    private RealMatrix scaleMatrix;
    private CholeskyDecomposition cholesky;
    private RandomGenerator random;

    /**
     * Builds a new Inverse Wishart distribution with the given scale and degrees of freedom.
     *
     * @param scaleMatrix scale matrix(Λ)
     * @param df degrees of freedom.
     */
    public InverseWishartDistribution(RealMatrix scaleMatrix, double df) {
        if (!scaleMatrix.isSquare()) {
            throw new RuntimeException("scaleMatrix must be square.");
        }

        this.scaleMatrix = scaleMatrix;
        this.df = df;
        this.random = new Well19937c();
        initialize();
    }

    private void initialize() {
        final int dim = scaleMatrix.getColumnDimension();

        // Build gamma distributions for the diagonal
        gammas = new GammaDistribution[dim];
        for (int i = 0; i < dim; i++) {

            gammas[i] = new GammaDistribution((df-i+0.0)/2, 2);
        }

        // Build the cholesky decomposition
        cholesky = new CholeskyDecomposition(inverseMatrix(scaleMatrix));
    }

    /**
     * Reseeds the random generator.
     *
     * @param seed new random seed.
     */
    public void reseedRandomGenerator(long seed) {
        random.setSeed(seed);
        for (int i = 0, len = scaleMatrix.getColumnDimension(); i < len; i++) {
            gammas[i].reseedRandomGenerator(seed+i);
        }
    }

    /**
     * Returns the inverse matrix.
     * @return inverted matrix.
     */
    public RealMatrix inverseMatrix(RealMatrix A) {
        RealMatrix result = new LUDecomposition(A).getSolver().getInverse();
        return result; 
    }

     /**
     * Returns a sample matrix from this distribution.
     * @return sampled matrix.
     */
     public RealMatrix sample() {

        for (int i=0; i<100; i++) {
            try {
                RealMatrix A = sampleWishart();
                RealMatrix result = inverseMatrix(A);
                LOG.log(Level.FINE, "Cov = {0}", result);
                return result;
            } catch (SingularMatrixException ex) {
                LOG.finer("Discarding singular matrix generated by the wishart distribution.");
            }
        }
        throw new RuntimeException("Unable to generate inverse wishart samples!");
    }

    private RealMatrix sampleWishart() {
        final int dim = scaleMatrix.getColumnDimension();

        // Build N_{ij}
        double[][] N = new double[dim][dim];
        for (int j = 0; j < dim; j++) {
            for (int i = 0; i < j; i++) {
                N[i][j] = random.nextGaussian();
            }
        }
        if (LOG.isLoggable(Level.FINEST)) {
            LOG.log(Level.FINEST, "N = {0}", Arrays.deepToString(N));
        }

        // Build V_j
        double[] V = new double[dim];
        for (int i = 0; i < dim; i++) {
            V[i] = gammas[i].sample();
        }
        if (LOG.isLoggable(Level.FINEST)) {
            LOG.log(Level.FINEST, "V = {0}", Arrays.toString(V));
        }

        // Build B
        double[][] B = new double[dim][dim];

        // b_{11} = V_1 (first j, where sum = 0 because i == j and the inner
        //               loop is never entered).
        // b_{jj} = V_j + \sum_{i=1}^{j-1} N_{ij}^2, j = 2, 3, ..., p
        for (int j = 0; j < dim; j++) {
            double sum = 0;
            for (int i = 0; i < j; i++) {
                sum += Math.pow(N[i][j], 2);
            }
            B[j][j] = V[j] + sum;
        }
        if (LOG.isLoggable(Level.FINEST)) {
            LOG.log(Level.FINEST, "B*_jj : = {0}", Arrays.deepToString(B));
        }

        // b_{1j} = N_{1j} * \sqrt V_1
        for (int j = 1; j < dim; j++) {
            B[0][j] = N[0][j] * Math.sqrt(V[0]);
            B[j][0] = B[0][j];
        }
        if (LOG.isLoggable(Level.FINEST)) {
            LOG.log(Level.FINEST, "B*_1j = {0}", Arrays.deepToString(B));
        }

        // b_{ij} = N_{ij} * \sqrt V_1 + \sum_{k=1}^{i-1} N_{ki}*N_{kj}
        for (int j = 1; j < dim; j++) {
            for (int i = 1; i < j; i++) {
                double sum = 0;
                for (int k = 0; k < i; k++) {
                    sum += N[k][i] * N[k][j];
                }
                B[i][j] = N[i][j] * Math.sqrt(V[i]) + sum;
                B[j][i] = B[i][j];
            }
        }
        if (LOG.isLoggable(Level.FINEST)) {
            LOG.log(Level.FINEST, "B* = {0}", Arrays.deepToString(B));
        }

        RealMatrix BMat = new Array2DRowRealMatrix(B);
        RealMatrix A = cholesky.getL().multiply(BMat).multiply(cholesky.getLT());
        if (LOG.isLoggable(Level.FINER)) {
            LOG.log(Level.FINER, "A* = {0}", Arrays.deepToString(N));
        }
        return A;
    }

}

其中因为commons.math中的卡方分布没有采样函数，所以我们借助于commons.math提供的Gamma分布进行采样，事实上，卡方分布和Gamma概率密度函数非常相近。上述采样的核心其实是先从威沙特分布中采样一个方阵，然后求其逆矩阵，则得到逆威沙特分布的一个样本。代码中inverseMatrix(scaleMatrix)是将逆威沙特分布的尺度矩阵求逆，这样就得到威沙特分布的尺度矩阵。