从高斯pm，pv到高斯qm，qv的Kullback-Leibler发散

Question

我正试图计算从Gaussian#1到Gaussian#2的Kullback-Leibler散度，我有两个高斯的平均和标准差，我尝试了1/rm1/python/sphinx/divergence.py的这段代码。

def gau_kl(pm, pv, qm, qv):
    """
    Kullback-Leibler divergence from Gaussian pm,pv to Gaussian qm,qv.
    Also computes KL divergence from a single Gaussian pm,pv to a set
    of Gaussians qm,qv.
    Diagonal covariances are assumed.  Divergence is expressed in nats.
    """
    if (len(qm.shape) == 2):
        axis = 1
    else:
        axis = 0
    # Determinants of diagonal covariances pv, qv
    dpv = pv.prod()
    dqv = qv.prod(axis)
    # Inverse of diagonal covariance qv
    iqv = 1./qv
    # Difference between means pm, qm
    diff = qm - pm
    return (0.5 *
            (numpy.log(dqv / dpv)            # log |\Sigma_q| / |\Sigma_p|
             + (iqv * pv).sum(axis)          # + tr(\Sigma_q^{-1} * \Sigma_p)
             + (diff * iqv * diff).sum(axis) # + (\mu_q-\mu_p)^T\Sigma_q^{-1}(\mu_q-\mu_p)
             - len(pm)))                     # - N

我使用均值和标准偏差作为输入，但是代码(len(pm))的最后一行会导致一个错误，因为平均值是一个数字，这里我不理解len函数。

请注意。这两个集合(即高斯)不相等，这就是我不能使用scipy.stats.entropy的原因

Answer 1

下面的函数计算任意两个多元正态分布(不需要协方差矩阵为对角)之间的KL-散度(其中numpy被导入为np)。

def kl_mvn(m0, S0, m1, S1):
    """
    Kullback-Liebler divergence from Gaussian pm,pv to Gaussian qm,qv.
    Also computes KL divergence from a single Gaussian pm,pv to a set
    of Gaussians qm,qv.
    

    From wikipedia
    KL( (m0, S0) || (m1, S1))
         = .5 * ( tr(S1^{-1} S0) + log |S1|/|S0| + 
                  (m1 - m0)^T S1^{-1} (m1 - m0) - N )
    """
    # store inv diag covariance of S1 and diff between means
    N = m0.shape[0]
    iS1 = np.linalg.inv(S1)
    diff = m1 - m0

    # kl is made of three terms
    tr_term   = np.trace(iS1 @ S0)
    det_term  = np.log(np.linalg.det(S1)/np.linalg.det(S0)) #np.sum(np.log(S1)) - np.sum(np.log(S0))
    quad_term = diff.T @ np.linalg.inv(S1) @ diff #np.sum( (diff*diff) * iS1, axis=1)
    #print(tr_term,det_term,quad_term)
    return .5 * (tr_term + det_term + quad_term - N)