log_prob.exp()值能否超过1？重构概率计算代码异常求助

问题概述 #

我用PyTorch编写了代码，计算输入样本在重构数据分布下的概率密度。代码中通过Normal分布的log_prob.exp()后取平均得到结果p，但输出的p超过了1。我想知道是代码存在问题，还是log_prob.exp()的值本身可以超过1？

代码实现 #

def reconstructed_probability(self, x: torch.Tensor) -> torch.Tensor:
    """
    Computes the probability density of the input samples under the learned
    distribution of reconstructed data.
    Args:
    x: Input data tensor of shape (batch_size, num_features).
    Returns:
    A tensor of shape (batch_size,) containing the probability densities of
    the input samples under the learned distribution of reconstructed data.
    """
    with torch.no_grad():
        pred = self.predict(x)
        recon_dist = Normal(pred['recon_mu'], pred['recon_sigma'])
        x = x.unsqueeze(0)
        print(pred['recon_mu'])
        print(pred['recon_sigma'])
        p = recon_dist.log_prob(x).exp().mean(dim=0).mean(dim=-1)  # vector of shape [batch_size]
        return p

示例输入 #

pred['recon_mu']：

torch.tensor([[[-3.4496e-02,  6.8990e-04,  2.6068e-02],[-1.6106e-02,  9.0834e-05,  2.5933e-02],[ 1.5478e-02, -1.0349e-02,  5.4951e-03],[ 4.3387e-03, -7.6342e-03, -6.9675e-03],[-1.1998e-02,  8.4998e-03,  9.7759e-04],[-8.3179e-03,  1.9121e-02, -2.9490e-02],[-2.2698e-02, -2.6095e-03, -2.8768e-02],[ 2.1862e-02,  4.6313e-03, -3.8202e-03],[-8.9038e-03, -6.5506e-03, -2.0708e-02],[ 3.4313e-03,  3.4495e-03, -3.2643e-03],[ 2.6851e-02,  2.2717e-03, -1.5377e-02],[ 1.8616e-02,  9.1497e-03, -1.2855e-02],[ 2.6439e-02,  2.0135e-02, -1.1923e-02],[ 1.7019e-02,  1.5564e-02, -1.6765e-02],[ 3.1842e-02,  1.9835e-02, -2.1093e-02],[ 2.8283e-02,  6.9692e-03,  2.7520e-02],[-3.7459e-02,  7.6441e-03,  7.5181e-03],[-2.4621e-02, -3.1863e-03, -1.5638e-02],[ 5.9735e-03,  1.2214e-02,  1.9666e-03],[ 2.5512e-02,  8.0318e-03,  1.1243e-02],[-3.1942e-02, -6.4672e-03,  3.7776e-03],[-1.1698e-02, -6.0982e-03, -6.1742e-03],[-3.3454e-02, -5.9270e-03,  5.0409e-03],[ 1.4640e-02,  7.8550e-03,  1.4246e-03],[ 8.3495e-03,  1.4253e-02, -3.1341e-02],[ 1.7591e-03,  1.0684e-02,  2.1749e-02],[ 3.3377e-02, -7.6833e-03,  9.6476e-03],[ 7.5541e-04,  5.7854e-04,  5.8191e-03],[ 1.6344e-02,  9.8728e-03,  4.5916e-03],[-6.3240e-03,  1.2885e-02, -1.6636e-02],[ 7.3146e-03,  1.0790e-02,  1.2983e-02],[-5.3678e-02,  3.5973e-03,  1.0571e-02]]])

pred['recon_sigma']：

torch.tensor([[[0.0560, 0.0183, 0.0262],[0.0772, 0.0272, 0.0386],[0.0513, 0.0143, 0.0225],[0.0559, 0.0163, 0.0242],[0.0496, 0.0144, 0.0211],[0.0351, 0.0087, 0.0141],[0.0281, 0.0060, 0.0103],[0.0582, 0.0179, 0.0251],[0.0572, 0.0174, 0.0258],[0.0940, 0.0359, 0.0481],[0.0503, 0.0132, 0.0197],[0.0968, 0.0390, 0.0491],[0.0406, 0.0102, 0.0160],[0.0844, 0.0293, 0.0393],[0.0815, 0.0293, 0.0371],[0.0448, 0.0113, 0.0204],[0.0699, 0.0232, 0.0318],[0.0546, 0.0168, 0.0241],[0.0452, 0.0138, 0.0205],[0.0798, 0.0303, 0.0396],[0.0610, 0.0196, 0.0297],[0.0722, 0.0220, 0.0330],[0.0483, 0.0143, 0.0211],[0.0699, 0.0243, 0.0339],[0.0391, 0.0098, 0.0154],[0.0586, 0.0194, 0.0247],[0.0433, 0.0103, 0.0165],[0.0630, 0.0190, 0.0283],[0.0597, 0.0193, 0.0281],[0.0577, 0.0192, 0.0257],[0.0610, 0.0215, 0.0304],[0.0445, 0.0125, 0.0197]]])

原因分析与修正 #

1. 概率密度可以大于1 #

首先明确：概率密度值本身完全可以大于1。概率密度（PDF）的定义是在区间上积分后得到概率（取值0到1），但单个点的密度值没有上限。比如你示例中recon_sigma存在0.0060这样的极小值，对应的正态分布在均值处的密度为1/(σ√(2π)) ≈ 66，远大于1，这是完全正常的。

2. 代码的核心错误：联合概率计算方式错误 #

你的代码中对各特征维度的密度取平均，这不是正确的联合概率密度计算方式：

• 若假设特征之间相互独立，多维正态分布的联合概率密度是各维度概率密度的乘积，对应log域是求和，而非平均。
• 你当前的计算mean(dim=-1)是把各维度的密度取平均，当单个维度密度大于1时，结果自然会超过1，且这根本不是样本的联合概率密度。

另外，x.unsqueeze(0)是多余的操作，因为pred['recon_mu']的第一个维度是冗余的1，直接squeeze后和x的形状匹配即可。

3. 修正后的代码 #

def reconstructed_probability(self, x: torch.Tensor) -> torch.Tensor:
    """
    Computes the probability density of the input samples under the learned
    distribution of reconstructed data.
    Args:
    x: Input data tensor of shape (batch_size, num_features).
    Returns:
    A tensor of shape (batch_size,) containing the probability densities of
    the input samples under the learned distribution of reconstructed data.
    """
    with torch.no_grad():
        pred = self.predict(x)
        # 移除冗余的第一维度，匹配x的形状
        recon_mu = pred['recon_mu'].squeeze(0)
        recon_sigma = pred['recon_sigma'].squeeze(0)
        recon_dist = Normal(recon_mu, recon_sigma)
        # 各维度log概率求和得到联合log概率，再exp得到联合概率密度
        joint_log_prob = recon_dist.log_prob(x).sum(dim=-1)
        p = joint_log_prob.exp()
        return p

修正后得到的p是样本的联合概率密度，虽然仍可能大于1（因为单个维度密度可以很大），但这是符合概率密度定义的正确结果。

内容的提问来源于stack exchange，提问作者Jiwon