论文标题
格拉斯曼形式主义中的多元二元概率分布
Multivariate binary probability distribution in the Grassmann formalism
论文作者
论文摘要
我们提出了多元二进制随机变量的概率分布。为此，我们使用格拉曼（Grassmann）号码，一个反通讯数字。在我们的模型中，分区函数，中央力矩以及边缘和条件分布通过类似于多变量高斯分布中的协方差矩阵的参数的矩阵来分析表达。也就是说，所有可能状态的总和对于获得分区函数和各种预期值并不是必需的，这是常规多元伯努利分布的问题。所提出的模型与多元高斯分布有许多相似之处。例如，边际和条件分布分别由参数矩阵及其反矩阵表示。也就是说，逆矩阵表示部分相关。边际和条件分布的分析表达式也可用于生成多变量二进制变量的随机数。因此，我们使用合成数据集验证了提出的方法。我们观察到，各种统计数据的采样分布与理论预测一致，估计值是一致的，渐近地正常。
We propose a probability distribution for multivariate binary random variables. For this purpose, we use the Grassmann number, an anti-commuting number. In our model, the partition function, the central moment, and the marginal and conditional distributions are expressed analytically by the matrix of the parameters analogous to the covariance matrix in the multivariate Gaussian distribution. That is, summation over all possible states is not necessary for obtaining the partition function and various expected values, which is a problem with the conventional multivariate Bernoulli distribution. The proposed model has many similarities to the multivariate Gaussian distribution. For example, the marginal and conditional distributions are expressed by the parameter matrix and its inverse matrix, respectively. That is, the inverse matrix expresses a sort of partial correlation. Analytical expressions for the marginal and conditional distributions are also useful in generating random numbers for multivariate binary variables. Hence, we validated the proposed method using synthetic datasets. We observed that the sampling distributions of various statistics are consistent with the theoretical predictions and estimates are consistent and asymptotically normal.
