Grassmann形式中的多元二元概率分布
Multivariate binary probability distribution in the Grassmann formalism
AI总结：
本文提出一种基于Grassmann数的多元二元概率分布模型，通过解析表达式解决传统多元伯努利分布的计算复杂度问题，并展示其在生成随机数和参数估计中的应用。
摘要：
我们提出了一种用于多元二元随机变量的概率分布。该概率分布表示为参数矩阵的主子式，这是一种类似于多元高斯分布中逆协方差矩阵的矩阵。在我们的模型中，分区函数、中心矩以及边缘和条件分布都以解析形式表达。也就是说，求和所有可能状态是不需要的，以获得分区函数和各种期望值，这是传统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，边缘和条件分布分别以参数矩阵及其逆矩阵表示。也就是说，逆矩阵代表了一种部分相关性。所提出分布可以使用Grassmann数、反交换数推导出来。解析表达式对于生成多元二元变量的随机数也是有用的。因此，我们使用合成数据集研究了参数估计的抽样分布。从观测数据中最大似然估计的计算复杂度与唯一观测状态的数量成正比，而不是像传统多元伯努利分布那样与所有可能状态的数量成正比。我们实证观察到，最大似然估计的抽样分布似乎一致且渐近正态。
英文摘要：
We propose a probability distribution for multivariate binary random variables. The probability distribution is expressed as principal minors of the parameter matrix, which is a matrix analogous to the inverse covariance matrix in the multivariate Gaussian distribution. In our model, the partition function, central moments, and the marginal and conditional distributions are expressed analytically. 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 in terms of the parameter matrix and its inverse matrix, respectively. That is, the inverse matrix represents a sort of partial correlation. The proposed distribution can be derived using Grassmann numbers, anticommuting numbers. Analytical expressions for the marginal and conditional distributions are also useful in generating random numbers for multivariate binary variables. Hence, we investigated sampling distributions of parameter estimates using synthetic datasets. The computational complexity of maximum likelihood estimation from observed data is proportional to the number of unique observed states, not to the number of all possible states as is required in the case of the conventional multivariate Bernoulli distribution. We empirically observed that the sampling distributions of the maximum likelihood estimates appear to be consistent and asymptotically normal.
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