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We propose a probability distribution for multivariate binary randomvariables. For this purpose, we use the Grassmann number, an anti-commutingnumber. In our model, the partition function, the central moment, and themarginal and conditional distributions are expressed analytically by the matrixof the parameters analogous to the covariance matrix in the multivariateGaussian distribution. That is, summation over all possible states is notnecessary 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 Gaussiandistribution. For example, the marginal and conditional distributions areexpressed by the parameter matrix and its inverse matrix, respectively. Thatis, the inverse matrix expresses a sort of partial correlation. Analyticalexpressions for the marginal and conditional distributions are also useful ingenerating random numbers for multivariate binary variables. Hence, wevalidated the proposed method using synthetic datasets. We observed that thesampling distributions of various statistics are consistent with thetheoretical predictions and estimates are consistent and asymptotically normal.

Multivariate binary probability distribution in the Grassmann formalism