Multivariate binary probability distribution in the Grassmann formalism.

Area of Science:

• Statistics
• Probability Theory
• Machine Learning

Background:

• Conventional multivariate Bernoulli distributions present computational challenges due to the need for summing over all possible states.
• Existing models lack efficient analytical expressions for key statistical properties.

Purpose of the Study:

• To propose a novel probability distribution for multivariate binary random variables.
• To develop a model with analytical solutions for partition functions, moments, and marginal/conditional distributions.
• To enhance computational efficiency in parameter estimation compared to traditional methods.

Main Methods:

• The proposed distribution is defined using principal minors of a parameter matrix, analogous to the inverse covariance matrix in Gaussian models.
• Analytical expressions are derived for the partition function, central moments, and marginal/conditional distributions.
• The model utilizes Grassmann numbers for derivation and facilitates random number generation.

Main Results:

• The partition function and expected values are obtained analytically, avoiding summation over all states.
• Marginal and conditional distributions are expressed using the parameter matrix and its inverse, revealing partial correlation structures.
• Maximum likelihood estimation shows computational complexity dependent on unique observed states, not total possible states.

Conclusions:

• The proposed multivariate binary distribution offers significant analytical and computational advantages over conventional methods.
• The model's structure and properties closely resemble the multivariate Gaussian distribution.
• Empirical analysis indicates that parameter estimates are consistent and asymptotically normal, supporting its utility.