Multivariate Bernoulli Models and Generation Techniques Open Access
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AbstractThe purpose of this thesis is to explore different modeling strategies for generation of high dimensional Bernoulli vectors. We discuss the basic properties of the Multivariate Bernoulli (MB) distribution and examine several existing models. Some methods in the literature, including fully specified, latent variable, mixture and conditional mean models are discussed. Mixture models allow for the indirect full specification of a MB distribution and yield closed form solutions for the probability mass function once a suitable prior joint distribution for success probabilities is available. We discuss a MB distribution based on the Dirichlet prior, obtain its joint pmf and describe a method of bit-flipping that can accommodate positive and negative correlations. The Dirichlet distribution provides a MB distribution whose parameter space is defined on a simplex. We present parameter implantation as a variant of mixture method that specifies the success probabilities as a given function of independent latent variables. We also propose a generalized normal mixture model whose higher order moments are not only functions of success probabilities and second order moments, but further controlled by latent variables. Several examples are provided. While the methods of Emrich and Piedmonte (1991) and Modarres (2010) are most flexible, the mixture models are more efficient methods, especially for large dimensions.