Distributions over random boolean vectors. Useful in computer science and piano rolls. Not quite the same as categorical distributions, although those can be written as distributions over boolean vectors, but in a multi-class classification case each realisation has only one class; in an \(n\)-class rv, there are \(n\) possible outcomes. In a multivariate Bernoulli distribution there are \(2^n\) possible outcomes.
Multivariate Gumbel-softmax tricks.
Not sure how these work but maybe related. See (Broderick, Pitman, and Jordan 2013; Zhang and Paisley 2019).
See, e.g. Lumbreras, Filstroff, and Févotte (2020)
Broderick, Tamara, Jim Pitman, and Michael I. Jordan. 2013. “Feature Allocations, Probability Functions, and Paintboxes.” Bayesian Analysis 8 (4): 801–36.
Dai, Bin, Shilin Ding, and Grace Wahba. 2013. “Multivariate Bernoulli Distribution.” Bernoulli 19 (4).
Lumbreras, Alberto, Louis Filstroff, and Cédric Févotte. 2020. “Bayesian Mean-Parameterized Nonnegative Binary Matrix Factorization.” Data Mining and Knowledge Discovery 34 (6): 1898–1935.
Miettinen, Pauli, and Stefan Neumann. 2020. “Recent Developments in Boolean Matrix Factorization.” In Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, 4922–28. Yokohama, Japan: International Joint Conferences on Artificial Intelligence Organization.
Reuter, Stephan, Ba-Tuong Vo, Ba-Ngu Vo, and Klaus Dietmayer. 2014. “The Labeled Multi-Bernoulli Filter.” IEEE Transactions on Signal Processing 62 (12): 15.
Rukat, Tammo, Chris C. Holmes, Michalis K. Titsias, and Christopher Yau. 2017. “Bayesian Boolean Matrix Factorisation.” In Proceedings of the 34th International Conference on Machine Learning, 2969–78. PMLR.
Teugels, Jozef L. 1990. “Some Representations of the Multivariate Bernoulli and Binomial Distributions.” Journal of Multivariate Analysis 32 (2): 256–68.
Vo, Ba Ngu, Ba Tuong Vo, and Hung Gia Hoang. 2017. “An Efficient Implementation of the Generalized Labeled Multi-Bernoulli Filter.” arXiv:1606.08350 [Stat], February.
Vo, Ba-Ngu, Ba-Tuong Vo, and Dinh Phung. 2014. “Labeled Random Finite Sets and the Bayes Multi-Target Tracking Filter.” IEEE Transactions on Signal Processing 62 (24): 6554–67.
Vo, Ba-Tuong, and Ba-Ngu Vo. 2013. “Labeled Random Finite Sets and Multi-Object Conjugate Priors.” IEEE Transactions on Signal Processing 61 (13): 3460–75.
Vo, Ba-Tuong, Ba-Ngu Vo, and Antonio Cantoni. 2009. “The Cardinality Balanced Multi-Target Multi-Bernoulli Filter and Its Implementations.” IEEE Transactions on Signal Processing 57 (2): 409–23.
Wang, Xi, and Junming Yin. 2020. “Relaxed Multivariate Bernoulli Distribution and Its Applications to Deep Generative Models.” In Proceedings of the 36th Conference on Uncertainty in Artificial Intelligence (UAI), 500–509. PMLR.
Zhang, Aonan, and John Paisley. 2019. “Random Function Priors for Correlation Modeling.” In International Conference on Machine Learning, 7424–33.
Zhou, Mingyuan, Lauren Hannah, David Dunson, and Lawrence Carin. 2012. “Beta-Negative Binomial Process and Poisson Factor Analysis.” In Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics, 1462–71. PMLR.
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