Robert Fluddβs piano rolls
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.
Continuous relaxations
Multivariate Gumbel-softmax tricks.
Paintbox models
Not sure how these work but maybe related. See (Broderick, Pitman, and Jordan 2013; Zhang and Paisley 2019).
References
Broderick, Tamara, Jim Pitman, and Michael I. Jordan. 2013. βFeature Allocations, Probability Functions, and Paintboxes.β Bayesian Analysis 8 (4): 801β36.
Dai, Bin, Shilin Ding, and Grace Wahba. 2013. βMultivariate Bernoulli Distribution.β Bernoulli 19 (4).
Lumbreras, Alberto, Louis Filstroff, and CΓ©dric FΓ©votte. 2020. βBayesian Mean-Parameterized Nonnegative Binary Matrix Factorization.β Data Mining and Knowledge Discovery 34 (6): 1898β1935.
Miettinen, Pauli, and Stefan Neumann. 2020. βRecent Developments in Boolean Matrix Factorization.β In Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, 4922β28. Yokohama, Japan: International Joint Conferences on Artificial Intelligence Organization.
Reuter, Stephan, Ba-Tuong Vo, Ba-Ngu Vo, and Klaus Dietmayer. 2014. βThe Labeled Multi-Bernoulli Filter.β IEEE Transactions on Signal Processing 62 (12): 15.
Rukat, Tammo, Chris C. Holmes, Michalis K. Titsias, and Christopher Yau. 2017. βBayesian Boolean Matrix Factorisation.β In Proceedings of the 34th International Conference on Machine Learning, 2969β78. PMLR.
Teugels, Jozef L. 1990. βSome Representations of the Multivariate Bernoulli and Binomial Distributions.β Journal of Multivariate Analysis 32 (2): 256β68.
Vo, Ba Ngu, Ba Tuong Vo, and Hung Gia Hoang. 2017. βAn Efficient Implementation of the Generalized Labeled Multi-Bernoulli Filter.β arXiv:1606.08350 [Stat], February.
Vo, Ba-Ngu, Ba-Tuong Vo, and Dinh Phung. 2014. βLabeled Random Finite Sets and the Bayes Multi-Target Tracking Filter.β IEEE Transactions on Signal Processing 62 (24): 6554β67.
Vo, Ba-Tuong, and Ba-Ngu Vo. 2013. βLabeled Random Finite Sets and Multi-Object Conjugate Priors.β IEEE Transactions on Signal Processing 61 (13): 3460β75.
Vo, Ba-Tuong, Ba-Ngu Vo, and Antonio Cantoni. 2009. βThe Cardinality Balanced Multi-Target Multi-Bernoulli Filter and Its Implementations.β IEEE Transactions on Signal Processing 57 (2): 409β23.
Wang, Xi, and Junming Yin. 2020. βRelaxed Multivariate Bernoulli Distribution and Its Applications to Deep Generative Models.β In Proceedings of the 36th Conference on Uncertainty in Artificial Intelligence (UAI), 500β509. PMLR.
Zhang, Aonan, and John Paisley. 2019. βRandom Function Priors for Correlation Modeling.β In International Conference on Machine Learning, 7424β33.
Zhou, Mingyuan, Lauren Hannah, David Dunson, and Lawrence Carin. 2012. βBeta-Negative Binomial Process and Poisson Factor Analysis.β In Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics, 1462β71. PMLR.
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