# Random binary vectors

The class of distributions that cause you to reinvent Shannon information if you stare at them long enough

February 20, 2017 — March 30, 2022

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.

## 1 Continuous relaxations

Multivariate Gumbel-softmax tricks.

## 2 Paintbox models

Not sure how these work but maybe related. See (Broderick, Pitman, and Jordan 2013; Zhang and Paisley 2019).

## 3 Matrix models

TBC.

See, e.g. Lumbreras, Filstroff, and Févotte (2020)

## 4 References

*Bayesian Analysis*.

*Bernoulli*.

*Data Mining and Knowledge Discovery*.

*Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence*.

*IEEE Transactions on Signal Processing*.

*Proceedings of the 34th International Conference on Machine Learning*.

*Journal of Multivariate Analysis*.

*IEEE Transactions on Signal Processing*.

*IEEE Transactions on Signal Processing*.

*arXiv:1606.08350 [Stat]*.

*IEEE Transactions on Signal Processing*.

*Proceedings of the 36th Conference on Uncertainty in Artificial Intelligence (UAI)*.

*International Conference on Machine Learning*.

*Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics*.