# Categorical random variates

February 20, 2017 — January 12, 2022

Distributions over categories.

## 1 Stick breaking tricks

Recommended reading: Machine Learning Trick of the Day (6): Tricks with Sticks— Shakir Mohammed.

TBC.

## 2 via random measures

See random measures.

## 3 Gumbel-max

See Gumbel-max tricks.

## 4 Pólya-Gamma augmentation

See Pólya-Gamma.

## 5 Softmax models

TBC

## 6 Multicategorical distributions

Can something belong to many categories? Then we are probably looking for Paintbox models (Broderick, Pitman, and Jordan 2013; Zhang and Paisley 2019) or some kind of multivariate Bernoulli model.

## 7 Dirichlet distribution

TBD. See Dirichlet distributions.

## 8 Dirichlet process

TBD. A distribution over an unknown number of categories. See also Gamma processes, which is how I learned to understand Dirichlet processes, insofar as I do.

## 9 Parametric distributions over non-negative integers

See count models.

## 10 Ordinal

If there is a natural ordering to the categories, then we are in a weird place. TBC.

## 11 Calibration

In the context of binary classification, calibration refers to the process of transforming the output scores from a binary classifier to class probabilities. If we think of the classifier as a “black box” that transforms input data into a score, we can think of calibration as a post-processing step that converts the score into a probability of the observation belonging to class 1.

The scores from some classifiers can already be interpreted as probabilities (e.g. logistic regression), while the scores from some classifiers require an additional calibration step before they can be interpreted as such (e.g. support vector machines).

He recommends the tutorial Huang et al. (2020) and associated github.

More general probabilistic calibration here.

## 12 Hierarchical

TBD

## 13 References

*An Introduction to Categorical Data Analysis*.

*Bayesian Analysis*.

*Journal of the American Statistical Association*.

*The Annals of Statistics*.

*Proceedings of the 31st International Conference on Machine Learning*.

*The Annals of Statistics*.

*Journal of the American Medical Informatics Association : JAMIA*.

*arXiv:2110.01515 [Cs, Stat]*.

*Canadian Journal of Statistics*.

*arXiv:1611.01144 [Cs, Stat]*.

*Bernoulli*.

*2011 International Conference on Computer Vision*.

*Journal of the American Statistical Association*.

*Proceedings of the 22nd International Conference on Neural Information Processing Systems*. NIPS’09.

*Artificial Intelligence and Statistics*.

*arXiv:1506.08180 [Cs, Stat]*.

*Proceedings of the 40th International Conference on Machine Learning*.

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

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

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

*arXiv:1503.08542 [Cs, Stat]*.

*International Conference on Machine Learning*.