# Deviance-style normalization for jointly overdispersed counts

arXiv: https://arxiv.org/abs/2606.26061 · HTML: https://mlsanity.com/paper/2606.26061/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 71 references that are in the map.
- First posted on arXiv 24 Jun 2026.

## Builds on (most cited first, 1 of 1)

- [Fast MLE Computation for the Dirichlet Multinomial](https://mlsanity.com/paper/1405.0099/) (2014)

## Abstract (arXiv)

We introduce a Dirichlet--multinomial (DM) deviance residualization for sparse, jointly overdispersed count matrices, the regime that dominates sequencing-based biochemical assays. The DM null treats each sample's count vector as a fixed-total composition with a single scalar concentration α0 governing overdispersion, and arises exactly by conditioning independent negative-binomial feature counts on the observed sample total -- making the DM the joint conditional analogue of standard feature-wise overdispersed count models. The resulting transform preserves exact sparsity, evaluates in constant time per nonzero entry, agrees with multinomial residuals on singleton counts, shrinks repeated-count residuals according to the overdispersion the null tolerates, and recovers the multinomial residual as α0→∞. The same fixed-dispersion comparison principle extends to ordered and tree-structured features via the generalized DM and the Dirichlet-tree multinomial, giving a single residual family that subsumes joint and feature-wise count nulls under a common compositional logic and is computationally lightweight enough to drop into existing sparse pipelines.