# NOTMAD: Estimating Bayesian Networks with Sample-Specific Structures and Parameters

arXiv: https://arxiv.org/abs/2111.01104 · HTML: https://mlsanity.com/paper/2111.01104/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 46 references that are in the map.
- First posted on arXiv 1 Nov 2021.

## Builds on (most cited first, 13 of 13)

- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [Neural Additive Models: Interpretable Machine Learning with Neural Nets](https://mlsanity.com/paper/2004.13912/) (2020)
- [Context-Specific Independence in Bayesian Networks](https://mlsanity.com/paper/1302.3562/) (2013)
- [Causal Structure Discovery from Distributions Arising from Mixtures of DAGs](https://mlsanity.com/paper/2001.11940/) (2020)
- [Contextual Explanation Networks](https://mlsanity.com/paper/1705.10301/) (2017)
- [Labeled Directed Acyclic Graphs: a generalization of context-specific independence in directed graphical models](https://mlsanity.com/paper/1310.1187/) (2013)
- [Maximum Likelihood Estimation for Learning Populations of Parameters](https://mlsanity.com/paper/1902.04553/) (2019)
- [Learning Mixtures of DAG Models](https://mlsanity.com/paper/1301.7415/) (2013)
- [Causal Discovery with a Mixture of DAGs](https://mlsanity.com/paper/1901.09475/) (2019)
- [Exact Estimation of Multiple Directed Acyclic Graphs](https://mlsanity.com/paper/1404.1238/) (2014)
- [Learning Sample-Specific Models with Low-Rank Personalized Regression](https://mlsanity.com/paper/1910.06939/) (2019)
- [Bayesian joint inference for multiple directed acyclic graphs](https://mlsanity.com/paper/2008.06190/) (2020)

## Built on by (most cited first, 1 of 1)

- [Contextualized Machine Learning](https://mlsanity.com/paper/2310.11340/) (2023)

## Abstract (arXiv)

Context-specific Bayesian networks (i.e. directed acyclic graphs, DAGs) identify context-dependent relationships between variables, but the non-convexity induced by the acyclicity requirement makes it difficult to share information between context-specific estimators (e.g. with graph generator functions). For this reason, existing methods for inferring context-specific Bayesian networks have favored breaking datasets into subsamples, limiting statistical power and resolution, and preventing the use of multidimensional and latent contexts. To overcome this challenge, we propose NOTEARS-optimized Mixtures of Archetypal DAGs (NOTMAD). NOTMAD models context-specific Bayesian networks as the output of a function which learns to mix archetypal networks according to sample context. The archetypal networks are estimated jointly with the context-specific networks and do not require any prior knowledge. We encode the acyclicity constraint as a smooth regularization loss which is back-propagated to the mixing function; in this way, NOTMAD shares information between context-specific acyclic graphs, enabling the estimation of Bayesian network structures and parameters at even single-sample resolution. We demonstrate the utility of NOTMAD and sample-specific network inference through analysis and experiments, including patient-specific gene expression networks which correspond to morphological variation in cancer.