# Loss as the Inconsistency of a Probabilistic Dependency Graph: Choose Your Model, Not Your Loss Function

arXiv: https://arxiv.org/abs/2202.11862 · HTML: https://mlsanity.com/paper/2202.11862/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 26 references that are in the map.
- First posted on arXiv 24 Feb 2022.

## Builds on (most cited first, 4 of 4)

- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [A survey of loss functions for semantic segmentation](https://mlsanity.com/paper/2006.14822/) (2020)
- [Probabilistic Dependency Graphs](https://mlsanity.com/paper/2012.10800/) (2020)

## Built on by (most cited first, 6 of 6)

- [A Pattern Language for Machine Learning Tasks](https://mlsanity.com/paper/2407.02424/) (2024)
- [Free Energy Risk Metrics for Systemically Safe AI: Gatekeeping Multi-Agent Study](https://mlsanity.com/paper/2502.04249/) (2025)
- [Local Inconsistency Resolution: The Interplay between Attention and Control in Probabilistic Models](https://mlsanity.com/paper/2604.17140/) (2026)
- [Inference for Probabilistic Dependency Graphs](https://mlsanity.com/paper/2311.05580/) (2023)
- [Safety from Honesty in a Disinterested AI Predictor](https://mlsanity.com/paper/2606.29657/) (2026)
- [On Neural Networks as Infinite Tree-Structured Probabilistic Graphical Models](https://mlsanity.com/paper/2305.17583/) (2023)

## Abstract (arXiv)

In a world blessed with a great diversity of loss functions, we argue that that choice between them is not a matter of taste or pragmatics, but of model. Probabilistic depencency graphs (PDGs) are probabilistic models that come equipped with a measure of "inconsistency". We prove that many standard loss functions arise as the inconsistency of a natural PDG describing the appropriate scenario, and use the same approach to justify a well-known connection between regularizers and priors. We also show that the PDG inconsistency captures a large class of statistical divergences, and detail benefits of thinking of them in this way, including an intuitive visual language for deriving inequalities between them. In variational inference, we find that the ELBO, a somewhat opaque objective for latent variable models, and variants of it arise for free out of uncontroversial modeling assumptions -- as do simple graphical proofs of their corresponding bounds. Finally, we observe that inconsistency becomes the log partition function (free energy) in the setting where PDGs are factor graphs.