# Direct Message Approximation (DMA): A Consistency-Based Framework for Tractable Approximate Inference on Factor Graphs

arXiv: https://arxiv.org/abs/2609.29466 · HTML: https://mlsanity.com/paper/2609.29466/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 29 references that are in the map.
- First posted on arXiv 24 Sep 2026.

## Builds on (most cited first, 6 of 6)

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [A Simple Baseline for Bayesian Uncertainty in Deep Learning](https://mlsanity.com/paper/1902.02476/) (2019)
- [Weight Uncertainty in Neural Networks](https://mlsanity.com/paper/1505.05424/) (2015)
- [Probabilistic Backpropagation for Scalable Learning of Bayesian Neural Networks](https://mlsanity.com/paper/1502.05336/) (2015)
- [Variational Learning is Effective for Large Deep Networks](https://mlsanity.com/paper/2402.17641/) (2024)
- [Stochastic Expectation Propagation](https://mlsanity.com/paper/1506.04132/) (2015)

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

- [Efficient Message Passing for Partial Differential Equation Priors](https://mlsanity.com/paper/2609.32956/) (2026)

## Abstract (arXiv)

Approximate message passing on factor graphs underlies two dominant families of probabilistic inference algorithms: expectation propagation (EP) and variational message passing (VMP). Both methods approximate the marginal at each factor edge, forcing an iterative round-robin schedule, risking negative-precision messages, and, for VMP, collapsing to point estimates at Dirac-delta factors. We introduce Direct Message Approximation (DMA), which approximates factor-to-variable messages directly rather than the marginal. For normalisable factors, we define a consistency condition (requiring exactness when all other incoming messages are Dirac deltas) to guide message construction. We prove a master theorem (proper messages, any graph) bounding marginal KL from message KL, with three structural corollaries: Dirac-input consistency, no EP-style inner-loop iteration, and no negative-precision messages. Further, we prove a complementary O(1/r2) guarantee for the inherently improper backward message of the product factor, whose closed-form treatment has resisted prior work. As a concrete instantiation, we derive explicit DMA messages for the product and leaky-ReLU factors and assemble a Bayesian neural network (BNN) inference algorithm with one forward/backward sweep per training example and no gradient learning-rate hyperparameter, validating that the structural guarantees translate to predictive uncertainty that widens in data-sparse regions, including under model mismatch.