# SOLBP: Second-Order Loopy Belief Propagation for Inference in Uncertain Bayesian Networks

arXiv: https://arxiv.org/abs/2208.07368 · HTML: https://mlsanity.com/paper/2208.07368/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 14 references that are in the map.
- First posted on arXiv 16 Aug 2022.

## Builds on (most cited first, 3 of 3)

- [A Differential Approach to Inference in Bayesian Networks](https://mlsanity.com/paper/1301.3847/) (2013)
- [Loopy Belief Propagation for Approximate Inference: An Empirical Study](https://mlsanity.com/paper/1301.6725/) (2013)
- [Handling Epistemic and Aleatory Uncertainties in Probabilistic Circuits](https://mlsanity.com/paper/2102.10865/) (2021)

## Abstract (arXiv)

In second-order uncertain Bayesian networks, the conditional probabilities are only known within distributions, i.e., probabilities over probabilities. The delta-method has been applied to extend exact first-order inference methods to propagate both means and variances through sum-product networks derived from Bayesian networks, thereby characterizing epistemic uncertainty, or the uncertainty in the model itself. Alternatively, second-order belief propagation has been demonstrated for polytrees but not for general directed acyclic graph structures. In this work, we extend Loopy Belief Propagation to the setting of second-order Bayesian networks, giving rise to Second-Order Loopy Belief Propagation (SOLBP). For second-order Bayesian networks, SOLBP generates inferences consistent with those generated by sum-product networks, while being more computationally efficient and scalable.