# Free Energy Manifold: Score-Based Inference for Hybrid Bayesian Networks

arXiv: https://arxiv.org/abs/2605.09839 · HTML: https://mlsanity.com/paper/2605.09839/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 34 references that are in the map.
- First posted on arXiv 11 May 2026.

## Builds on (most cited first, 15 of 15)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere](https://mlsanity.com/paper/2005.10242/) (2020)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Gradient Surgery for Multi-Task Learning](https://mlsanity.com/paper/2001.06782/) (2020)
- [GradNorm: Gradient Normalization for Adaptive Loss Balancing in Deep Multitask Networks](https://mlsanity.com/paper/1711.02257/) (2017)
- [Multi-Task Learning as Multi-Objective Optimization](https://mlsanity.com/paper/1810.04650/) (2018)
- [A note on the evaluation of generative models](https://mlsanity.com/paper/1511.01844/) (2015)
- [Conditional Density Estimation with Neural Networks: Best Practices and Benchmarks](https://mlsanity.com/paper/1903.00954/) (2019)
- [Arbitrary Conditional Distributions with Energy](https://mlsanity.com/paper/2102.04426/) (2021)

## Abstract (arXiv)

We introduce the Free Energy Manifold (FEM), a score-trained conditional energy model specialized for inference in hybrid Bayesian networks with discrete and continuous variables. FEM represents each conditional factor as an energy landscape over learned discrete-parent embeddings and continuous observations, enabling posterior evaluation, generative sampling, and compositional inference across multiple continuous leaves by energy addition under conditional independence. A central finding is the mode-bridge artifact: standard conditional energy models can create low-energy ridges between separated modes of the same class, producing overconfident posteriors at off-data interior points. We analyze this failure and propose valley regularization, an off-data calibration term that restores near-uniform posteriors in such regions while preserving in-data fit. Across synthetic multimodal hybrid-BN benchmarks, FEM substantially reduces KL divergence relative to classical baselines and a vanilla conditional EBM, including large gains at mode-bridge midpoint queries and in multi-leaf evidence composition. We also evaluate high-cardinality discrete-parent settings and a UCI Breast Cancer sanity check, showing that FEM is most useful when multimodal or compositional Bayesian-network inference is required, while discriminative classifiers remain preferable for closed-world classification tasks.