# Memory by Design: Bayesian Layers for Sequence Modeling

arXiv: https://arxiv.org/abs/2605.31163 · HTML: https://mlsanity.com/paper/2605.31163/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 33 of its 42 references that are in the map.
- First posted on arXiv 29 May 2026.

## Builds on (most cited first, 30 of 33)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention](https://mlsanity.com/paper/2006.16236/) (2020)
- [RULER: What's the Real Context Size of Your Long-Context Language Models?](https://mlsanity.com/paper/2404.06654/) (2024)
- [Transformers are SSMs: Generalized Models and Efficient Algorithms Through Structured State Space Duality](https://mlsanity.com/paper/2405.21060/) (2024)
- [RWKV: Reinventing RNNs for the Transformer Era](https://mlsanity.com/paper/2305.13048/) (2023)
- [Compressive Transformers for Long-Range Sequence Modelling](https://mlsanity.com/paper/1911.05507/) (2019)
- [Retentive Network: A Successor to Transformer for Large Language Models](https://mlsanity.com/paper/2307.08621/) (2023)
- [Gated Delta Networks: Improving Mamba2 with Delta Rule](https://mlsanity.com/paper/2412.06464/) (2024)
- [Gated Linear Attention Transformers with Hardware-Efficient Training](https://mlsanity.com/paper/2312.06635/) (2023)
- [Linear Transformers Are Secretly Fast Weight Programmers](https://mlsanity.com/paper/2102.11174/) (2021)
- [Parallelizing Linear Transformers with the Delta Rule over Sequence Length](https://mlsanity.com/paper/2406.06484/) (2024)
- [xLSTM: Extended Long Short-Term Memory](https://mlsanity.com/paper/2405.04517/) (2024)
- [Griffin: Mixing Gated Linear Recurrences with Local Attention for Efficient Language Models](https://mlsanity.com/paper/2402.19427/) (2024)
- [Simple linear attention language models balance the recall-throughput tradeoff](https://mlsanity.com/paper/2402.18668/) (2024)
- [Zoology: Measuring and Improving Recall in Efficient Language Models](https://mlsanity.com/paper/2312.04927/) (2023)
- [Kimi Linear: An Expressive, Efficient Attention Architecture](https://mlsanity.com/paper/2510.26692/) (2025)
- [Eagle and Finch: RWKV with Matrix-Valued States and Dynamic Recurrence](https://mlsanity.com/paper/2404.05892/) (2024)
- [RWKV-7 "Goose" with Expressive Dynamic State Evolution](https://mlsanity.com/paper/2503.14456/) (2025)
- [HGRN2: Gated Linear RNNs with State Expansion](https://mlsanity.com/paper/2404.07904/) (2024)
- [The Mamba in the Llama: Distilling and Accelerating Hybrid Models](https://mlsanity.com/paper/2408.15237/) (2024)
- [The Bayesian Learning Rule](https://mlsanity.com/paper/2107.04562/) (2021)
- [DeltaProduct: Improving State-Tracking in Linear RNNs via Householder Products](https://mlsanity.com/paper/2502.10297/) (2025)
- [ATLAS: Learning to Optimally Memorize the Context at Test Time](https://mlsanity.com/paper/2505.23735/) (2025)
- [Longhorn: State Space Models are Amortized Online Learners](https://mlsanity.com/paper/2407.14207/) (2024)
- [MoM: Linear Sequence Modeling with Mixture-of-Memories](https://mlsanity.com/paper/2502.13685/) (2025)
- [MesaNet: Sequence Modeling by Locally Optimal Test-Time Training](https://mlsanity.com/paper/2506.05233/) (2025)
- [Liger: Linearizing Large Language Models to Gated Recurrent Structures](https://mlsanity.com/paper/2503.01496/) (2025)

## Abstract (arXiv)

We introduce the design-model framework: a way to derive efficient recurrent sequence maps from explicit assumptions about memory. A design model writes evidence into memory by exact Bayesian filtering; a query- dependent readout produces a predictive distribution whose mean is the layer output. In our linear-Gaussian instantiation, the Bayesian Layer propagates both a mean and a covariance: the covariance tracks uncertainty over stored associations, steering writes toward uncertain directions, attenuating gains as evidence accumulates, and preserving confident memories. The same framework unifies several sub-quadratic recurrences: linear attention, GLA, and Mamba-2/SSD are exact filters under a latent-input design model, whereas DeltaNet and related Delta-rule models are covariance-reset reductions of the Bayesian Layer's design model. Restoring covariance propagation yields closed-form predictions for retrieval dynamics, which we verify empirically, and improves robustness beyond the training regime in controlled collision studies, learned associative recall, and the Zoology MQAR benchmark. Training from scratch on WikiText-103 under matched state budgets lowers perplexity on associative-recall hits. Distilling Bayesian Layers into a pretrained 340M Gated DeltaNet improves RULER long-context retrieval over a matched-compute control, at a 2.5--2.7% held-out perplexity cost.