# A probabilistic population code based on neural samples

arXiv: https://arxiv.org/abs/1811.09739 · HTML: https://mlsanity.com/paper/1811.09739/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Nov 2018.

## Built on by (most cited first, 4 of 4)

- [Neural Sampling in Hierarchical Exponential-family Energy-based Models](https://mlsanity.com/paper/2310.08431/) (2023)
- [How does the brain compute with probabilities?](https://mlsanity.com/paper/2409.02709/) (2024)
- [A Unified Theory of Exact Inference and Learning in Exponential Family Latent Variable Models](https://mlsanity.com/paper/2404.19501/) (2024)
- [JIR-Arena: The First Benchmark Dataset for Just-in-time Information Recommendation](https://mlsanity.com/paper/2505.13550/) (2025)

## Abstract (arXiv)

Sensory processing is often characterized as implementing probabilistic inference: networks of neurons compute posterior beliefs over unobserved causes given the sensory inputs. How these beliefs are computed and represented by neural responses is much-debated (Fiser et al. 2010, Pouget et al. 2013). A central debate concerns the question of whether neural responses represent samples of latent variables (Hoyer & Hyvarinnen 2003) or parameters of their distributions (Ma et al. 2006) with efforts being made to distinguish between them (Grabska-Barwinska et al. 2013). A separate debate addresses the question of whether neural responses are proportionally related to the encoded probabilities (Barlow 1969), or proportional to the logarithm of those probabilities (Jazayeri & Movshon 2006, Ma et al. 2006, Beck et al. 2012). Here, we show that these alternatives - contrary to common assumptions - are not mutually exclusive and that the very same system can be compatible with all of them. As a central analytical result, we show that modeling neural responses in area V1 as samples from a posterior distribution over latents in a linear Gaussian model of the image implies that those neural responses form a linear Probabilistic Population Code (PPC, Ma et al. 2006). In particular, the posterior distribution over some experimenter-defined variable like "orientation" is part of the exponential family with sufficient statistics that are linear in the neural sampling-based firing rates.