# Efficient non-conjugate Gaussian process factor models for spike count data using polynomial approximations

arXiv: https://arxiv.org/abs/1906.03318 · HTML: https://mlsanity.com/paper/1906.03318/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jun 2019.

## Built on by (most cited first, 8 of 8)

- [Neural Latents Benchmark '21: Evaluating latent variable models of neural population activity](https://mlsanity.com/paper/2109.04463/) (2021)
- [Multi-modal Gaussian Process Variational Autoencoders for Neural and Behavioral Data](https://mlsanity.com/paper/2310.03111/) (2023)
- [Nonnegative spatial factorization](https://mlsanity.com/paper/2110.06122/) (2021)
- [Structured Recognition for Generative Models with Explaining Away](https://mlsanity.com/paper/2209.05212/) (2022)
- [Conditionally-Conjugate Gaussian Process Factor Analysis for Spike Count Data via Data Augmentation](https://mlsanity.com/paper/2405.11683/) (2024)
- [Learning Coupled Subspaces for Multi-Condition Spike Data](https://mlsanity.com/paper/2410.19153/) (2024)
- [Scalable inference of functional neural connectivity at submillisecond timescales](https://mlsanity.com/paper/2510.20966/) (2025)
- [A New Family of Poisson Non-negative Matrix Factorization Methods Using the Shifted Log Link](https://mlsanity.com/paper/2601.05845/) (2026)

## Abstract (arXiv)

Gaussian Process Factor Analysis (GPFA) has been broadly applied to the problem of identifying smooth, low-dimensional temporal structure underlying large-scale neural recordings. However, spike trains are non-Gaussian, which motivates combining GPFA with discrete observation models for binned spike count data. The drawback to this approach is that GPFA priors are not conjugate to count model likelihoods, which makes inference challenging. Here we address this obstacle by introducing a fast, approximate inference method for non-conjugate GPFA models. Our approach uses orthogonal second-order polynomials to approximate the nonlinear terms in the non-conjugate log-likelihood, resulting in a method we refer to as textitpolynomial approximate log-likelihood (PAL) estimators. This approximation allows for accurate closed-form evaluation of marginal likelihoods and fast numerical optimization for parameters and hyperparameters. We derive PAL estimators for GPFA models with binomial, Poisson, and negative binomial observations and find the PAL estimation is highly accurate, and achieves faster convergence times compared to existing state-of-the-art inference methods. We also find that PAL hyperparameters can provide sensible initialization for black box variational inference (BBVI), which improves BBVI accuracy. We demonstrate that PAL estimators achieve fast and accurate extraction of latent structure from multi-neuron spike train data.