# Autoencoders and Probabilistic Inference with Missing Data: An Exact Solution for The Factor Analysis Case

arXiv: https://arxiv.org/abs/1801.03851 · HTML: https://mlsanity.com/paper/1801.03851/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jan 2018.

## Built on by (most cited first, 6 of 6)

- [Variational Gibbs Inference for Statistical Model Estimation from Incomplete Data](https://mlsanity.com/paper/2111.13180/) (2021)
- [Tractable Representation Learning with Probabilistic Circuits](https://mlsanity.com/paper/2507.04385/) (2025)
- [Inference for Generative Capsule Models](https://mlsanity.com/paper/2103.06676/) (2021)
- [Inference and Learning for Generative Capsule Models](https://mlsanity.com/paper/2209.03115/) (2022)
- [Generative Adversarial Classification Network with Application to Network Traffic Classification](https://mlsanity.com/paper/2303.10681/) (2023)
- [Improving Variational Autoencoder Estimation from Incomplete Data with Mixture Variational Families](https://mlsanity.com/paper/2403.03069/) (2024)

## Abstract (arXiv)

Latent variable models can be used to probabilistically "fill-in" missing data entries. The variational autoencoder architecture (Kingma and Welling, 2014; Rezende et al., 2014) includes a "recognition" or "encoder" network that infers the latent variables given the data variables. However, it is not clear how to handle missing data variables in this network. The factor analysis (FA) model is a basic autoencoder, using linear encoder and decoder networks. We show how to calculate exactly the latent posterior distribution for the factor analysis (FA) model in the presence of missing data, and note that this solution implies that a different encoder network is required for each pattern of missingness. We also discuss various approximations to the exact solution. Experiments compare the effectiveness of various approaches to filling in the missing data.