# Imputation estimators for unnormalized models with missing data

arXiv: https://arxiv.org/abs/1903.03630 · HTML: https://mlsanity.com/paper/1903.03630/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Mar 2019.

## Built on by (most cited first, 2 of 2)

- [Score Matching With Missing Data](https://mlsanity.com/paper/2506.00557/) (2025)
- [Missing Data Imputation by Reducing Mutual Information with Rectified Flows](https://mlsanity.com/paper/2505.11749/) (2025)

## Abstract (arXiv)

Several statistical models are given in the form of unnormalized densities, and calculation of the normalization constant is intractable. We propose estimation methods for such unnormalized models with missing data. The key concept is to combine imputation techniques with estimators for unnormalized models including noise contrastive estimation and score matching. In addition, we derive asymptotic distributions of the proposed estimators and construct confidence intervals. Simulation results with truncated Gaussian graphical models and the application to real data of wind direction reveal that the proposed methods effectively enable statistical inference with unnormalized models from missing data.