# Posterior Projection for Inference in Constrained Spaces

arXiv: https://arxiv.org/abs/1812.05741 · HTML: https://mlsanity.com/paper/1812.05741/

## Key facts

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

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

- [Conditioning Gaussian Processes on Almost Anything](https://mlsanity.com/paper/2605.21041/) (2026)
- [LatentFlow: A General Framework for Conditioning Stochastic Processes](https://mlsanity.com/paper/2607.12922/) (2026)
- [Bayesian Inference for Consistent Predictions in Overparameterized Nonlinear Regression](https://mlsanity.com/paper/2404.04498/) (2024)
- [Empirical Likelihood with Generative AI](https://mlsanity.com/paper/2606.00425/) (2026)

## Abstract (arXiv)

Estimation of parameters that obey specific constraints is crucial in statistics and machine learning; for example, when parameters are required to satisfy boundedness, monotonicity, or linear inequalities. Traditional approaches impose these constraints via constraint-specific transformations or sampling approaches, or by truncating the posterior distribution. Such methods often result in computational challenges, limited flexibility, and a lack of generality. We propose a generalized framework for constrained Bayesian inference by projecting the unconstrained posterior distribution into the space of the parameter constraints, providing a computationally efficient and easily implementable solution for a large class of problems. We rigorously establish the theoretical foundations of the projected posterior distribution, as well as providing asymptotic results for posterior consistency, posterior contraction, and optimal coverage properties. Our methodology is validated through both theoretical arguments and practical applications.