# Semiparametric inference using fractional posteriors

arXiv: https://arxiv.org/abs/2301.08158 · HTML: https://mlsanity.com/paper/2301.08158/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 55 references that are in the map.
- First posted on arXiv 19 Jan 2023.

## Builds on (most cited first, 16 of 16)

- [Understanding disentangling in β-VAE](https://mlsanity.com/paper/1804.03599/) (2018)
- [User-friendly introduction to PAC-Bayes bounds](https://mlsanity.com/paper/2110.11216/) (2021)
- [A General Framework for Updating Belief Distributions](https://mlsanity.com/paper/1306.6430/) (2013)
- [On the properties of variational approximations of Gibbs posteriors](https://mlsanity.com/paper/1506.04091/) (2015)
- [Inconsistency of Bayesian Inference for Misspecified Linear Models, and a Proposal for Repairing It](https://mlsanity.com/paper/1412.3730/) (2014)
- [Concentration of tempered posteriors and of their variational approximations](https://mlsanity.com/paper/1706.09293/) (2017)
- [Assigning a value to a power likelihood in a general Bayesian model](https://mlsanity.com/paper/1701.08515/) (2017)
- [General Bayesian Updating and the Loss-Likelihood Bootstrap](https://mlsanity.com/paper/1709.07616/) (2017)
- [Robust Bayesian inference via coarsening](https://mlsanity.com/paper/1506.06101/) (2015)
- [Calibrating general posterior credible regions](https://mlsanity.com/paper/1509.00922/) (2015)
- [Fast Rates for General Unbounded Loss Functions: from ERM to Generalized Bayes](https://mlsanity.com/paper/1605.00252/) (2016)
- [Improving Explorability in Variational Inference with Annealed Variational Objectives](https://mlsanity.com/paper/1809.01818/) (2018)
- [A comparison of learning rate selection methods in generalized Bayesian inference](https://mlsanity.com/paper/2012.11349/) (2020)
- [On the Robustness to Misspecification of α-Posteriors and Their Variational Approximations](https://mlsanity.com/paper/2104.08324/) (2021)
- [Empirical priors for prediction in sparse high-dimensional linear regression](https://mlsanity.com/paper/1903.00961/) (2019)
- [An asymptotic analysis of distributed nonparametric methods](https://mlsanity.com/paper/1711.03149/) (2017)

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

- [Posterior and variational inference for deep neural networks with heavy-tailed weights](https://mlsanity.com/paper/2406.03369/) (2024)
- [Concentration of a sparse Bayesian model with Horseshoe prior in estimating high-dimensional precision matrix](https://mlsanity.com/paper/2406.14269/) (2024)
- [Deep Horseshoe Gaussian Processes](https://mlsanity.com/paper/2403.01737/) (2024)
- [Adaptive posterior concentration rates for sparse high-dimensional linear regression with random design and unknown error variance](https://mlsanity.com/paper/2405.19016/) (2024)

## Abstract (arXiv)

We establish a general Bernstein--von Mises theorem for approximately linear semiparametric functionals of fractional posterior distributions based on nonparametric priors. This is illustrated in a number of nonparametric settings and for different classes of prior distributions, including Gaussian process priors. We show that fractional posterior credible sets can provide reliable semiparametric uncertainty quantification, but have inflated size. To remedy this, we further propose a textitshifted-and-rescaled fractional posterior set that is an efficient confidence set having optimal size under regularity conditions. As part of our proofs, we also refine existing contraction rate results for fractional posteriors by sharpening the dependence of the rate on the fractional exponent.