# Sequential Gibbs Posteriors with Applications to Principal Component Analysis

arXiv: https://arxiv.org/abs/2310.12882 · HTML: https://mlsanity.com/paper/2310.12882/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 73 references that are in the map.
- First posted on arXiv 19 Oct 2023.

## Builds on (most cited first, 20 of 20)

- [Exploring Generalization in Deep Learning](https://mlsanity.com/paper/1706.08947/) (2017)
- [Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data](https://mlsanity.com/paper/1703.11008/) (2017)
- [A General Framework for Updating Belief Distributions](https://mlsanity.com/paper/1306.6430/) (2013)
- [A Primer on PAC-Bayesian Learning](https://mlsanity.com/paper/1901.05353/) (2019)
- [Inconsistency of Bayesian Inference for Misspecified Linear Models, and a Proposal for Repairing It](https://mlsanity.com/paper/1412.3730/) (2014)
- [A Strongly Quasiconvex PAC-Bayesian Bound](https://mlsanity.com/paper/1608.05610/) (2016)
- [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)
- [Calibrating general posterior credible regions](https://mlsanity.com/paper/1509.00922/) (2015)
- [A comparison of learning rate selection methods in generalized Bayesian inference](https://mlsanity.com/paper/2012.11349/) (2020)
- [A multi-resolution approximation for massive spatial datasets](https://mlsanity.com/paper/1507.04789/) (2015)
- [Direct Gibbs posterior inference on risk minimizers: construction, concentration, and calibration](https://mlsanity.com/paper/2203.09381/) (2022)
- [Bayesian Inference on Matrix Manifolds for Linear Dimensionality Reduction](https://mlsanity.com/paper/1606.04478/) (2016)
- [Can we trust the bootstrap in high-dimension?](https://mlsanity.com/paper/1608.00696/) (2016)
- [Gibbs posterior inference on multivariate quantiles](https://mlsanity.com/paper/2002.01052/) (2020)
- [A generalized Bayes framework for probabilistic clustering](https://mlsanity.com/paper/2006.05451/) (2020)
- [Learning Subspaces of Different Dimension](https://mlsanity.com/paper/1404.6841/) (2014)
- [Robust and rate-optimal Gibbs posterior inference on the boundary of a noisy image](https://mlsanity.com/paper/1606.08400/) (2016)
- [Bayesian Inference over the Stiefel Manifold via the Givens Representation](https://mlsanity.com/paper/1710.09443/) (2017)
- [Bayesian Modular and Multiscale Regression](https://mlsanity.com/paper/1809.05935/) (2018)

## Built on by (most cited first, 1 of 1)

- [Adaptive Bayesian Online Learning via Expert Aggregation](https://mlsanity.com/paper/2607.20239/) (2026)

## Abstract (arXiv)

Gibbs posteriors are proportional to a prior distribution multiplied by an exponentiated loss function, with a key tuning parameter weighting information in the loss relative to the prior and providing a control of posterior uncertainty. Gibbs posteriors provide a principled framework for likelihood-free Bayesian inference, but in many situations, including a single tuning parameter inevitably leads to poor uncertainty quantification. In particular, regardless of the value of the parameter, credible regions have far from the nominal frequentist coverage even in large samples. We propose a sequential extension to Gibbs posteriors to address this problem. We prove the proposed sequential posterior exhibits concentration and a Bernstein-von Mises theorem, which holds under easy to verify conditions in Euclidean space and on manifolds. As a byproduct, we obtain the first Bernstein-von Mises theorem for traditional likelihood-based Bayesian posteriors on manifolds. All methods are illustrated with an application to principal component analysis.