# General Bayesian Updating and the Loss-Likelihood Bootstrap

arXiv: https://arxiv.org/abs/1709.07616 · HTML: https://mlsanity.com/paper/1709.07616/

## Key facts

- Cited by 34 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Sep 2017.

## Built on by (most cited first, 30 of 34)

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- [Concentration properties of fractional posterior in 1-bit matrix completion](https://mlsanity.com/paper/2404.08969/) (2024)
- [A sparse PAC-Bayesian approach for high-dimensional quantile prediction](https://mlsanity.com/paper/2409.01687/) (2024)
- [Enhancing Transfer Learning with Flexible Nonparametric Posterior Sampling](https://mlsanity.com/paper/2403.07282/) (2024)
- [Mitigating Statistical Bias within Differentially Private Synthetic Data](https://mlsanity.com/paper/2108.10934/) (2021)
- [Semiparametric inference using fractional posteriors](https://mlsanity.com/paper/2301.08158/) (2023)
- [A Semi-Bayesian Nonparametric Estimator of the Maximum Mean Discrepancy Measure: Applications in Goodness-of-Fit Testing and Generative Adversarial Networks](https://mlsanity.com/paper/2303.02637/) (2023)
- [Robust Bayesian Inference for Measurement Error Misspecification: The Berkson and Classical Cases](https://mlsanity.com/paper/2306.01468/) (2023)
- [Reproducible Parameter Inference Using Bagged Posteriors](https://mlsanity.com/paper/2311.02019/) (2023)
- [Interventional Processes for Causal Uncertainty Quantification](https://mlsanity.com/paper/2410.14483/) (2024)
- [Amortized Simulation-Based Inference in Generalized Bayes via Neural Posterior Estimation](https://mlsanity.com/paper/2601.22367/) (2026)
- [A Bayesian Non-parametric Approach to Generative Models: Integrating Variational Autoencoder and Generative Adversarial Networks using Wasserstein and Maximum Mean Discrepancy](https://mlsanity.com/paper/2308.14048/) (2023)
- [Sequential Gibbs Posteriors with Applications to Principal Component Analysis](https://mlsanity.com/paper/2310.12882/) (2023)
- [Adaptive posterior concentration rates for sparse high-dimensional linear regression with random design and unknown error variance](https://mlsanity.com/paper/2405.19016/) (2024)
- [Temperature Optimization for Bayesian Deep Learning](https://mlsanity.com/paper/2410.05757/) (2024)
- [When Is Generalized Bayes Bayesian? A Decision-Theoretic Characterization of Loss-Based Updating](https://mlsanity.com/paper/2602.01573/) (2026)
- [Supercharging Bayesian Inference with Reliable AI-Informed Priors](https://mlsanity.com/paper/2605.09834/) (2026)
- [Generalised Robust Bayes for Joint Inference of Model and Contamination](https://mlsanity.com/paper/2607.25665/) (2026)
- [Nonparametric posterior learning for emission tomography with multimodal data](https://mlsanity.com/paper/2108.00866/) (2021)
- [A Deep Bayesian Nonparametric Framework for Robust Mutual Information Estimation](https://mlsanity.com/paper/2503.08902/) (2025)
- [A weighted-likelihood framework for class imbalance in Bayesian prediction models](https://mlsanity.com/paper/2504.17013/) (2025)
- [General Bayesian Policy Learning](https://mlsanity.com/paper/2602.23672/) (2026)
- [Bayesian X-Learner: Calibrated Posterior Inference for Heterogeneous Treatment Effects under Heavy-Tailed Outcomes](https://mlsanity.com/paper/2604.27394/) (2026)

## Abstract (arXiv)

In this paper we revisit the weighted likelihood bootstrap, a method that generates samples from an approximate Bayesian posterior of a parametric model. We show that the same method can be derived, without approximation, under a Bayesian nonparametric model with the parameter of interest defined as minimising an expected negative log-likelihood under an unknown sampling distribution. This interpretation enables us to extend the weighted likelihood bootstrap to posterior sampling for parameters minimizing an expected loss. We call this method the loss-likelihood bootstrap. We make a connection between this and general Bayesian updating, which is a way of updating prior belief distributions without needing to construct a global probability model, yet requires the calibration of two forms of loss function. The loss-likelihood bootstrap is used to calibrate the general Bayesian posterior by matching asymptotic Fisher information. We demonstrate the methodology on a number of examples.