# General Bayesian Updating and the Loss-Likelihood Bootstrap

**Authors:** Simon Lyddon, Chris Holmes, Stephen Walker  
**arXiv:** [1709.07616](https://arxiv.org/abs/1709.07616)  
**Date:** 2017-09-22  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/1709.07616v2/

## Key points

- Authors Simon Lyddon, Chris Holmes, and Stephen Walker revisit the weighted likelihood bootstrap method for generating samples from an approximate Bayesian posterior of a parametric model.
- The method is extended to apply to a Bayesian nonparametric model without approximation.
- The parameter of interest is defined as minimizing an expected negative log-likelihood under an unknown sampling distribution.
- This leads to the development of the loss-likelihood bootstrap method for posterior sampling for parameters that minimize an expected loss.
- The authors establish a connection between this approach and general Bayesian updating, involving calibration of two forms of loss function.
- The loss-likelihood bootstrap is used to calibrate the general Bayesian posterior by matching asymptotic Fisher information.
- Various examples are provided in the paper to illustrate the methodology's application in advancing methods for parameter estimation.

## Summary

In their paper "General Bayesian Updating and the Loss-Likelihood Bootstrap," authors Simon Lyddon, Chris Holmes, and Stephen Walker revisit the weighted likelihood bootstrap method to generate samples from an approximate Bayesian posterior of a parametric model. They demonstrate that this method can also be applied to a Bayesian nonparametric model without approximation. The parameter of interest is defined as minimizing an expected negative log-likelihood under an unknown sampling distribution in this framework. This reinterpretation allows for the extension of the weighted likelihood bootstrap to posterior sampling for parameters that minimize an expected loss, leading to the development of the loss-likelihood bootstrap method. The authors establish a connection between this approach and general Bayesian updating, which involves updating prior belief distributions without constructing a global probability model but requires calibration of two forms of loss function. The loss-likelihood bootstrap is utilized to calibrate the general Bayesian posterior by matching asymptotic Fisher information. The methodology is illustrated through various examples in the paper. Overall, this research contributes to advancing methods for  and parameter estimation in both  and .