General Bayesian updating and the loss-likelihood bootstrap

By Simon Lyddon, Christopher C. Holmes, Stephen G. Walker. Biometrika, 2017

155 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/839872faaa71c532a06ced4a56f2c15262145773

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Christopher C. Holmes
2. Edwin Fong
3. Stephen G. Walker
4. Catia Scricciolo
5. Yun Yang
6. Martyn Plummer
7. Joshua S. Speagle
8. Ryan Martin

In Statistics and Probability

1. Christopher C. Holmes
2. Stephen G. Walker
3. Edwin Fong
4. Catia Scricciolo
5. Ryan Martin
6. Yun Yang
7. Martyn Plummer
8. Joshua S. Speagle

Citations in the corpus

• Martingale posterior distributions (2021, background): “While misspecification under our framework is still an open question, the Bayesian bootstrap has particularly desirable theoretical and practical properties under misspecification (Lyddon et al., 2018, 2019; Fong et al., 2019).”
• Robust generalised Bayesian inference for intractable likelihoods (2021, unknown): “Here we aim to pick 𝛽 such that the scale of the asymptotic precision matrix of the generalised posterior ( H ∗ ; Theorem 2) matches that of the minimum KSD point estimator ( H ∗ J − 1 ∗ H ∗ ; Lemma 4), an approach proposed in Lyddon et al. (2019).”