Bayes linear methods

Some kind of approximate Bayes thing

August 17, 2022 — March 28, 2024

Hilbert space
how do science
linear algebra
Monte Carlo
Figure 1

Some kind of approximate Bayes method. Utility unclear. Mainstream status: marginal.

From a brief glance, it seems that by assuming “linear beliefs” in some sense, we can construct all necessary posterior updates in terms of covariance matrices and means, without actually stipulating that the prior or likelihood (or anything) is Gaussian.

The results look suspiciously like the standard Gaussian posterior updates, in that the resulting estimator is frequently fancy least squares optimisation and lots of the same machinery is recovered, e.g. Matheron updates can be justified in this framework.

I suspect that I can find mainstream acceptance by simply making explicit Gaussian approximations, and thence avoiding controversy about this slightly esoteric option. But introducing fewer assumptions is always nice?

The best summary is presumably the textbook (Goldstein and Wooff 2007).

1 References

Chen, and Oliver. 2013. Levenberg–Marquardt Forms of the Iterative Ensemble Smoother for Efficient History Matching and Uncertainty Quantification.” Computational Geosciences.
Goldstein, and Wooff. 1995. Bayes Linear Computation: Concepts, Implementation and Programs.” Statistics and Computing.
Goldstein, and Wooff. 2007. Bayes Linear Statistics: Theory and Methods. Wiley Series in Probability and Statistics.
Oliver. 2022. Hybrid Iterative Ensemble Smoother for History Matching of Hierarchical Models.” Mathematical Geosciences.
White, Hunt, Fienen, et al. 2020. Approaches to Highly Parameterized Inversion: PEST++ Version 5, a Software Suite for Parameter Estimation, Uncertainty Analysis, Management Optimization and Sensitivity Analysis.” USGS Numbered Series 7-C26. Techniques and Methods.
Wilkinson. 1995. Bayes Linear Covariance Matrix Adjustment.”
Wooff, D. A. 1995. Bayes Linear Methods II-An Example with an Introduction to [B/D].” Bayes Linear Methodology,” Unpublished Draft.
Wooff, David, and Goldstein. 2000. The Bayes Linear Programming Language [B/D].” Journal of Statistical Software.