A Community Monte Carlo Approach for Quantifying Subjectivity-Driven Ensemble Uncertainty in Inverse Problems

Solving inverse problems allows for the estimation of system properties or model parameters that cannot be measured directly. However, the models arising from various experts differ more than their individual uncertainty estimates might suggest [1]. A crucial reason for this is the combined impact of many small subjective choices undertaken during the inversion procedure. This includes a) the selected subset of data, b) the model space parametrisation, c) the type of forward model, d) the chosen numerical and optimisation methods, and e) regularisation.

In this work, we present an adapted Bayesian inference method that explicitly incorporates these subjective choices – collectively referred to as the control parameters – as a random variable to obtain estimates of ensemble statistics. It can be shown that, while the ensemble model m^e may be computed simply by averaging over N individual models m_i (i = 1, 2, ..., N), the ensemble covariance C^e consists of a sum of two terms,

The first term represents the mean of individual posterior covariances C_i, and the second term represents the variance of the mean models.

The theoretical developments are illustrated with a novel small-scale "Community Monte Carlo" experiment, where a group of experts was asked to select suitable regularisation (tuning) parameters to obtain a solution to a linear straight-ray tomography problem. The regularisation parameters include, e.g., the data prior standard deviation σ_D and the model prior standard deviation σ_M.

Crucially, the computation of ensemble estimates reveals that the average of individual covariances – the first term in eq. (1) – dominates the ensemble covariance. This is due to individuals favouring smaller values of σ_M, resulting in similar-looking models that deviate minimally from the prior, and larger data errors σ_D, leading to comparatively large posterior covariance matrices tending towards the prior model covariance.

Our proof-of-concept suggests that the field of seismic tomography should not strive for consensus among models, which risks condensing the ensemble and producing overly optimistic uncertainty estimates. Instead, the diversity of expert-derived models can be seen as an opportunity for "Community Monte Carlo," emphasising the need to actively explore a broader range of plausible subjective choices and rigorously quantify their effect on model uncertainty.

References:

[1] Andreas Fichtner, Jeroen Ritsema, Solvi Thrastarson; A high-resolution discourse on seismic tomography. Proc. A 1 August 2025; 481 (2320): 20240955. https://doi.org/10.1098/rspa.2024.0955