Bayesian Knowledge Infusion for Studying Historical Sunspot Numbers

A scientific method that proposes a value Y to estimate a target value ρ is often subject to some level of uncertainty. In the Bayesian framework, the level of uncertainty can be measured by the width of the 68% interval, which is the range of the middle 68% of the ranked ρ values sampled from the posterior distribution p(ρ|Y). This paper considers Bayesian knowledge infusion (BKI) to reduce the uncertainty of the posterior distribution p(ρ|Y) based on additional knowledge that an event A happens. BKI is achieved by using a conditional prior distribution p(ρ|A) in the Bayes theorem, assuming that given the true ρ, its error-contaminated value Y is independent of event A. We use two examples to illustrate how to study whether or not it is possible to reduce uncertainty from 14C reconstruction (Y) of the annual sunspot number (SSN) (ρ) by infusing additional information (A) using BKI. Information (A) that SSN is from a year that has a Far Eastern record of naked eye sunspots is found to be not so effective in reducing the uncertainty. In contrast, information that SSN is from a year at a cycle minimum is found to be very effective, producing much narrower 68% intervals. The resulting Bayesian point estimates of SSN (the posterior medians of ρ) are cross-validated and tested on a subset of telescopically observed SSNs that were unused in the process of Bayes computation.