Focused Correlation, confirmation, and the jigsaw puzzle of variable evidence

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Schlosshauer, Maximilian; Wheeler, Gregory
preprint description
Focused correlation compares the degree of association within an evidence set to the degree of association in that evidence set given that some hypothesis is true. A difference between the confirmation lent to a hypothesis by one evidence set and the confirmation lent to that hypothesis by another evidence set is robustly tracked by a difference in focused correlations of those evidence sets on that hypothesis, provided that all the individual pieces of evidence are equally, positively relevant to that hypothesis (Wheeler and Scheines 2010). If focused correlation is interpreted as a `coherence' measure, this result is theoretically significant because it establishes conditions under which a difference in coherence entails a difference in confirmation. However, the result is of limited practical benefit because the *equal relevance* condition is a very strong restriction. In this essay we prove tracking results for focused correlation analogous to Wheeler and Scheines's results but for cases involving unequal relevance. Our result is robust as well, and we retain conditions for bidirectional tracking between incremental confirmation measures and focused correlation. The cases in which tracking fail are instructive, suggesting that focused correlation is a stronger measure of evidential support than classical incremental confirmation measures. We illustrate this point with examples, situate our results in relationship to the noted impossibility results concerning Bayesian epistemic coherence measures, and offer an additional reason for distinguishing classical, incremental confirmation from other notions of confirmation.