Information Threshold, Bayesian Inference and Decision-Making
This work addresses decision-making under uncertainty, offering a benchmark for confidence in binary classification, though it appears incremental as it builds on existing Bayesian frameworks.
The paper tackles the problem of determining when additional information ceases to significantly improve Bayesian classification reliability by defining an 'information threshold' as the point of maximum curvature in the prior-posterior curve, and reports a mathematical relationship where at this threshold, the prior and posterior sum to 1.
We define the information threshold as the point of maximum curvature in the prior vs. posterior Bayesian curve, both of which are described as a function of the true positive and negative rates of the classification system in question. The nature of the threshold is such that for sufficiently adequate binary classification systems, retrieving excess information beyond the threshold does not significantly alter the reliability of our classification assessment. We hereby introduce the "marital status thought experiment" to illustrate this idea and report a previously undefined mathematical relationship between the Bayesian prior and posterior, which may have significant philosophical and epistemological implications in decision theory. Where the prior probability is a scalar between 0 and 1 given by $φ$ and the posterior is a scalar between 0 and 1 given by $ρ$, then at the information threshold, $φ_e$: $φ_e + ρ_e = 1$ Otherwise stated, given some degree of prior belief, we may assert its persuasiveness when sufficient quality evidence yields a posterior so that their combined sum equals 1. Retrieving further evidence beyond this point does not significantly improve the posterior probability, and may serve as a benchmark for confidence in decision-making.