Projection / capability. Update belief from evidence, quantify information, combine uncertain prediction/measurement, and keep knowledge separate from authority.
Question this mathematics answers. What should this observation change about our belief or estimate—and what does it not authorize?
Equation family
Bayes: P(H|E)=P(E|H)P(H)/P(E)Odds form: O(H|E)=O(H)×LR, LR=P(E|H₁)/P(E|H₀)Self-information: I(E)=−ln P(E)Shannon entropy: H=−Σ pᵢ ln pᵢScalar Kalman: K=Σ⁻/(Σ⁻+R); x⁺=x⁻+K(z−x⁻)
Symbols. H = hypothesis (or entropy by context—notation must be local); E = evidence; LR = likelihood ratio; pᵢ = probability; x⁻/x⁺ = prior/posterior estimate; Σ⁻ = prior variance; z = measurement; R = measurement-noise variance.
Assumptions. Bayes requires a specified hypothesis/event model. Kalman equations shown are the scalar linear-Gaussian/direct-observation case. Information in nats uses natural logarithms.
Where the inputs come from. Priors may come from prevalence/history; likelihoods from validated sensor/test performance; z from measurement; variances from calibration/model uncertainty. These are inputs with provenance, not aesthetic slider values.
Short derivation / justification
Bayes: P(H,E)=P(E|H)P(H)=P(H|E)P(E). Rearranging yields the rule. Kalman intuition: K approaches 1 when prior uncertainty dominates measurement noise and approaches 0 when measurement noise dominates.
Worked example
Prior odds 1:4 (0.25) and LR=6 give posterior odds 1.5:1. The evidence reverses the balance, but it does not grant permission to act.
Cadet or real-world application. Diagnostics, sensor fusion, fault isolation, scientific inference, screening tests.
What to look for. Observation ≠ measurement ≠ model ≠ inference. Ask which component failed before trusting a precise posterior.
Common misinterpretation. “A larger posterior or lower variance makes the decision authorized.”
Break / domain boundary. Epistemic confidence and institutional/ethical authority are different state variables.