Methodology & Statistical Rigor

Risk Bridge formalizes risk prediction transport as a constrained optimization problem. This section details the underlying statistical theory, likelihood derivations, solver ladder mechanics, and empirical calibration metrics.

Topics Covered

• Constrained MLE Formulation: Mathematical derivation of the joint likelihood \(L(Y, Z \mid X)\), the truncated-lognormal distribution for \(Z\), and the reference calibration moment constraints.
• Solver Ladder & Optimization: Algorithmic strategy transitioning from unconstrained warm starts (BFGS) to interior-point constrained optimization (trust-constr) and Sequential Least Squares Programming (SLSQP).
• Calibration Metrics & Validation: Precise mathematical definitions of Calibration-in-the-Large (CITL), calibration slope, observed/expected ratio, and Brier score.