Estimate Murphy decomposition without explicit recalibration

Develop a method to estimate the resolution and miscalibration components of Murphy’s Bregman-score decomposition when the recalibrated unit premium m_Q(Π) cannot be computed explicitly from the population model.

Background

The manuscript’s Murphy decomposition separates the Bregman score of a unit premium rule into resolution and miscalibration terms through the exposure-weighted recalibrated premium m_Q(Π)=E_Q[Y|Π]. In the synthetic example, this recalibrated premium is available analytically, allowing the author to compute the miscalibration component directly.

In real-world applications, the population distribution is unknown and m_Q(Π) must be estimated. The paper later considers isotonic-regression estimates and two finite-sample estimators of miscalibration, but the author identifies the general problem of obtaining a reliable decomposition when recalibration is not explicitly available as unresolved.

References

The previous example essentially benefits from the fact that we can explicitly compute the recalibrated unit premium $m_{}(\Pi)$. Only this allows us to obtain Murphy's decomposition of the Bregman score into the resolution term and the miscalibration term, in particular, this allows us to compute miscalibration estimation 00. What can we do in a real-world situation where the recalibrated unit premium cannot be computed explicitly?

miscalibration estimation 00:

MCB^φ(Π)=S^φ(Y,m(Π))−S^φ(Y,Π).\widehat{\mathrm{MCB}}_\varphi(\Pi)= \widehat{S}_\varphi(Y, m_{}(\Pi))- \widehat{S}_\varphi(Y, \Pi).

— A Practical Guide on Graphical Model Validation  (2609.26445 - Wüthrich, 22 Sep 2026) in Section 6, immediately after Example “Gamma deviance scoring” and before subsection “Graphical illustration of the Bregman score”

In our numerical example MCB numbers, the true value is in between the two estimates, however, we do not know whether this holds more generally or only in this example.

— A Practical Guide on Graphical Model Validation  (2609.26445 - Wüthrich, 22 Sep 2026) in Section 6, subsection “Sample computation of resolution and miscalibration”