:PROPERTIES:
:ID: 0543ac39-af9b-40fd-9973-e60576a20695
:END:

#+title: thesis/method
#+filetags: :project: :knowledge: :method:

:PROPERTIES:
:ID: thesis-method
:END:

#+title: thesis/method
#+filetags: :project: :knowledge: :method:

Chapter 3 of the BGU MSc thesis on ROLL. Core contribution chapter. Currently ~30 pages.
Parent: [[id:6294e2be-6189-4473-b363-a1dd9a75fb9b][thesis]]

** Guidelines

- sec:problem-formulation is now notation-only — motivation deferred to Ch. 2
  via \Cref{sec:imbalanced-tpr-fpr,sec:neyman-pearson}
- sec:imbalanced-tpr-fpr and sec:neyman-pearson were removed from this chapter and live in Ch. 2
- Both objectives (TPR@FPR and FPR@TPR) are equivalent by negating scores and swapping labels;
  all derivations given for TPR@FPR only — FPR@TPR follows by transformation

** Section Structure

| Section | Label | Status |
|---------|-------|--------|
| Problem Formulation (notation only) | sec:problem-formulation | Done |
| The ROLL Framework | sec:roll-framework | Written |
| Differentiability Problem | sec:differentiability-problem | Written |
| Score Distribution Fitting | sec:score-distribution-fitting | Written |
| General ROLL Formulation and Derivation | sec:roll-formulation | Written — core theoretical contribution |
| Gaussian ROLL (forward + gradient) | sec:roll-gaussian | Written |
| Beta ROLL (forward + gradient) | sec:roll-beta | Stub — TBD |
| KDE ROLL (forward + gradient) | sec:roll-kde | Written |
| Properties | sec:roll-properties | Written — gradient balance + locality |
| Custom Backward Pass | sec:roll-backward | Stub — TBD |
| Numerical Stability | sec:roll-numerical-stability | Stub — TBD |
| Bandwidth Estimation (ISJ + scheduling) | sec:kde-bandwidth | Written |

** Gotchas

- KDE backward pass uses inverse-function-theorem approach deliberately; implicit differentiation
  was considered and rejected — user needs to explain the derivation in their own words
- Correct gradient of threshold w.r.t. negative-class score:
  σ'(τ − x_i) / Σ_j σ'(τ − x_j) where sum is over ALL of B_0 — sympy-verified
- Beta ROLL: translation-invariance does NOT hold when scores pass through sigmoid before fitting;
  gradient-balance property breaks for Beta — documented in sec:roll-properties
- Gradient-balance proof uses a uniform-shift argument; requires translation-invariant CDF estimator