Technical Summary: One Score, Two Decisions: Selective Prediction on the Rare-Disease Tail

1. Problem Definition

The paper addresses the challenge of selective prediction in rare-disease diagnosis, a task framed as ranking thousands of possible diseases based on a patient's phenotypes (structured clinical findings). The core problem is that diagnostic systems must decide not only which disease to predict but when to endorse that prediction versus deferring to a human expert.

Current practice typically applies a single threshold to the top score (the confidence of the highest-ranked candidate) to make two distinct decisions:

1. Correctness: Is the top-ranked candidate the correct diagnosis?
2. Presence: Does the candidate list contain any correct diagnosis (i.e., is the system capable of answering this case at all)?

The authors argue that using the same score and the same threshold for both decisions is fundamentally flawed, particularly in the "ultra-rare" tail of disease prevalence where diagnostic support is most needed but data is sparse.

2. Methodology and Framework

The paper proposes a three-step framework for selective prediction over ranked outputs:

A. Feasibility Check (The Oracle Ceiling)

Before attempting to calibrate confidence, the authors establish a necessary condition for feasibility. If a predictor has a base accuracy $p$ (Recall@1) and the system answers a fraction $c$ of cases, the maximum achievable selective accuracy is bounded by:
$$\text{Acc}_{\text{sel}} \leq \min\left(1, \frac{p}{c}\right)$$
If the required target accuracy $\tau$ exceeds this bound (i.e., $p < \tau \cdot c$), no amount of confidence recalibration or thresholding can achieve the goal because the correct answers do not exist in the top-ranked predictions to be selected.

B. Score Decomposition and Signal Selection

The authors decompose the scoring function $s(d|x)$ for a candidate $d$ and case $x$ into two components:
$$s(d|x) = b(x) + r(d|x)$$

• $b(x)$: A case-level offset shared by all candidates for a specific patient (e.g., general confidence, query complexity).
• $r(d|x)$: A candidate-specific residual distinguishing the candidates.

They argue that different decisions require different signals:

• For Correctness (Top-1): The top-two margin ($s_1 - s_2$) is superior. It cancels the shared component $b(x)$, isolating the separation between the leader and the runner-up.
• For Presence (Answerability): The top score ($s_1$) is required. The shared component $b(x)$ indicates whether the entire candidate set fits the input (e.g., whether a retriever has relevant documents). Subtracting $b(x)$ via a margin removes this critical information, potentially causing the system to miss cases where an answer exists but is not the top choice.

C. Theoretical Limit (Unidentifiability)

The paper proves that unlabelled scores alone cannot determine whether switching from a top-score gate to a margin gate will improve performance. Two different joint distributions of scores and labels can yield identical unlabelled score distributions and base accuracy, yet result in opposite gains from switching signals. Therefore, the choice of signal requires labelled validation.

3. Experimental Setup

• Data: 10,374 real patient records from the Phenopacket Store, stratified by Orphanet prevalence classes (Ultra-rare: <1/10^6 or 1–9/10^6; Less-rare: $\ge$1/10^5).
• Models:
  • 8 small open-weight LLMs (Qwen, Llama, Mistral, etc.).
  • 5 medical-specialized LLMs.
  • Frontier models (DeepSeek-V4).
  • Phenotype rankers: An information-content (IC) overlap retriever and Exomiser (phenotype-only mode).
• Tasks:
  • Rare-disease diagnosis (Recall@1).
  • SciFact retrieval (passage ranking).
  • Biomedical entity linking (BC5CDR, MedMentions).
• Controls: Strict "leave-source-out" decontamination to prevent curation leakage between patient records and disease profiles.

4. Key Results

A. Feasibility in the Ultra-Rare Tail

• Small LLMs Fail the Feasibility Check: Across 8 small open-weight LLMs, Recall@1 on ultra-rare diseases is at most 4.6%. At 10% coverage, the theoretical ceiling for selective accuracy is $\le 46\%$. Even with perfect confidence ranking, these models cannot reach a 50% selective accuracy target.
• Advanced Models Pass the Check: Frontier models (e.g., DeepSeek-V4-Pro) and specialized medical models achieve higher base rates (15.8%–22.4%), lifting the feasibility ceiling to $\approx 100\%$, making the target theoretically attainable.
• Phenotype Rankers Outperform: A dedicated phenotype-overlap retriever achieves 25.6% Recall@1 on ultra-rare cases, significantly outperforming LLMs.

B. Margin vs. Top Score

• Correctness: On phenotype rankers (Exomiser and the IC-retriever), the top-two margin significantly outperforms the raw top score for identifying correct top-1 predictions.
  • Exomiser: Margin selects 10% of cases at 29.0% accuracy vs. 13.3% base rate; the top score provides no usable operating point due to score saturation/ties.
  • IC-Retriever: Margin achieves 81.0% accuracy at 10% coverage, compared to 45.8% for the raw score.
• Presence (Answerability): The margin performs poorly for detecting whether any correct candidate exists.
  • In entity linking (BC5CDR) and SciFact retrieval, the margin is often the worst statistic for detecting "NIL" (no valid candidate) cases because it discards the case-level signal $b(x)$.
  • The raw score (or a case-side estimate like query IDF mass) is required to detect presence.

C. Mechanism Validation

• Shared Component Removal: Experiments subtracting an estimate of the case-level offset $b(x)$ from the retriever's scores confirmed that the raw score's poor performance is driven by the shared component. Removing it lifted the raw score's accuracy to match the margin's level.
• Unidentifiability: The paper confirms that without labelled data, one cannot predict from the score distribution alone whether a margin or raw score will be better for a specific system.

5. Significance and Claims

The paper makes three primary claims:

1. Feasibility is a Prerequisite: Selective prediction cannot rescue a ranker that rarely places the correct answer first. For ultra-rare diseases, many current LLMs fail the feasibility check entirely; no confidence calibration can fix a lack of correct top-ranked predictions.
2. Signal-Decision Mismatch: For fixed candidate sets with unnormalized scores, the top-two margin is the correct signal for correctness (is the leader right?), while the raw score (or case-level signal) is required for presence (is there an answer?). Using the margin for presence discards necessary information.
3. No Free Lunch in Gate Selection: Unlabelled scores cannot determine the optimal gating strategy. The gain from switching signals is unidentifiable from the score distribution alone; labelled validation is strictly required to choose between a margin-based or score-based gate.

Clinical Implication: The authors propose a "retriever-first" triage approach for rare diseases. A phenotype-based retriever should be used to filter cases, and its margin should be used to gate the most confident predictions. However, they emphasize that this is a triage aid, not an autonomous decision-maker, as even the best gates on the ultra-rare tail operate with significant error rates (e.g., ~20% error at 10% coverage) and require human oversight.

Limitations Acknowledged:

• The absolute accuracy of the retriever is partially identified due to potential curation leakage, though the relative lift of the margin over the base rate is robust.
• The gate is biased toward patients with more recorded phenotypic terms, potentially excluding patients at first presentation (equity concern).
• Thresholds are retrospective estimates and not clinically certified guarantees for new patients.