Budget-Sensitive Discovery Scoring: A Formally Verified Framework for Evaluating AI-Guided Scientific Selection

Abhinaba Basu; Pavan Chakraborty

cs.LGMar 12, 2026

Budget-Sensitive Discovery Scoring: A Formally Verified Framework for Evaluating AI-Guided Scientific Selection

Authors: Abhinaba Basu, Pavan Chakraborty

Organizations: Indian Institute of Information Technology Allahabad (IIITA) and National Institute of Electronics & Information Technology (NIELIT). · Indian Institute of Information Technology Allahabad (IIITA).

Abstract

Scientific discovery increasingly relies on AI systems to select candidates for expensive experimental validation, yet no principled, budget-aware evaluation framework exists for comparing selection strategies -- a gap intensified by large language models (LLMs), which generate plausible scientific proposals without reliable downstream evaluation. We introduce the Budget-Sensitive Discovery Score (BSDS), a formally verified metric -- 20 theorems machine-checked by the Lean 4 proof assistant -- that jointly penalizes false discoveries (lambda-weighted FDR) and excessive abstention (gamma-weighted coverage gap) at each budget level. Its budget-averaged form, the Discovery Quality Score (DQS), provides a single summary statistic that no proposer can inflate by performing well at a cherry-picked budget. As a case study, we apply BSDS/DQS to: do LLMs add marginal value to an existing ML pipeline for drug discovery candidate selection? We evaluate 39 proposers -- 11 mechanistic variants, 14 zero-shot LLM configurations, and 14 few-shot LLM configurations -- using SMILES representations on MoleculeNet HIV (41,127 compounds, 3.5% active, 1,000 bootstrap replicates) under both random and scaffold splits. Three findings emerge. First, the simple RF-based Greedy-ML proposer achieves the best DQS (-0.046), outperforming all MLP variants and LLM configurations. Second, no LLM surpasses the Greedy-ML baseline under zero-shot or few-shot evaluation on HIV or Tox21, establishing that LLMs provide no marginal value over an existing trained classifier. Third, the proposer hierarchy generalizes across five MoleculeNet benchmarks spanning 0.18%-46.2% prevalence, a non-drug AV safety domain, and a 9x7 grid of penalty parameters (tau >= 0.636, mean tau = 0.863). The framework applies to any setting where candidates are selected under budget constraints and asymmetric error costs.

Figures & tables

Figure 1: Evaluation pipeline for 39 proposers across five categories: baselines (2), mechanistic ablations (5), direct optimization (BSDS-Recursive), ablation controls (3), and real LLMs (28). All proposers are evaluated by the formally verified

\mathrm{BSDS}

\mathrm{DQS}

metric with 1,000 bootstrap replicates. Multi-round proposers receive evaluation feedback (dashed arrow).

Mechanistic Ablations
Proposer	ML	Structure	Stochastic	Multi-round	Coverage
Random	No	No	Yes	No	Full
Greedy-ML	Yes	No	No	No	Full
Informed-Prior	Yes	Yes	No	No	Full
Retrieval	Yes	Yes	No	No	Full
Generative	Yes	No	Yes	No	Full

Table 1: Summary of the 39 proposer strategies. “ML” indicates use of pre-trained RF scores; “Structure” indicates use of fingerprints or descriptors; “Stochastic” indicates non-deterministic selection; “Multi-round” indicates iterative refinement with label feedback.

Proposer	$B/N=0.01$	$B/N=0.05$	$B/N=0.20$	$\mathrm{DQS}_{\mathrm{boot}}$ [95% CI]
Random	$-0.953$	$-0.895$	$-0.782$	$-0.819$ $[-0.834,-0.803]$
Greedy-ML	$\mathbf{-0.003}$	$\mathbf{-0.010}$	$\mathbf{-0.121}$	$\mathbf{-0.046}$ $[-0.076,-0.019]$
Informed-Prior	$-0.297$	$-0.232$	$-0.303$	$-0.178$ $[-0.210,-0.146]$
Retrieval	$-0.056$	$-0.033$	$-0.138$	$-0.091$ $[-0.121,-0.062]$
Generative	$-0.906$	$-0.822$	$-0.548$	$-0.670$ $[-0.691,-0.649]$
BSDS-Guided	$-0.236$	$-0.184$	$-0.248$	$-0.234$ $[-0.264,-0.202]$

Table 2: Per-budget

\mathrm{BSDS}

at seed 0 (point estimates) and bootstrap-averaged

\mathrm{DQS}

with 95% BCa confidence intervals (1,000 seeds). Best value per column in bold.

Variant	$\mathrm{DQS}$	$\mathrm{BSDS}_{0.05}$	What is ablated
BSDS-Recursive (full)	$-0.323$	$-0.298$	— (reference)
BSDS-NoAug	$-0.296$	$-0.269$	No recursive feature augmentation
BSDS-1Round	$-0.258$	$-0.223$	Single optimization round only
Greedy-MLP-NN	$-0.315$	$-0.281$	BCE loss (standard ML)
Greedy-ML (RF)	$\mathbf{-0.046}$	$\mathbf{-0.010}$	RF baseline

Table 3: Ablation analysis: isolating the contribution of each BSDS-Recursive component. All models share the same 5-fold CV protocol and 10-dimensional feature set.

Figure 2:

\mathrm{BSDS}

as a function of budget fraction

B/N

for all 11 mechanistic proposers (2 baselines, 5 ablations, BSDS-Recursive, and 3 ablation controls), with 95% bootstrap confidence bands. Greedy-ML dominates all proposers across budget levels.

Figure 3: Bootstrap distributions of

\mathrm{DQS}

for all mechanistic proposers (1,000 replicates). Greedy-ML has the lowest variance; Retrieval achieves the most favorable individual budget-level scores.

Proposer	$\mathrm{HR}$	$\mathrm{FDR}$	$\mathrm{PPV}$	TP	$\mathrm{BSDS}$
Random	0.062	0.957	0.043	89	$-0.895$
Greedy-ML	0.581	0.592	0.408	839	$\mathbf{-0.010}$
Informed-Prior	0.451	0.683	0.317	651	$-0.232$
Retrieval	0.568	0.601	0.399	820	$-0.033$
Generative	0.105	0.927	0.073	151	$-0.822$
BSDS-Guided	0.521	0.634	0.366	752	$-0.184$

Table 4: Component metrics at

B/N=0.05

(seed 0).

\mathrm{BSDS}=\mathrm{HR}-\lambda\,\mathrm{FDR}-\gamma\,(1-\mathrm{Cov})

, with

\lambda=1.0

\gamma=0.3

\mathrm{PPV}

= positive predictive value; TP = true positive count.

Model	Mode	$\mathrm{DQS}$	$\mathrm{BSDS}_{0.05}$
ChatGPT-5.2	Direct	$-0.683$	$-0.788$
ChatGPT-5.2	Rerank	$-0.159$	$-0.165$
Claude	Direct	$-0.678$	$-0.813$
Claude	Rerank	$-0.161$	$-0.170$
Gemini	Direct	$-0.585$	$-0.668$
Gemini	Rerank	— †	— †

Table 5: LLM proposer performance: budget-averaged

\mathrm{DQS}

(1,000 bootstrap seeds) and

\mathrm{BSDS}

B/N=0.05

. Direct = zero-shot LLM ranking; Rerank = LLM reranking of Greedy-ML shortlist. Best LLM value per column in bold.

Model	Mode	$\mathrm{DQS}$	$\mathrm{BSDS}_{0.05}$
ChatGPT-5.2	FewDirect	$-0.624$	$-0.720$
ChatGPT-5.2	FewRerank	$-0.138$	$-0.160$
Claude	FewDirect	$-0.547$	$-0.656$
Claude	FewRerank	$-0.141$	$-0.132$
Gemini	FewDirect	— †	— †
Gemini	FewRerank	— †	— †

Table 6: Few-shot LLM performance (

k{=}3

active +

k{=}3

inactive examples). FewDirect = few-shot without ML scores; FewRerank = few-shot with RF scores. Values in italics indicate improvement over the corresponding zero-shot mode (cf. Table 5 ). Best few-shot value per column in bold.

Model	Mode	HIV $\mathrm{DQS}$	Tox21 $\mathrm{DQS}$
DeepSeek-V3.1	Direct	$-0.774$	$-0.401$
	Rerank	$-0.318$	$-0.071$
	FewDirect	$-0.747$	$-0.383$
	FewRerank	$-0.236$	$-0.065$
Llama-4-Maverick	Direct	$-0.802$	$-0.363$
	Rerank	— †	$-0.042$

Table 7: LLM performance on Tox21 NR-AR-LBD (3.5% prevalence). Same protocol as HIV (Tables 5 – 6 ). Greedy-ML DQS on Tox21

=+0.086

. No LLM surpasses the RF baseline.

Model	$\mathrm{DQS}$	$\mathrm{BSDS}_{0.05}$	AUROC	EF@1%
Greedy-ML (ECFP4 + RF)	$-0.046$	$-0.010$	$0.854$	$22.1\times$
Greedy-MolFormer	$-0.150$	$-0.178$	$0.806$	$20.5\times$
Greedy-ChemBERTa	$-0.184$	$-0.200$	$0.785$	$19.6\times$
BSDS-Recursive (ECFP4)	$-0.323$	$-0.298$	$0.776$	$22.1\times$

Table 8: Foundation model baselines: frozen pretrained [CLS] embeddings + RF with greedy (top-

B

) selection. Same 5-fold CV protocol. Seed-0 (full dataset) values.

Figure 4:

\mathrm{DQS}

as a function of softmax temperature

\tau

for the Generative proposer. Performance degrades monotonically with increasing

\tau

; even the optimal

\tau^{*}=0.1

yields negative

\mathrm{DQS}

Figure 5: NL framing invariance. Dots show mean $\mathrm{DQS}$ across six requirement phrasings; horizontal bars show the min–max spread. All proposers except Informed-Prior exhibit zero spread ( $\Delta<0.01$ ), confirming that $\mathrm{DQS}$ is invariant to NL framing.

Proposer	$(0.1,0.1)$	$(1.0,0.3)$	$(2.0,0.5)$	$(5.0,1.0)$
Greedy-ML	$\mathbf{+0.515}$	$\mathbf{-0.046}$	$\mathbf{-0.670}$	$\mathbf{-2.540}$
Retrieval	$+0.484$	$-0.091$	$-0.729$	$-2.644$
Ensemble	$+0.396$	$-0.102$	$-0.649$	$-2.270$
Informed-Prior	$+0.429$	$-0.178$	$-0.851$	$-2.873$
BSDS-Guided	$+0.415$	$-0.234$	$-0.950$	$-3.077$

Table 9: DQS for the top-5 proposers at four representative

(\lambda,\gamma)

pairs spanning the sensitivity grid. Greedy-ML maintains the top position across all settings.

Figure 6: Parameter sensitivity analysis across a $9\times 7$ grid of $(\lambda,\gamma)$ values. (a) DQS of BSDS-Recursive: absolute performance varies with parameters (positive at low penalties, negative at high), but the star marks the default $(1.0,0.3)$ . (b) Kendall $\tau$ of the full proposer ranking vs. the default ranking: $\tau\geq 0.636$ across all 63 parameter pairs (mean $\tau=0.863$ ), confirming that the proposer hierarchy is robust to parameter choice.

Proposer	HIV	Tox21	ClinTox	MUV-466	SIDER-Ear
	3.5%	3.5%	7.5%	0.18%	46.2%
Random	$-0.819$	$-0.817$	$-0.776$	$-0.849$	$-0.390$
Greedy-ML	$\mathbf{-0.046}$	$+0.086$	$\mathbf{-0.278}$	$\mathbf{-0.628}$	$\mathbf{+0.019}$
Informed-Prior	$-0.178$	$-0.046$	$-0.428$	$-0.703$	$-0.097$
Retrieval	$-0.091$	$+0.058$	$-0.389$	$-0.698$	$-0.083$
Generative	$-0.670$	$-0.624$	$-0.701$	$-0.842$	$-0.330$

Table 10: Cross-dataset DQS for all 11 mechanistic proposers across five MoleculeNet benchmarks spanning 0.18%–46.2% prevalence. Greedy-ML consistently ranks among the top proposers across prevalence regimes. All MLP variants perform worse than Greedy-ML on every dataset.

Figure 7: Cross-dataset DQS comparison for 11 mechanistic proposers across five MoleculeNet benchmarks and AV Safety (non-drug domain). Error bars show 95% bootstrap CIs. Greedy-ML consistently ranks among the top proposers across prevalence regimes from 0.18% to 46.2%. Proposers unavailable for AV Safety (requiring molecular features) are omitted. Horizontal line at DQS

=0

Proposer	AV Safety DQS
Random	$-0.819$
Greedy-ML	$+0.179$
Generative	$+0.018$
BSDS-Recursive	$+0.145$
BSDS-NoAug	$+0.101$
BSDS-1Round	$+0.192$

Table 11: DQS for 7 generic proposers on AV Safety triage (30,000 scenarios, 3.41% safety-critical). Greedy-MLP-NN achieves the highest DQS, diverging from the drug-discovery hierarchy where Greedy-ML leads.

Strategy	$B$	Hits	Hit Rate	Cost ($K)	ROI
Random	50	2	3.4%	250	$-$ 66%
RF (Greedy-ML)	50	48	96.0%	250	860%
BSDS-Recursive	50	31	62.0%	250	520%
Greedy-MLP-NN	50	39	78.0%	250	680%
Random	200	7	3.5%	1000	$-$ 65%
RF (Greedy-ML)	200	172	86.0%	1000	760%

Table 12: Deployment simulation on MoleculeNet HIV: hits, hit rate, and return on investment (ROI) for four ranking strategies across experimental budgets

B

Proposer	Random DQS	Scaffold DQS
Greedy-ML	$\mathbf{-0.046}$	$\mathbf{-0.116}$
Retrieval	$-0.091$	$-0.163$
Ensemble	$-0.102$	$-0.182$
Informed-Prior	$-0.178$	$-0.260$
BSDS-Guided	$-0.234$	$-0.310$
BSDS-1Round	$-0.258$	$-0.321$

Table 13: Random vs. scaffold split comparison on MoleculeNet HIV. DQS values are budget-averaged over 1,000 bootstrap replicates. Greedy-ML remains the best proposer under both split types.

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