A Mathematical Treatise on the Non-identifiability of Initial Metric Improvements and Overall Success Probabilities in Drug Discovery Processes

GhostDrift Mathematical Research Institute

Abstract

This treatise formalizes the drug discovery process as a stochastic process equipped with filtration on a probability space. We rigorously establish that the conventional axiom—that enhancing early-stage metrics (e.g., spectral characteristics, binding enthalpies) through quantum computing or advanced heuristics logically implies a commensurate increase in overall success probability—is fundamentally non-deducible without the articulation of explicit latent conditions. Transcending mere refutation, this paper delineates a "Verifiable Audit Protocol" synthesized from the proof logic. This protocol establishes the necessary criteria (PASS/FAIL) for legitimizing technological claims of "drug discovery success" versus mere "local KPI optimization," substantiated by non-identifiability theorems and the formal implications of Goodhart's Law.

Introduction and Formal Definitions

Enhancements in early-stage metrics, such as binding affinities facilitated by quantum algorithms or AI-driven modeling, do not inherently constitute a guarantee for downstream drug discovery success. We elucidate this disconnect through formal measure theory and define the structural requirements for technological validation. We formalize the discovery process as a sequence of discrete serial filtrations and establish the following mathematical foundation:

Discovery Process and Filtration Structure

Given a probability space $(\Omega, \mathcal{F}, P)$, let $A_i \in \mathcal{F}$ denote the success event at each stage $i=1, \dots, n$ of the discovery pipeline. The $\sigma$-algebra (filtration) representing the information set available at the $k$-th stage ($1 \le k < n$) is defined as: \[ \mathcal{F}_k := \sigma(A_1, \dots, A_k) \subset \mathcal{F} \] The cumulative success event $S$ is defined as the joint intersection of all sequential stage-wise success events: \[ S := \bigcap_{i=1}^n A_i \]

Henceforth, the initial-phase probability space up to stage $k$ is denoted as $(\Omega_k,\mathcal{F}_k,P_k)$, where $\Omega_k:=\Omega$ and $P_k$ represents the restriction of the measure $P$ to the sub-$\sigma$-algebra $\mathcal{F}_k$. Metrics observed during the initial phase (KPIs, affinity scores, etc.) are formalized as $\mathcal{F}_k$-measurable random variables $M: \Omega \to \mathbb{R}$.

Taxonomy of "Improvement"

Within this framework, "improvement" is categorized into two distinct modalities based on the analytical context:

(Fixity of the Initial Measure): Maintaining the joint distribution on $\mathcal{F}_k$ (specifically $M$ and $A_1,\dots,A_k$) as invariant. This is utilized in Theorem 1 to demonstrate that the specification of initial information fails to constrain subsequent success probabilities.
(Optimization of Metric Expectation): A transformation that increases the expected value $\mathbb{E}_\mu[M]$ of metric $M$ under a fixed population distribution $\mu$. This serves the analysis of selection bias and "Winner's Curse" in Section 3.

Main Theorem: Non-identifiability via Measure Extension

We prove that the proposition "an improvement in the initial metric $M$ necessitates an improvement in the overall success probability $P(S)$" is logically invalid. The following theorem demonstrates that overall success probability remains an unconstrained parameter that can be manipulated arbitrarily while preserving the complete stochastic profile of the initial phase.

Non-identifiability through Measure Extension

Let $(\Omega_k, \mathcal{F}_k, P_k)$ be the probability space representing the first $k$ stages, and let $M$ be an $\mathcal{F}_k$-measurable variable. For any parameter $q \in [0, 1]$, there exists an extended probability space $(\widetilde{\Omega}, \widetilde{\mathcal{F}}, \widetilde{P})$, a measurable projection $\pi:\widetilde{\Omega}\to\Omega_k$, and a sequence of downstream events $A_{k+1},\dots,A_n\in\widetilde{\mathcal{F}}$ satisfying the following:

(Measure Preservation) For any $E\in\mathcal{F}_k$, let $\widehat{E}:=\pi^{-1}(E)$. Then $\widetilde{P}(\widehat{E})=P_k(E)$, and the distribution of the pulled-back metric $\widehat{M}:=M\circ\pi$ is identical to $M$.
(Independence of Success Probability) The global success event $\widetilde{S}:=\bigcap_{i=1}^n A_i$ satisfies: \[ \widetilde{P}(\widetilde{S}) = q \cdot P_k\left(\bigcap_{i=1}^k A_i\right) \]

Proof. The existence is established through the explicit construction of product measures. Consider an auxiliary probability space $(\Omega',\mathcal{F}',P') := (\{0,1\}, 2^{\{0,1\}}, \nu_q)$, where $\nu_q(\{1\})=q$. We define the extended space as: \[ (\widetilde{\Omega}, \widetilde{\mathcal{F}}, \widetilde{P}) := (\Omega_k \times \{0,1\},\ \mathcal{F}_k \otimes 2^{\{0,1\}},\ P_k \otimes \nu_q) \] Setting the projection $\pi(\omega,b):=\omega$ and the downstream events $A_{k+1}=\cdots=A_n := \Omega_k\times\{1\}$, the global success event is $\widetilde{S} = \bigl(\cap_{i=1}^k A_i\bigr)\times\{1\}$. Consequently: \[ \widetilde{P}(\widetilde{S}) = P_k\!\left(\bigcap_{i=1}^k A_i\right) \cdot \nu_q(\{1\}) = q \cdot P_k\!\left(\bigcap_{i=1}^k A_i\right) \] This construction proves that fixing the initial stochastic information fails to determine the success rate of subsequent phases. □

The Selection Paradox: Mathematical Goodhart's Law

We now examine the dynamic selection process where candidates are prioritized based on metric $M$. We rigorously demonstrate a paradoxical regime where metric "optimization" inherently degrades the terminal success probability.

Expectation Inversion and Selection Bias

Define the selection rule as $\theta^\star = \arg\max_{\theta\in\Theta} m(\theta)$. There exist non-trivial configurations where the true success rate $s(\theta^\star)$ of the selected candidate monotonically decreases as the aggregate metric expectation increases ($\mathbb{E}_{\mu}[m'] > \mathbb{E}_{\mu}[m]$).

Proof. Let $\Theta=\{G,B\}$ with a prior $\mu(G)=\mu(B)=\tfrac12$, and true success rates $s(G)=1, s(B)=0$.

Baseline: $m(G)=0.9, m(B)=0.8 \implies \theta^\star=G$ ($s=1$). Expected metric is $0.85$.
Metric "Optimization": $m'(G)=0.9, m'(B)=0.95 \implies \theta^\star=B$ ($s=0$). Expected metric is $0.925$.

Thus, maximizing aggregate scores can select for candidates with zero success probability, invalidating the metric as a surrogate for global success. □

Bottleneck Lemma: Stochastic Upper Bounds

Upper Bound via Sequential Bottlenecks

Consider a subset of process indices $T$. If the conditional success probability of each stage is bounded by a bottleneck parameter $\epsilon$ (i.e., $\forall j \in T, P(A_j \mid \cap_{i=1}^{j-1} A_i) \le \epsilon$), then: \[ P(S) \le \epsilon^{|T|} \]

Proof. Applying the chain rule of probability: $P(S) = P(A_1) \prod_{i=2}^n P(A_i \mid \cap_{l=1}^{i-1} A_l)$. By bounding each conditional term with $\epsilon$ or unity, we derive the geometric decay $P(S) \le \epsilon^{|T|}$. □

Conditional Validity: Mathematical Requirements for Success

The preceding theorems refute the *unconditional* guarantee of success. However, technological improvements can be validated if specific structural couplings are established. We define the following positive proposition as a necessary gate for legitimizing technological claims:

Success Theorem for Monotonic Estimators

Assume the metric $M$ functions as a monotonic estimator of the conditional success probability $P(S|\mathcal{F}_k)$, modeled as $M = f(P(S|\mathcal{F}_k)) + \epsilon$ where $f'$ is positive. If a computational method significantly reduces the variance of the error $\epsilon$ and enhances the discriminatory power to identify high-probability candidates, then—and only then—will the expected success probability in the selected ensemble increase.

This theorem establishes the minimal mathematical threshold for asserting the efficacy of quantum or AI methodologies.

Audit Protocol: The PASS/FAIL Framework

Synthesizing the above theorems, the legitimacy of technological implementation in drug discovery is governed by the following three-pillar audit protocol. These criteria are mandatory for any claim of "enhanced success rate."

A: Identification Audit

Is there a formal, statistically coupled, or structural causal link established between the initial metric $F_k$ and the downstream success $S$? (Required to negate the non-identifiability in Theorem 1)

B: Goodhart Robustness Audit

Does the optimization logic include constraints to exclude "expectation inversion regions" where score maximization compromises the true success rate? (Required to negate Theorem 2)

C: Bottleneck Audit

Has the method demonstrated a quantitative breakthrough (enhancement of the bottleneck parameter $\epsilon$) at the rate-limiting stage of the process, rather than merely exhibiting localized computational speedups? (Required to satisfy the Lemma)

VERDICT: PASS If all three conditions are satisfied, the claim that the technology "increases drug discovery success probability" is mathematically substantiated and assertible under Theorem 3.

VERDICT: FAIL If any condition is absent, the claim is a logical fallacy. While the technology may improve "local KPIs," it is mathematically impermissible to represent this as an improvement in "discovery success rates."

Audit Conclusion

To assert that quantum computing or AI methodologies elevate the "global success probability" of drug discovery, an implementation must satisfy (i) Identification, (ii) Goodhart Robustness, and (iii) Bottleneck Breakthrough. Claims failing these criteria are over-generalizations that lack mathematical support. Conversely, implementation designs that meet these criteria may be legitimately categorized as contributors to discovery success rather than mere local optimizations.

References

Proxy Optimization & Reward Hacking

[1] Skalse, J., Howe, N. H., Krasheninnikov, D., & Krueger, D. (2022). Defining and Characterizing Reward Hacking. arXiv preprint arXiv:2201.07683.
[2] Zhuang, S., & Hadfield-Menell, D. (2020). Consequences of Misaligned AI. Advances in Neural Information Processing Systems (NeurIPS), 33, 15763-15773.
[3] Karwowski, J., et al. (2024). Goodhart's Law in Reinforcement Learning. International Conference on Learning Representations (ICLR).
[4] Hennessy, J., & Goodhart, C. (2023). Goodhart's Law and Machine Learning: A Structural Perspective. International Economic Review.

Selection Bias & Winner's Curse

[5] Andrews, I., Kitagawa, T., & McCloskey, A. (2024). Inference on Winners. The Quarterly Journal of Economics, 139(1), 31-75.
[6] Zrnic, T., & Fithian, William. (2024). A Flexible Defense Against the Winner's Curse. arXiv preprint arXiv:2211.05051.

Empirical Discovery Metrics & Bottlenecks

[7] Schuhmacher, A., et al. (2025). Benchmarking R&D success rates and drug development costs: A look at FDA approvals (2006–2022). Drug Discovery Today.
[8] Sun, D., et al. (2022). Why 90% of clinical drug development fails and how to improve it. Acta Pharmaceutica Sinica B, 12(7), 3049-3062.
[9] Amorim, M. J., et al. (2024). Advancing Drug Safety in Drug Development. Chemical Research in Toxicology.

Quantum Computing & Advanced Heuristics

[10] Zinner, M., et al. (2021). Quantum computing’s potential for drug discovery. Drug Discovery Today, 26(7), 1680-1686.
[11] Quantum Machine Learning in Drug Discovery. (2024). ACS Chemical Reviews. DOI: 10.1021/acs.chemrev.4c00678.

Mathematical Foundations

[12] Billingsley, P. (2012). Probability and Measure (Anniversary Edition). Wiley.
[13] Durrett, R. (2019). Probability: Theory and Examples (Vol. 49). Cambridge University Press.

[Nature and Scope of this Audit Report] This treatise is formalized as an engineering audit protocol rather than a pursuit of abstract mathematical truth. It prioritizes "Systemic Accountability" and "Empirical Verifiability under Finite Resources" as its primary evaluative benchmarks. The mathematical modeling is specifically constructed to maximize the transparency of operational and structural risks in industrial implementation.

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