Optimal Sequential Diagnosis Algorithm

• The paper’s main contribution is the formulation of an optimal sequential diagnosis algorithm that minimizes cost while maximizing diagnostic accuracy using decision-theoretic models and Bayesian updates.
• It details methodologies such as decision-analytic Bellman recursion and information-gain strategies to efficiently select tests and queries across various diagnostic domains.
• Empirical evaluations show the approach outperforms human heuristics, offering significant cost savings and improved accuracy in medical, industrial, and technical fault diagnosis.

Sequential diagnosis is a class of decision-theoretic algorithms dedicated to the iterative identification of system faults or disease conditions, subject to incomplete information and cost constraints. The optimal sequential diagnosis algorithm aims to select, at each stage, the next measurement, test, or query such that overall diagnostic cost is minimized while diagnostic accuracy is maximized. This task is exemplified in domains ranging from clinical medicine, circuit fault isolation, and industrial troubleshooting, to sequential change-detection in stochastic processes.

1. Formal Problem Definition

The archetypal sequential diagnosis problem can be cast as a partially observable Markov decision process (POMDP) in which the system's true state (disease label or fault configuration) is hidden, and the agent acts by selecting among several types of queries or tests. At time $t$, the agent chooses an action from three classes: free-text clinical questions, diagnostic tests (e.g., laboratory, imaging procedures), or commitment to a final diagnosis. Observations are generated in response to each action by a deterministic or probabilistic gatekeeper, and the agent maintains a Bayesian belief state over candidate diagnoses, updating with each observation via Bayes rule (Nori et al., 27 Jun 2025).

The objective is to maximize the expected utility: $$\max_{\pi} \,\, \mathbb{E}_{D,\, trajectory \sim \pi} \Big[ \mathbb{I}[\text{Judge}(D,\, \hat{d}) \ge 4] - \lambda \sum_{t: a_t=\text{Test}} \text{Cost}(\tau_t) \Big]$$ where $\pi$ is a policy mapping histories to actions, $\lambda$ regulates cost-accuracy tradeoff, and the reward is determined by external evaluation (e.g., clinical rubric). This general structure is seen in both medicine and technical fault-diagnosis (Nori et al., 27 Jun 2025, Kalagnanam et al., 2013).

2. Algorithmic Approaches and Canonical Methods

Decision-Analytic Bellman Recursion

In the traditional single-fault model, diagnosis can be formalized as finite-horizon dynamic programming. The agent maintains a belief vector $(b_1, \dots, b_n)$ over $n$ fault hypotheses, performing tests $t$ with known cost $c_t$ and informative outcomes. The cost-to-go function $V(b)$ satisfies the Bellman equation: $$V(b) = \begin{cases} 0 & \text{if some } b_i = 1 \ \min_{t} \{ c_t + \sum_{o} P(o|b,t) V(b_{t,o}) \} & \text{otherwise} \end{cases}$$ where $b_{t,o}$ is the post-test belief (Kalagnanam et al., 2013). The optimal policy recursively selects the test minimizing expected total future cost.

C/P Rule

In the special case of deterministic tests with single faults, the optimal test sequence is given by ordering components by increasing $c_i / P(h_i)$ ("cost-over-probability") and testing in that order until failure is found (Kalagnanam et al., 2013).

POMDP-Style and Information-Gain Approaches

For systems with expensive or diverse queries, and especially in the context of clinical diagnostics with large ontologies, the action-selection is cast as myopic, greedy optimization of expected information gain per unit cost: $$IG(\tau) = H[b_t] - \mathbb{E}_r[ H[b_t | \tau \rightarrow r] ]$$ where $H[\cdot]$ is entropy, and actions are ranked by $IG(\tau) / \text{Cost}(\tau)$ (Nori et al., 27 Jun 2025).

Subadditive Sequential Testing

When test costs are governed by general subadditive functions (e.g., batching, routing, machine activation), optimal test selection generalizes beyond greedy rules. The "Greedy-with-Truncation" algorithm selects batches minimizing $C(B) / [1-P(B)]$, truncates the sequence when batch completion is more cost-effective, and achieves $(4\rho+\gamma)$-approximation assuming appropriate oracles for cost and ratio minimization (Harris et al., 29 Jan 2025).

Cost Structure	Value Oracle γ	Ratio Oracle ρ	Approximation Factor
Additive/cardinality	1	1	5
Tree/hierarchical	1+ε	1+ε	5+O(ε)
Machine activation	ln n	1	4+ln n
Routing	1.5	2+ε	9.5+O(ε)

3. Multi-Agent Orchestration (MAI-DxO Framework)

The MAI Diagnostic Orchestrator (MAI-DxO) adapts optimal sequential diagnosis to the LLM context in evidence-based medicine. It simulates a physician panel decomposed into five roles: Dr. Hypothesis (maintains top-K differential), Dr. Test-Chooser (ranks actions by information-gain-per-cost), Dr. Challenger (proposes falsification tests), Dr. Stewardship (vetos low-value actions), and Dr. Checklist (checks standard nomenclature and reasoning consistency). The consensus determines whether to Ask, Test, or Diagnose. Bayesian updates maintain a normative disease belief distribution (Nori et al., 27 Jun 2025).

Pseudocode snippet:

For t = 1,2,… until “Diagnose”: Hypothesis: select top-K candidates Test-Chooser: rank by IG(τ)/Cost(τ) Challenger: propose diagnostic falsification Stewardship: veto poor value Checklist: ensure standard test vocabulary Consensus: determine next action Execute and Bayesian update

Empirical evaluation on 304 challenging NEJM-CPC cases with cost and diagnostic accuracy benchmarks demonstrated that MAI-DxO, paired with advanced LLMs, achieves 80–85.5% accuracy, exceeding median physician performance, and delivers robust cost-efficiency (Nori et al., 27 Jun 2025).

4. Decoupled Query Optimization and Complexity

In knowledge-base and model-based diagnosis, query generation is decomposed into two independent steps: (1) minimize expected diagnostic effort (e.g., entropy over diagnosis set), and (2) among all minimal-effort discriminating queries, select the one of minimal physical or cognitive cost. This insight allows reduction of query search to combinatorial operations over candidate diagnoses, often avoiding calls to expensive logical inference engines (Rodler et al., 2017, Rodler et al., 2016, Rodler et al., 2017).

• Canonical queries are built by minimal hitting sets over "traits" (unique distinguishing sets).
• The optimal partition is located in $O(2^{|D|})$ time, but practical methods prune the search to $O(|D|^2)$.
• For scalability, the hitting-set search and query-enrichment (if desired) remain polynomial in the number of diagnoses and query sentences.

5. Extensions: Multi-Class, Change Detection, and Abstraction

Multi-Class Sequential Diagnosis

Optimal strategies for multi-hypothesis diagnosis (e.g., in settings with exponentially correlated observation distributions) exploit underlying low-rank structure in the belief manifold. Matrix factorization yields diagnostic statistics reducing the DP dimension, enabling computation of optimal policies for large $N$ (number of classes) (Wang, 2015). For exponential family models, beliefs lie on $k$-dimensional manifolds, and policy boundaries are efficiently computable via grid-based backward induction.

Sequential Change Diagnosis

In online change-point detection with identification among $K$ alternatives, recursive schemes such as Adaptive Matrix CuSum mitigate pre-change data contamination in identification statistics (Warner et al., 2022). These algorithms provide explicit control on false alarms, misidentification, and Lorden-type delay, and achieve first-order asymptotic optimality. Threshold selection is guided by analytic expressions involving KL-divergences and exponential bounds on errors (Warner et al., 2023, 0710.4847).

Diagnosis by Abstraction and Active Testing

For large systems (e.g., circuits), abstraction via cones and component cloning shrinks the effective search space. Bayesian networks compiled to d-DNNF enable entropy-based or hybrid heuristics for measurement selection, scaling diagnosis to thousands of components with performance within 5–10% of theoretical optima (Siddiqi et al., 2014, Feldman et al., 2014).

6. Empirical Performance and Theoretical Guarantees

Sequential diagnosis algorithms consistently outperform human expert heuristics (by ≈14%) and naive black-box search, with robust cost savings preserved under moderate uncertainty in underlying model parameters (Kalagnanam et al., 2013, Nori et al., 27 Jun 2025). Approaches supporting subadditive and submodular costs provide constant-factor approximations, with ETH-based hardness ruling out general poly-logarithmic approximations for submodular cases (Harris et al., 29 Jan 2025).

Benchmark evaluations utilize real-world diagnostic cases, technical fault sets, and industrial-scale knowledge bases, demonstrating algorithmic scalability and cost-effectiveness. For example, MAI-DxO achieves up to 85.5% diagnostic accuracy at reduced mean cost, generalizing across diverse LLMs and outperforming physicians (Nori et al., 27 Jun 2025).

7. Operational Modes, Pareto Frontier, and Future Directions

Optimal sequential diagnosis frameworks support several operational configurations designed to traverse the Pareto frontier in accuracy–cost space, including:

1. Instant Answer: lowest cost, minimal accuracy.
2. Question Only: moderate cost, intermediate accuracy.
3. Budgeted: constrained spending, high efficiency.
4. No Budget: unconstrained cost, maximal achievable accuracy.
5. Ensemble: highest accuracy at increased cost (Nori et al., 27 Jun 2025).

Current trends focus on multi-agent orchestration for cognitive bias reduction, robust query selection under general cost models, and scalable algorithms for high-cardinality diagnosis scenarios. ETH-based complexity insights have shaped expectations in submodular environments, and the integration of model-abstraction, query-enrichment, and practitioner panel simulation extends applicability to both clinical and industrial contexts.

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References:

• "Sequential Diagnosis with LLMs" (Nori et al., 27 Jun 2025)
• "A Comparison of Decision Analysis and Expert Rules for Sequential Diagnosis" (Kalagnanam et al., 2013)
• "Sequential Testing with Subadditive Costs" (Harris et al., 29 Jan 2025)
• "Inexpensive Cost-Optimized Measurement Proposal for Sequential Model-Based Diagnosis" (Rodler et al., 2017)
• "Scalable Computation of Optimized Queries for Sequential Diagnosis" (Rodler et al., 2016)
• "Optimal Sequential Multi-class Diagnosis" (Wang, 2015)
• "Bayesian sequential change diagnosis" (0710.4847)
• "Worst-Case Misidentification Control in Sequential Change Diagnosis using the min-CuSum" (Warner et al., 2023)
• "Sequential Diagnosis by Abstraction" (Siddiqi et al., 2014)
• "A Model-Based Active Testing Approach to Sequential Diagnosis" (Feldman et al., 2014)
• "A Generally Applicable, Highly Scalable Measurement Computation and Optimization Approach to Sequential Model-Based Diagnosis" (Rodler et al., 2017)
• "Towards Optimizing Reiter's HS-Tree for Sequential Diagnosis" (Rodler, 2019)
• "Asymptotically Optimal Anomaly Detection via Sequential Testing" (Cohen et al., 2014)
• "Sequential Change Diagnosis Revisited and the Adaptive Matrix CuSum" (Warner et al., 2022)