Optimizing Sequential Diagnostic Testing under Uncertainty: Models, Heuristics, and Bounds for Differential Diagnosis

Physicians regularly face the challenge of differential diagnosis-distinguishing between diseases with overlapping symptoms-by conducting a sequence of diagnostic tests to identify the underlying condition. While AI-based tools have improved the estimation of prior probabilities of diseases, there is little research on how to select and sequence diagnostic tests, which are often disease-specific, imperfect, and asymmetrically costly or rewarding. We model this decision-making problem as a finite-horizon Markov Decision Process and establish structural results on both the optimal test sequence and stopping criterion. For symmetric and perfectly accurate tests, we identify prior distributions where it is optimal to forgo testing, as well as those where it is optimal to continue testing in descending order of prior until a positive result occurs or all but one test have been run. We also characterize priors where the optimal policy is to stop upon a positive result or after an intermediate threshold number of negative outcomes, with the threshold derived from first-order conditions. When tests are imperfect or rewards are asymmetric, standard rules-such as testing in descending order of prior probability or diagnostic reward-may fail: it can be optimal to begin with the second most probable or rewarding disease. Since solving the MDP exactly becomes intractable with many diseases, we develop a tractable heuristic, the economic index policy (EIP), that mimics the optimal threshold policy while sequencing tests using an economic index informed by our structural results. To evaluate its performance when the number of diseases is large, we derive a novel upper bound on the value function by combining information relaxation with regression-based penalty functions, resulting in a tight and scalable mixed-integer linear program. Numerical experiments show that EIP can outperform standard limited lookahead and approximate dynamic programming heuristics, particularly when test accuracy is high and diagnostic rewards and penalties are asymmetric. Its performance closely tracks the optimal policy even in large-scale settings, highlighting the value of leveraging structure to design efficient and robust heuristics for sequential decision problems.