Bayesian sequential change diagnosis

Sequential change diagnosis is the joint problem of detection and identification of a sudden and unobservable change in the distribution of a random sequence. In this problem, the common probability law of a sequence of i.i.d. random variables suddenly changes at some disorder time to one of finitely many alternatives. This disorder time marks the start of a new regime, whose fingerprint is the new law of observations. Both the disorder time and the identity of the new regime are unknown and unobservable. The objective is to detect the regime-change as soon as possible, and, at the same time, to determine its identity as accurately as possible. Prompt and correct diagnosis is crucial for quick execution of the most appropriate measures in response to the new regime, as in fault detection and isolation in industrial processes, and target detection and identification in national defense. The problem is formulated in a Bayesian framework. An optimal sequential decision strategy is found, and an accurate numerical scheme is described for its implementation. Geometrical properties of the optimal strategy are illustrated via numerical examples. The traditional problems of Bayesian change-detection and Bayesian sequential multi-hypothesis testing are solved as special cases. In addition, a solution is obtained for the problem of detection and identification of component failure(s) in a system with suspended animation.

💡 Research Summary

The paper introduces a unified Bayesian framework for the problem of sequential change diagnosis, which simultaneously addresses the detection of an abrupt, unobservable change in the distribution of a data stream and the identification of the new regime among a finite set of alternatives. Formally, a sequence of i.i.d. observations {Xₙ} follows an initial law f₀ until an unknown disorder time τ, after which it switches to one of M possible laws f₁,…,f_M. Both τ and the index D of the new law are treated as random variables with prior probabilities π₀ (no change) and π_i (change to regime i). As data arrive, a Bayesian filter updates the posterior belief vector pₙ = (pₙ⁰, pₙ¹,…,pₙᴹ), where pₙ⁰ is the probability that the change has not yet occurred and pₙᶦ is the joint probability that the change has happened and the new regime is i.

The decision maker incurs three types of cost: (i) a false‑alarm cost c₀ if a change is declared before it actually occurs, (ii) a delay cost c₁ per time unit that the change remains undetected after τ, and (iii) a mis‑identification loss L_i if the final declared regime Ď differs from the true D. The objective is to choose a stopping time T and a terminal decision rule Ď that minimize the expected total Bayesian risk R = E