Bayesian Sequential Change Diagnosis

• Bayesian sequential change diagnosis is a rigorous framework for the rapid detection and identification of abrupt, unobservable shifts in probability distributions.
• It employs dynamic programming and posterior probability updates to optimize detection strategies by balancing delay and misclassification costs.
• The method is widely applicable in fields like fault detection, signal processing, and target tracking through efficient geometric partitioning of decision regions.

Bayesian Sequential Change Diagnosis is the formal theory and methodology underlying the rapid detection and identification of abrupt, unobservable changes in the probability distribution of sequential observations, with explicit modeling of uncertainty in both the change time and post-change regime. This topic is foundational in applications such as fault detection and isolation in engineering systems, anomaly detection in signal processing, and target detection in defense, where it is crucial to quickly and accurately infer not only when a regime shift has occurred but also which new regime is in effect. The distinctive feature of Bayesian sequential change diagnosis is the unified minimization of a Bayes risk integrating both detection delay and misclassification costs, executed via optimal policies leveraging dynamic programming, posterior probability recursion, and geometric analysis of decision regions.

1. Bayesian Problem Formulation and Sufficient Statistics

The prototypical setting models observations $(X_1, X_2, \ldots)$ as i.i.d. draws from a known pre-change law $P_0$ with density $f_0$. At a random, unobservable time $\theta$, the distribution abruptly shifts to one of $M$ alternatives $P_\mu$ (with densities $f_1, \ldots, f_M$, index $\mu$ unknown). The core objective is to devise a sequential decision strategy $\delta = (\tau, d)$, where $\tau$ is an $\mathcal{F}$-adapted stopping time and $d$ an $\mathcal{F}_\tau$-measurable terminal choice, to minimize the Bayes risk

$$R(\delta) = c \cdot \mathbb{E}\left[(\tau - \theta)^+\right] + \mathbb{E}[ a_{0, d} \cdot 1_{\{\tau < \theta\}} + a_{\mu, d} \cdot 1_{\{\theta \leq \tau < \infty\}} ],$$

with $c > 0$ the cost per unit detection delay, $a_{0,d}$ the false alarm penalty, and $a_{\mu,d}$ the misidentification penalty (with $a_{ii} = 0$).

The sufficient statistic for the sequential decision problem comprises the posterior probability vector

$$\Pi_n = (\Pi_n^{(0)}, \Pi_n^{(1)}, \ldots, \Pi_n^{(M)}) \in S^M,$$

where $$S^M = \{ \pi \in [0,1]^{M+1} : \sum_{i=0}^{M} \pi_i = 1 \}$$ is the standard probability simplex, and

$$\Pi_n^{(0)} = \mathbb{P}\{ \theta > n ~|~ \mathcal{F}_n \}, \quad \Pi_n^{(i)} = \mathbb{P}\{ \theta \leq n, \mu = i ~|~ \mathcal{F}_n \}, \quad i \in \{1, \ldots, M\}.$$

This vector is recursively updated as a Markov process by

$$\Pi_{n+1}^{(i)} = \frac{D_i(\Pi_n, X_{n+1})}{D(\Pi_n, X_{n+1})},$$

where

$$D_0(\pi, x) = (1 - p) \pi_0 f_0(x), \qquad D_i(\pi, x) = [\pi_i + \pi_0 p \nu_i] f_i(x) ~\forall i \in M,$$

$D(\pi, x) = \sum_{j=0}^M D_j(\pi,x).$

The parameters $p$ and $\nu_i$ encode the base priors on the disorder time and postchange alternative.

2. Optimal Stopping Reformulation and Dynamic Programming Characterization

The Bayesian sequential change diagnosis problem reduces to an optimal stopping problem on the process $\{\Pi_n\}$. The Bayes risk for a stopping strategy is

$$R(\tau) = \mathbb{E} \left[ \sum_{n=0}^{\tau-1} c (1 - \Pi_n^{(0)}) + h(\Pi_\tau) \right],$$

with terminal cost $h(\pi) = \min_{j \in M} h_j(\pi)$ and $h_j(\pi) = \sum_{i=0}^M \pi_i a_{ij}$. The optimal terminal decision is myopic: $d^* = \arg\min_j h_j(\Pi_\tau)$.

Dynamic programming yields the value function $V_0(\pi)$ as the unique solution to

$$V_0(\pi) = \min \Big\{ h(\pi),~ c (1 - \pi_0) + (T V_0)(\pi) \Big\},$$

where the Bayesian integral operator $T$ is

$$(T f)(\pi) = \int_{\mathbb{E}} D(\pi, x)~ f\left( \frac{D_0(\pi,x)}{D(\pi,x)}, \ldots, \frac{D_M(\pi, x)}{D(\pi,x)} \right) m(dx).$$

At each step, the procedure compares the immediate cost of stopping ($h(\pi)$) to the cost of sampling another observation ($c(1-\pi_0)$ plus expected future cost).

3. Geometry of Stopping and Continuation Regions

A signature result is the geometric interpretation of the stopping rule as a partition of $S^M$: $\Gamma = \{ \pi \in S^M : V_0(\pi) = h(\pi) \}$ is the “stopping region,” and, for each $j \in M$,

$$\Gamma^{(j)} = \{ \pi \in \Gamma : h(\pi) = h_j(\pi) \}$$

are the diagnosis-specific partitions. The main properties established are:

• Each $\Gamma^{(j)}$ is nonempty, closed, convex, and, under horizon truncation, forms a decreasing sequence of convex sets.
• The continuation region $\Gamma^c$ (where it is optimal to keep sampling) may be disconnected—even for moderate $M$, reflecting trade-offs between immediate action and evidence accumulation.
• The simplex’s extreme points $e_j$ (where $\Pi_n = e_j$ indicates certainty about hypothesis $j$) always lie within the stopping region.
• These geometric regularities facilitate efficient numerical approximation by discretization and low-dimensional projection of $S^M$.

Numerical illustrations (e.g., for $M=2$ and $M=3$) show how the topology of $\Gamma$ (connectedness, overlap at boundaries) depends on misclassification penalties and delay cost parameters.

4. Implementation, Computational Considerations, and Algorithmic Summary

The optimal sequential diagnosis algorithm follows:

1. Initialize posterior: $\Pi_0 = (1 - p_0, p_0 \nu_1, \ldots, p_0 \nu_M )$.
2. For each $n \ge 0$, recursively update $\Pi_{n+1}$ given $X_{n+1}$.
3. At each $n$, evaluate $V_0(\Pi_n)$ by value iteration (using cubic spline or similar methods to represent the stopping region in local coordinates).
4. Stop sampling at earliest $n = \sigma$ such that $\Pi_\sigma \in \Gamma$.
5. Declare $d^* = \arg\min_{j \in M} h_j(\Pi_\sigma)$.

Efficient implementation relies on local parameterizations of the simplex (e.g., polar coordinates for $M=2$ projected into $\mathbb{R}^2$), and spline interpolation along the boundaries $\partial \Gamma^{(j)}$.

For moderately large $M$ or high-dimensional data, mapping $S^M$ into lower-dimensional Euclidean spaces using linear projections allows visualization and accelerates the nearest-neighbor search for region membership.

5. Relation to Classical Problems and Specializations

Classical Bayesian change detection (Shiryaev’s problem) and sequential multi-hypothesis testing (Wald-Wolfowitz) are natural limiting cases of Bayesian sequential change diagnosis. Specifically:

• Setting $M=1$, with $a_{0,1}=1$ and $a_{1,1}=0$, yields the canonical risk $$R(\delta) = c \mathbb{E}[(\tau-\theta)^+] + \mathbb{P}\{\tau < \theta\}$$ with the classical optimal stopping solution.
• Degenerating the change-time prior ($p_0=1$), so that $\theta=0$ a.s., produces the classical one-stage multi-hypothesis testing scenario.

The framework also models compound failure in systems exhibiting “suspended animation”: after the first failure, only the defective component’s new law is relevant, and the remainder are “suspended.” The framework generalizes to identification of either the count or type of failures, and cost functions $a_{ij}$ can be modified to penalize various diagnostics.

6. Practical Examples and Numerical Characterization

The geometric structure enables practical illustration and quantification of the stopping regions for various parametric choices. For $M=2$, simulation results show:

• In settings with symmetric costs, the $\Gamma^{(1)}$ and $\Gamma^{(2)}$ regions can overlap, so the decision is not unique on their intersection.
• Asymmetric costs or delay parameters lead to disconnected stopping and continuation regions, with boundaries that are nontrivial—these can be stably learned by cubic spline fitting and queried in real time during online operation.

For $M=3$, visualization is achieved by mapping $S^3 \subset \mathbb{R}^4$ into $\mathbb{R}^3$ while preserving geometric “distances” to the simplex faces, aiding interpretation and fast runtime region-checking.

This recursive and numerically efficient strategy is operationally important in high-frequency domains such as radar, biosignal processing, and industrial asset monitoring, where rapid and reliable change isolation is critical.

7. Significance and Impact

Bayesian sequential change diagnosis synthesizes Bayesian updating, optimal stopping theory, and geometric properties of Markov sufficient statistics to produce a mathematically rigorous and practically deployable strategy in settings requiring joint detection and rapid identification of regime changes with explicit delay and error trade-offs. The geometric characterization supports construction of fast, numerically adaptive algorithms applicable to high-stakes applications—fault detection, target tracking, biomedical monitoring—where both promptness and accuracy are paramount. The reduction to a dynamic programming equation on the simplex ensures theoretical tractability and provides a template for extending the method to more complex observation models and prior structures.

The unified framework described here (0705.0043, 0710.4847) has led to downstream developments, including extensions to multiple data sources, Markov chain models, networks, and practical online implementation, and continues to serve as the foundational paradigm for Bayesian regime-change diagnosis under uncertainty.