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A quickest detection problem with false negatives

Tiziano De Angelis, Jhanvi Garg, Quan Zhou

TL;DR

The paper develops a novel variant of Bayesian quickest detection that incorporates false negatives from multiple costly inspections. The observer updates a posterior process after each test and solves an optimal multiple stopping problem by establishing an exact equivalence with a recursive stopping problem, which is then solved via a free boundary analysis. The main contribution is an explicit, closed-form characterization of the optimal testing strategy: perform inspections when the posterior hits a threshold a^*, and update the posterior via the function g at negative test outcomes; the value function is piecewise given by applying a recursive operator to a base function, with a^* uniquely determined by a boundary condition. In the special case ε=0, the model reduces to Shiryaev’s classical problem up to a scaling of the delay-cost parameter, highlighting a precise link between the new and classical formulations and providing practical insight for calibration and interpretation.

Abstract

We formulate and solve a variant of the quickest detection problem which features false negatives. A standard Brownian motion acquires a drift at an independent exponential random time which is not directly observable. Based on the observation in continuous time of the sample path of the process, an optimizer must detect the drift as quickly as possible after it has appeared. The optimizer can inspect the system multiple times upon payment of a fixed cost per inspection. If a test is performed on the system before the drift has appeared then, naturally, the test will return a negative outcome. However, if a test is performed after the drift has appeared, then the test may fail to detect it and return a false negative with probability $ε\in(0,1)$. The optimisation ends when the drift is eventually detected. The problem is formulated mathematically as an optimal multiple stopping problem, and it is shown to be equivalent to a recursive optimal stopping problem. Exploiting such connection and free boundary methods we find explicit formulae for the expected cost and the optimal strategy. We also show that when $ε= 0$ our expected cost is an affine transformation of the one in Shiryaev's classical optimal detection problem with a rescaled model parameter.

A quickest detection problem with false negatives

TL;DR

The paper develops a novel variant of Bayesian quickest detection that incorporates false negatives from multiple costly inspections. The observer updates a posterior process after each test and solves an optimal multiple stopping problem by establishing an exact equivalence with a recursive stopping problem, which is then solved via a free boundary analysis. The main contribution is an explicit, closed-form characterization of the optimal testing strategy: perform inspections when the posterior hits a threshold a^*, and update the posterior via the function g at negative test outcomes; the value function is piecewise given by applying a recursive operator to a base function, with a^* uniquely determined by a boundary condition. In the special case ε=0, the model reduces to Shiryaev’s classical problem up to a scaling of the delay-cost parameter, highlighting a precise link between the new and classical formulations and providing practical insight for calibration and interpretation.

Abstract

We formulate and solve a variant of the quickest detection problem which features false negatives. A standard Brownian motion acquires a drift at an independent exponential random time which is not directly observable. Based on the observation in continuous time of the sample path of the process, an optimizer must detect the drift as quickly as possible after it has appeared. The optimizer can inspect the system multiple times upon payment of a fixed cost per inspection. If a test is performed on the system before the drift has appeared then, naturally, the test will return a negative outcome. However, if a test is performed after the drift has appeared, then the test may fail to detect it and return a false negative with probability . The optimisation ends when the drift is eventually detected. The problem is formulated mathematically as an optimal multiple stopping problem, and it is shown to be equivalent to a recursive optimal stopping problem. Exploiting such connection and free boundary methods we find explicit formulae for the expected cost and the optimal strategy. We also show that when our expected cost is an affine transformation of the one in Shiryaev's classical optimal detection problem with a rescaled model parameter.
Paper Structure (23 sections, 17 theorems, 186 equations, 4 figures)

This paper contains 23 sections, 17 theorems, 186 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Classical quickest detection
  3. Motivating examples for our study
  4. Our model and mathematical contribution
  5. Structure of the paper
  6. Setting
  7. Bayesian formulation
  8. Main results
  9. A recursive problem and its equivalence with the quickest detection
  10. Explicit solution of the recursive problem
  11. Free boundary problem: formulation and verification theorem
  12. Solution of the free boundary problem
  13. Solution of the recursive optimal stopping problem
  14. Solution of the quickest detection problem and further remarks
  15. Properties of the optimal stopping rule
  16. ...and 8 more sections

Key Result

Proposition 2.3

Let $(\tau_k)_{k=0}^\infty$ be an admissible sequence of stopping times. Then, for $k\in\mathbb N$ and $t\ge 0$, with and where, for $s\ge 0$ is a $((\mathcal{G}^{k}_{\tau_k+s})_{s\ge 0},{\mathsf P}_\pi)$-Brownian motion.

Figures (4)

  • Figure 1: A numerical example of the optimally stopped posterior probability process for parameters set to $\lambda = 2$, $\beta = 1.5$, $\mu=1$, $\sigma = 1$, $\epsilon = 0.4$, $\pi = 0.1$. The optimizer performs a test on the system whenever the posterior probability hits $a^* = 0.792$. If the test returns a negative outcome the new posterior becomes $g(a^*)=0.603$. For this simulated example, $\theta = 1.92$ (indicated by the straight vertical line), and in total the observer performs 7 tests (indicated by the vertical dotted lines). In particular, the drift is detected after one false negative. It may be worth noticing that the realisation of $\theta$ is about 3 standard deviations away from the mean. That is why several inspections happen before the drift has actually appeared.
  • Figure 2: Value function $\hat{V}$ with $\lambda = 2$, $\gamma = 0.5$ and $\beta = 1$.
  • Figure 3: Optimal stopping rule $a^*$ for $\lambda = 2$, $\gamma = 0.5$ and $\epsilon \in [0, 0.99]$. The line type indicates the choice of $\beta$. Left: $\epsilon \mapsto a^*(\epsilon)$; middle: $\epsilon \mapsto g_\epsilon(a^*(\epsilon))$; right: $\epsilon \mapsto a^*(\epsilon) - g_{\epsilon}(a^*(\epsilon))$.
  • Figure 4: Optimal stopping times for $\lambda = 2$, $\gamma = 0.5$ and $\epsilon \in [0, 0.99]$. The line type indicates the choice of $\beta$. Left: ${\mathsf E}_0[N_Y] = ( (1-\epsilon) a^* )^{-1}$, the expected number of tests needed to detect the drift-change; middle: $\bar{{\mathsf E}}_{g(a^*)}[\tau_{a^*}]$, the expected waiting time between consecutive tests; right: ${\mathsf E}_0[\tau^*_{N_Y}]$, the expected time of the drift-change detection, with $\pi = 0$.

Theorems & Definitions (43)

  • Definition 2.1: Admissible sequence
  • Remark 2.2: Intuition about problem formulation
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Remark 2.7
  • Proposition 3.1
  • proof
  • ...and 33 more