Post-detection inference for sequential changepoint localization

TL;DR

This work introduces a detector-agnostic, nonparametric framework for post-detection inference of sequential changepoints by constructing conditional confidence sets for T given data up to a stopping time τ. The core approach inverts level-α tests for each potential changepoint t using universal thresholds built from forward/backward e-processes, yielding finite-sample conditional coverage guaranteed to hold regardless of the detection method. It extends to composite pre-change via least-favorable distributions and offers a simulation-based adaptive threshold scheme in fully parametric settings, with additional methods to form confidence sets for pre- and post-change parameters. Extensive simulations, including a SST-2 sentiment-change example, validate nonasymptotic coverage and reveal tradeoffs between universal and adaptive approaches, confirming broad applicability and robustness of the proposed framework.

Abstract

This paper addresses a fundamental but largely unexplored challenge in sequential changepoint analysis: conducting inference following a detected change. We develop a very general framework to construct confidence sets for the unknown changepoint using only the data observed up to a data-dependent stopping time at which an arbitrary sequential detection algorithm declares a change. Our framework is nonparametric, making no assumption on the composite post-change class, the observation space, or the sequential detection procedure used, and is non-asymptotically valid. We also extend it to handle composite pre-change classes under a suitable assumption, and also derive confidence sets for the change magnitude in parametric settings. We provide theoretical guarantees on the width of our confidence intervals. Extensive simulations demonstrate that the produced sets have reasonable size, and slightly conservative coverage. In summary, we present the first general method for sequential changepoint localization, which is theoretically sound and broadly applicable in practice.

Figures (6)

• Figure 3.1: Pre-change and post-change parts are shown in black and blue, respectively. Outside the time segment $[t, \dots, {\hat{T}-1}]$ when $t<\hat{T}$ (or, $[\hat{T}, \dots, {t-1}]$ when $t>\hat{T}$) both $H_{0,t}$ and $H_{0,\hat{T}}$ agree on the distribution of the observations.
• Figure 6.1: Setting I: The first $T-1$ observations are drawn from $N(0,1)$ and the rest from $N(1,1)$. The point estimates \ref{['eq:known-pre-post']} are shown in a vertical red dashed line, and confidence sets (adaptive \ref{['eq:ci-simple']}) are shown in red points, with $B=N=100$, $\alpha=0.05, L=\infty$. Results of $5$ independent simulations are shown.
• Figure 6.2: First $T-1$ samples are drawn from $N(0,1)$ and the remaining samples from $N(1,1), T=100$. The point estimates \ref{['eq:unknown-pre-post']} are shown in a vertical red dashed line. Confidence sets (universal \ref{['eq:ci-nonpara']}) are shown in red points. $N=100$, $\alpha=0.1$. Results of $5$ random simulations are shown.
• Figure 6.3: Vertical red dashed lines show the point estimates \ref{['eq:t-hat']}. The confidence set \ref{['eq:ci-nonpara']} is marked in red points. Results of $5$ independent simulations are shown.
• Figure 6.4: Vertical red dashed lines show the point estimate. The confidence set \ref{['eq:ci-nonpara']} is marked in red points. The true changepoint $T=500$ is marked with the black line. Texts are plotted from time $t=470$ upto detection time, $\tau=522$.

Theorems & Definitions (41)

• Proposition 2.1
• Proposition 2.2
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Remark 3.4
• Definition 3.6
• Proposition 3.7
• Theorem 3.8
• Theorem 3.9