Beyond Martingale Estimators: Structured Estimators for Maximizing Information Freshness in Query-Based Update Systems

This paper investigates information freshness in a remote estimation system in which the remote information source is a continuous-time Markov chain (CTMC). For such systems, estimators have been mainly restricted to the class of martingale estimators in which the remote estimate at any time is equal to the value of the most recently received update. This is mainly due to the simplicity and ease of analysis of martingale estimators, which however are far from optimal, especially in query-based (i.e., pull-based) update systems. In such systems, maximum a-posteriori probability (MAP) estimators are optimal. However, MAP estimators can be challenging to analyze in continuous-time settings. In this paper, we introduce a new class of estimators, called structured estimators, which can seamlessly shift from a martingale estimator to a MAP estimator, enabling them to retain useful characteristics of the MAP estimate, while still being analytically tractable. Particularly, we introduce a new estimator termed as the $p$-MAP estimator which is a piecewise-constant approximation of the MAP estimator with finitely many discontinuities, bringing us closer to a full characterization of MAP estimators when modeling information freshness. In fact, we show that for time-reversible CTMCs, the MAP estimator reduces to a $p$-MAP estimator. Using the binary freshness (BF) process for the characterization of information freshness, we derive the freshness expressions and provide optimal state-dependent sampling policies (i.e., querying policies) for maximizing the mean BF (MBF) for pull-based remote estimation of a single CTMC information source, when structured estimators are used. Moreover, we provide optimal query rate allocation policies when a monitor pulls information from multiple heterogeneous CTMCs with a constraint on the overall query rate.

💡 Research Summary

The paper tackles the problem of maintaining up‑to‑date information about a remote source that evolves as a continuous‑time Markov chain (CTMC). Traditional analyses of information freshness—particularly the binary freshness (BF) metric and its long‑term average, mean binary freshness (MBF)—have relied on the martingale estimator (ME), which simply holds the most recent received state as the current estimate. While analytically convenient, the ME is far from optimal in pull‑based (query‑driven) systems because it can lock the estimate onto a rarely occurring state until the next query arrives, dramatically reducing freshness.

To overcome this limitation, the authors introduce a new class of structured estimators that can transition smoothly between the ME and the maximum‑a‑posteriori (MAP) estimator. Within this class they define the p‑MAP estimator, a piecewise‑constant approximation of the MAP estimator with a finite number of discontinuities (transition points). The key theoretical insight is that for time‑reversible CTMCs—which include many practical models such as CPU state machines and birth‑death processes—the MAP estimator becomes constant after a finite settling time τ* and coincides exactly with a suitably chosen p‑MAP estimator. Even when the MAP estimator oscillates indefinitely (e.g., when the stationary distribution is uniform), increasing the number of transition points makes the p‑MAP arbitrarily close to the true MAP.

Using the BF process Δ(t)=1{X(t)=ĤX(t)} and the definition MBF=E