Filtering and Tracking Survival Propensity (Reconsidering the Foundations of Reliability)

Abstract

The work described here was motivated by the need to address a long standing problem in engineering, namely, the tracking of reliability growth. An archetypal scenario is the performance of software as it evolves over time. Computer scientists are challenged by the task of when to release software. The same is also true for complex engineered systems like aircraft, automobiles and ballistic missiles. Tracking problems also arise in actuarial science, biostatistics, cancer research and mathematical finance.

A natural approach for addressing such problems is via the control theory methods of filtering, smoothing and prediction. But to invoke such methods, one needs a proper philosophical foundation, and this has been lacking. The first three sections of this paper endeavour to fill this gap. A consequence is the point of view proposed here, namely, that reliability not be interpreted as a probability. Rather, reliability should be conceptualized as a dynamically evolving propensity in the sense of Pierce and Popper. Whereas propensity is to be taken as an undefined primitive, it manifests as a chance (or frequency) in the sense of de Finetti. The idea of looking at reliability as a propensity also appears in the philosophical writings of Kolmogorov. Furthermore, survivability which quantifies ones uncertainty about a propensity should be the metric of performance that needs to be tracked. The first part of this paper is thus a proposal for a paradigm shift in the manner in which one conceptualizes reliability, and by extension, survival analysis. This message is also germane to other areas of applied probability and statistics, like queueing, inventory and time series analysis.

The second part of this paper is technical. Its purpose is to show how the philosophical material of the first part can be incorporated into a framework that leads to a methodological package. To do so, we focus on the problem which motivated the first part, and develop a mathematical model for describing the evolution of an item’s propensity to survive. The item could be a component, a system or a biological entity. The proposed model is based on the notion of competing risks. Models like this also appear in biostatistics under the label of cure models. Whereas the competing risks scenario is instructive, it is not the only way to describe the phenomenon of growth; its use here is illustrative. All the same, one of its virtues is that it paves the path towards a contribution to the state of the art of filtering by considering the case of censored observations. Even though censoring is the hallmark of survival analysis, it could also arise in time series analysis and control theory, making the development here of a broader and more general appeal.