Betting on residual life: The caveats of conditioning

Abstract

Assessing conditionals based on any specified probability model is straightforward and unique when the conditioning event is in the subjunctive mood; that is, supposing that the conditioning event were to occur. The matter becomes problematic, however, when the conditioning event actually does occur as observed data, and thus becomes a reality. We illustrate this point by considering a commonly occurring scenario in the actuarial sciences, engineering reliability, survival analysis, and in general, any type of an activity that involves filtering. We argue that there could be more than one way to bet on residual life. Our message is that it is the likelihood—not Bayes’ Law—which is the tail that wags the dog!

This paper should appeal to both probabilists and statisticians who are interested in foundational issues. It has been written to honour Richard Johnson whose Editorship of Statistics and Probability Letters has provided a platform for dialogue between probabilists, statisticians, and those who strive to be both.

Introduction

In the process of using marker data to assess the lifetime of an item experiencing failure due to ageing, we were confronted by a dilemma that sneaked upon us as a matter of course (see Singpurwalla, 2006a). It turns out that the scenario leading to the dilemma is quite common and can arise when addressing practical issues of conditioning in the actuarial, the engineering, and the biomedical sciences. Stripped to its essentials, the scenario goes as follows.

Suppose that an item's lifetime X is judged to have a distribution function $G(x)=P(X⩽x)$, and a survival function $\overline{G}(x)\stackrel{\mathrm{def}}{=}1-G(x)=P(X>x)$. We suppose that lifetime can be continuously monitored so that $x⩾0$. Were this item supposed to survive until x, its residual (or remaining) lifetime will be $X-x$. We are required to make statements of uncertainty about $(X-x)$, so that actuarial, engineering, or medical decisions about the item can be made. That is, we are required to specify $P(X-x>u|X>x)$, for all $u>0$. Our interpretation of probability is de Finnetian (see de Finetti, 1937), in the sense that probability reflects one's disposition to a two-sided bet. Thus, probability assessments can be seen as a device for hedging our bets on the item's survival, or some other unknown quantity of interest, such as parameters in probability models.

A solution to the problem posed is elementary and unique, given a distribution function G. Specifically, for any $u>0$

$$P(X-x>u|X>x)=P(X>x+u|X>x)={\displaystyle \frac{P(X>x+u)}{P(X>x)}}={\displaystyle \frac{\overline{G}(x+u)}{\overline{G}(x)}}\text{.}$$

Suppose now, that instead of the subjunctive, “were the item to survive until x”, we are told that the item actually did survive to x. That is, the event $(X>x)$ is no more an uncertain event; $(X>x)$ has now become observed data. What then would our assessment of the uncertainty about the residual life $(X-x)$ be? In other words, how would we bet on the event $(X-x>u)$, for $u>0$? Would it continue to be $\overline{G}(x+u)/\overline{G}(x)$, or could it be something else? If the latter, would the number to bet be unique? For a discussion of these and related questions, one may visit Freedman and Purves (1969). A more recent discourse on the different kinds of conditional beliefs is in Joyce (1999, Chapter 6).

Intuitively, it seems that there ought to be some distinction between looking at $(X>x)$ as a possibility, versus looking at it as a fact that is revealed as data. Thus, $\overline{G}(x+u)/\overline{G}(x)$ need not be the correct answer. Yet many individuals when faced with this problem would simply mimic the steps leading to Eq. (1.1) and continue to declare $\overline{G}(x+u)/\overline{G}(x)$ as their answer. In doing so they do not appear to be making a distinction between $(X>x)$ as a supposition versus a reality. Alternatively put, they may be failing to recognize the connotation that in a conditional probability statement, the word “given” does not indicate a fact; rather it indicates a supposition that the conditioning event is true. Thus, are those who declare $\overline{G}(x+u)/\overline{G}(x)$ as their answer—irrespective of the character of the conditioning event—in error, or is there a rationale for their answer?

We claim that the rationale cannot completely be within the calculus of probability, because the notion of probability—at least from a subjectivistic point of view—is germane only when the disposition of all events in question is unknown. Thus, for example, it may not make sense to say that the probability that a coin with heads on both faces when flipped will land heads, is one. This is because the disposition of the outcome is known before the flip. Consequently, a two-sided bet on the outcome heads has to be $1, which will be exchanged for a $1 when the coin lands heads, which it will. The two-sided bet of $1 is thus meaningless. The rationale therefore must come from concepts in statistics wherein the notion of a likelihood plays a signal role. By all accounts the notion of a likelihood appears to be alien to probability theory.

In what follows we point out that there are both philosophical and technical arguments which support $\overline{G}(x+u)/\overline{G}(x)$ as an answer, but that this answer is one among other possible answers. This is the main point of this article. Arguments about conditioning are common among philosophers of science. That such arguments could also be relevant to reliability, survival analysis, filtering, and forecasting seems to not have been recognized.