Probability, chance and the probability of chance.

1. Probability and chance

1.1. Introduction: Statement of the problem and objectives

Consider the following archetypal problem that commonly arises in the contexts of biomedicine, engineering and the physical sciences.

Suppose that at some reference time [tau], the "now time," YOU are asked to predict the time to failure T of some physical or biological unit. The capitalized YOU is to emphasize the fact that it is a particular individual, namely yourself, that has been asked to make the prediction. To facilitate prediction, you examine the unit carefully and learn all that you can about its genesis: how, when and where it was made. You denote this information by H([tau]), for history at time [tau]. In the case of biological units, H([tau]) would pertain to genetic and/or medical information. Suppose, as is generally true, that based on H([tau]) you conclude that prediction with certainty is not possible. Consequently, you are now faced with two options: walk away from the problem, or make an informed guess about T.

Suppose that you choose the second option and are prepared to make guesses about the event (T [greater than or equal to]), for some t > 0. In reliability, t > 0 is known as the "mission time." There are several additional caveats to this basic problem that go into forming our overall framework; these will be presented in Sections 2 and 3. In Section 2, we introduce the caveat of data, and in Section 3 the caveat of surrogate information.

To keep the mathematics simple, you introduce a counter, say X, and adopt the convention that X = 1 (a "success") whenever T [greater than or equal to] t, and X = 0 (a "failure"), otherwise. Thus, the events (T [greater than or equal to] t) and (X = 1) are isomorphic; however, there is a loss of granularity in going from T to X. This is because X continues to equal one, even when T [greater than or equal to] t + a, for any and all a > 0. With the introduction of X, informed guesses about (T [greater than or equal to] t) boil down to informed guesses about (X = 1). But what do we mean by an informed guess, and how shall we make this operational? Do the terms probability, chance and likelihood constitute an informed guess, or does each of these terms connote a distinct notion? Furthermore, do these terms cover all the scenarios of uncertainty that one can possibly encounter or are there scenarios that call for additional notions such as "belief" and "plausibility"? The aim of this paper is to show that each of the above terms encapsulates a distinct notion, so that their indiscriminate use should not be a matter of course.

1.2. Personal probability: Making guesses operational

By informed guess, we mean a quantified measure of your uncertainty about the event (X = 1) in the light of H([tau]), and subsequent to a thoughtful evaluation of its consequences. Now, it is generally well acknowledged that probability is a satisfactory way to quantify uncertainty, and to some, such as Lindley (1982), the only satisfactory way. There are several interpretations of probability (c.f. Good (1965)). The one we shall adopt is personal probability, also known as subjective probability. Here, you quantify your uncertainty about the event (X = 1), based on H([tau]), by your personal probability denoted:

[P.sub.Y](X = 1;H([tau])). (1)

The subscript indexing P emphasizes the fact that the specified probability is that of a particular individual, namely, you. For convenience, we set [tau] = 0 and denote H(0) by simply H. Henceforth, we also omit the subscript associated with P, so that Equation (1) is written:

P(X = 1;H) = p, (2)

where 0 < p < 1. The p so specified is a personal probability because it is not unique to all persons; more important, it can change with time for the same individual. This is because the background history for this person also changes, and it is the history that plays a key role in specifying a personal probability. Thus, an informed guess is tantamount to specifying a p, where p is a personal probability.

To make an informed guess operational, that is, to make a pragmatic use of it, we need to interpret p. For this we appeal to De Finetti (1974) who proposed that p represent the amount you--the specifier of p--is willing to stake in a two-sided bet (or gamble) about the event (X = 1). That is, should X turn out to be one, you receive as a reward one monetary unit against the p staked out by you. Should X turn out to be zero, then the amount staked, namely p, is lost. By a two-sided bet, we mean the willingness to stake p for the event (X = 1), or an amount (1 - p) for the event (X = 0). That is, you are indifferent between the two gambles: one monetary unit in exchange for p if (X = 1), or one monetary unit in exchange for (1 - p) if (X = 0). It is useful to bear in mind that in keeping with the spirit of the individual nature of personal probability, the amount p represents your stake. For the same event (X = 1), your colleague may choose to stake a different amount [~.p], with [~.p] [not equal to] p. It is also important to note that with p interpreted as a gamble, the bet will only be settled when X reveals itself. Thus, bets can only be made operational for events that are ultimately observed. We do not consider here the disposition of the second party in the bet; we assume that the second party is willing to accept any bet put forth by you.

Thus, to summarize, in the context of this paper, the word "probability" is used to denote the amount an individual is prepared to stake in a two-sided bet about an uncertain event. This probability can be specified based on H alone, and it is not essential that H contain data on items judged to be similar to the item in question. That is, personal probabilities can be specified without the benefit of having observed data.

1.3. Chance or propensity: A useful abstraction

Whereas specifying a personal probability can be done solely by introspection considering H, a more systematic approach, which involves breaking the problem into smaller, easier problems, begins with invoking the law of total probability on the event (X = 1; H). Specifically, for some unknown quantity [theta], 0 < [theta] < 1, and an entity [pi] ([theta]; H), whose interpretation is given later in Section 1.4:

P(X = 1,H) = [[integral].sub.0.sup.1]P(X = 1|[theta];H)[pi]([theta];H)d[theta], (3)

= [[integral].sub.0.sup.1]P(X = 1|[theta])[pi]([theta];H)d[theta], (4)

if you assume that X is independent of H given [theta]. That is, were you to know [theta], then knowledge of H is unnecessary. The meaning of [theta], known as a parameter, remains to be discussed, but for now we state that in the language of personal probability, Equation (3) implies an extension of the conversation from P(X = 1; H) to P(X = 1 | [theta]; H). The idea here is that after invoking the assumption of independence, you may find it easier to quantify your uncertainty about (X = 1) were you to know [theta], than quantifying the uncertainty based on H. Whereas the dimension of H can be very large, the dimension of [theta] is one. Thus, the role of the parameter [theta] is to simplify the process of uncertainty quantification by imparting to X independence from H.

In Equation (4), the quantity p(X = 1 | [theta]) is known as a probability model for the binary X. Following Bernoulli, you let p(X = 1 | [theta]) = [theta], where p(X = 1 | [theta]) represents your bet (personal probability) about the event (X = 1) were you to know [theta]. This brings us to the question of what does [theta] mean? That is, how should we interpret [theta]?

The meaning of [theta] was made transparent by De Finetti (c.f. Lindley and Phillips (1976)) in his now famous theorem on binary exchangeable sequences. Loosely speaking, this theorem says that if a large number of units judged similar to each other (the technical term is exchangeable) and to the unit in question were to be observed for their survival or failure until t, and if [X.sub.i] = 1 if the ith item survived until t ([X.sub.i] = 0 otherwise), then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

that is [theta] is the average of the [X.sub.i]s, when the number of [X.sub.i]s is infinite. De Finetti refers to this [theta] as a chance or propensity. Note that there is no personal element involved in defining [theta], other than the fact that [theta] derives from the behavior of exchangeable sequences, and exchangeability is a judgment. What you judge to be exchangeable may not sit well with your colleagues. Because [theta] connotes the limit of an exchangeable binary sequence, [theta] can be seen as an objective entity. More important, since [theta] cannot be actually observed (n in the Equation (5) is infinite), we claim that chance is an abstract construct. It is a useful abstraction all the same, because in writing P(X = 1 | [theta]) = [theta], you are saying that your stake on the uncertain event (X = 1) is [theta], were you to know [theta]. But no one can possibly tell you what [theta] is, and this is what leads us to the next section. But before we do so, it may be of interest to mention a few words about two other interpretations of [theta].

One is due to Laplace, who in keeping with the scientific climate of his time, and being influenced by Newton, was concerned with cause and effect relationships. Accordingly, to Laplace, [theta] was the cause of an effect, namely, the event (X = 1). The second interpretation of [theta] stems from the relative frequency interpretation of probability. Indeed, here [theta] is taken to be the probability that X = 1.

Finally, even though the notion of chance introduced here has been in the context of binary variables, a parallel notion also exists for other kinds of variables.

1.4. Probability of chance: Taking chances with chance

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