Parametric Failure Data Analysis
5.2 ASSESSING PREDICTIVE DISTRIBUTIONS IN THE ABSENCE OF DATA
5.2.3 The Bernoulli as a Chance Distribution
The Bernoulli, as a chance distribution, was introduced in section 3.1.3 in connection with de Finetti-type representation theorems for zero-one exchangeable sequences. In the context of the set-up of this section, suppose that the event T > t has as its proxy an event X=1, and that X=0 is a proxy for the event T≤t. A motivation for introducing the binary X in place of the continuous T is simplification. Specifically, we are able to replace multi- parameter chance distributions by a single parameter chance distribution. To see why, we note
from (5.1) that when the continuous T is replaced by the Bernoulli X with a parameter , we have
P X=1=
P X=1 F d
= 1 0
F d (5.10)
since has support01. Thus werePT > t •to be a Weibull chance distribution of the form exp−t/, as was done in section 5.2.2, the representation of (5.10) would entail a chance distribution consisting of a single parameter. Whereasencapsulates the effect of both and, there is a loss of granularity in going from the representation of (5.1) to (5.10). This is because with the former one can place bets on the eventT > tfor a range of values of t, whereas the latter enables bets on only a single event defined byX.
In section 3.1.2, a beta distribution with parametersandwas chosen as the prior for, 0< <1. For >0, its density atis of the form
+
−11−−1 (5.11)
where=
0 x−1e−xdx. The mean and variance of this beta distribution are/+and /+2++1, respectively.
This distribution is attractive in the sense that by a suitable choice ofand, various forms of subjective judgments aboutcan be expressed. For example,==1 results in a uniform density for, whereas the choice=2,=1 makes the density of the form 2– a triangle with an apex of 2 at=1. Identical values of >1 and >1 make the density symmetrical about =1/2, which is where it also attains its maximum. Values of < >1 and > <1 make the density L-shaped or J-shaped (Figure 5.1). The former is appropriate when one has a strong prior opinion about the outcomeX being 0; the latter when the prior opinion thatXwhen revealed will be 1 is strong. Values ofandless than one enable us to describe a U-shaped density for(Figure 5.1). The U-shaped density foris appropriate when one opines that in repeated trials of the phenomenon that generatesX, the values of theXs when revealed will be a long sequence of either zeros or ones. When < =1 and=<1 the density is L(J)-shaped, but unlike the illustrations of Figure 5.1, it does have a probability mass at=10. A hierarchical two-stage approach for specifying a prior foris in section 2 of Chen and Singpurwalla (1996).
L-shaped
α<1, β>1
0 1 θ
J-shaped
α>1, β<1
0 1 θ
U-shaped
α<1, β<1
0 1 θ
Figure 5.1 L, J, and U-shaped densities for the prior of.
ASSESSING PREDICTIVE DISTRIBUTIONS IN THE ABSENCE OF DATA 131 A lot has been written about a suitable choice for the prior distribution of . Much of the discussion is philosophical and focused on densities that encapsulate little or no knowledge, or those that best allow the data – when available – to dominate the effect of the prior. Such priors are known under several names as ‘objective’, ‘default’, ‘reference’ and ‘non-informative’, and are used when one wishes to adopt an impartial stance. Geisser (1984), following Bayes and Laplace, advocates the uniform as a noninformative prior for; Laplace used the principle of indifference for justifying his use of this prior. Others who have written on this topic are Bernardo (1979a), Haldane (cf. Jeffreys, 1961; p. 123) and Zellner (1977). Bernardo and Zellner advocate (default) proper priors that are proportional to−1/21−−1/2and1−1−, respectively. However, some of the proposed priors forare improper, and in some cases they depend on the nature of the experiment that generatesX; such priors are objectionable. For example, Haldane proposes that the density for be proportional to1−−1which is improper because its integral is not one. Lindley (1997a) refers to the property of the failure of the density to integrate to one as impropriety. The desire to produce priors that are universal, and consequently non-subjective, has yielded distributions having the objectionable features mentioned above. More about this will be said in section 5.3.2.
A subjectivistic assessment of based on expert testimony has been proposed by Chaloner and Duncan (1983), much in the spirit of eliciting expert testimonies on Weibull lifetimes that result in (5.9). An alternative approach is to induce a prior distribution forusing the elicited priors on the parameters of the chance distributions of sections 5.2.1 and 5.2.2. This task is easier in the case of the exponential than the Weibull. Specifically, forPT > t=exp−t, I have proposed the gamma distribution with parametersandelicited via expert testimonies on the mean time to failureET as a suitable choice. Since
Px=1==PT > t=exp−t
a prior oninduces a prior on. It can be seen – details left as an exercise for the reader – that a gamma prior onwith scaleand shaperesults in the following as a prior density for, 0< <1:
/t−1
t
ln
1
−1
(5.12)
It is important to bear in mind that the mission timetis a parameter of the above prior. This is to be expected since depends ont, taking the value 10whent=0. In the case of the Weibull with scale and shape,=exp−t/. This form makes the task of inducing a prior onbased on a joint prior onandcumbersome; it will have to be done by simulation.
Similarly, for the case of a gamma chance distribution.
SincePX=1=E(equation (5.10)) we have the result that when the prior onis a beta density of the form given by (5.11),PX=1=/+, the mean of. When the prior on is an induced density of the form given in (5.12), thenPX=1=/+t, the mean of under this density; its variance is/+2t−/+t2. Note that thein/+ is not the samein/+t; the former is a parameter of the beta density for, whereas the latter is the shape parameter of a gamma prior for. Since/+tis also the survival function of a Pareto distribution (equation (5.8)) we have the interesting observation that the mean value function of the density of (5.12) is the survival function of a Pareto distribution.
Whereas (5.10) pertains to the special case ofX=1, its more general version entailingX=x, forx=1 or 0, takes the form
PX=x= 1 0
x1−1−xFd
IfFis taken to be the beta distribution with parametersand, then
P X=x= 1 0
x1−1−x+
−11−−1d
=++x−x+1
+ (5.13)
Verify that whenx=1, the above expression becomes /+, and that when x=0 it is /+. Furthermore, if the prior onwere to be a uniform distribution, i.e. were==1, (5.13) would yieldPX=1=PX=2=1/2. This result implies that under the uniform prior for, the reliability of the unit in question is 1/2.
Series Systems with Bernoulli Chance Distributions
Now suppose that we have two such units and that they are connected to form a series system.
What is the reliability of this system supposing that their associatedXis,i=12, are condition- ally – given– independent? The conventional answer, namely1
2
·1
2
=14, would be incorrect!
The correct answer is 1/3. To see why, we start by considering, forxi=0 or 1,i=12,
P X1=x1 X2=x2= 1 0
P X1=x1 X2=x2 Fd
= 1 0
P X1=x1PX2=x2 Fd
sinceX1andX2are conditionally independent, given. Thus
P X1=x1 X2=x2= 1 0
x11−1−x1x21−1−x2Fd
= 1 0
x1+x21−2−x1−x2d (5.14)
sinceFis a uniform distribution on01. Simplification of the above yields, P X1=x1 X2=x2=x1+x2+13−x1−x2
4
which forx1=x2=1 becomes 314 =13
The incorrect answer is a consequence of invoking the assumption of independence after averaging out. This is flawed becauseX1andX2are independent conditional on. Uncondi- tionally, they are not independent; they are exchangeable. This example highlights two features:
the assumption of unconditional independence that is commonly made in practice underestimates the reliability of series systems and the importance of when to average out with respect to the unknown parameters. The latter point is also illustrated in Singpurwalla (2000), who considers the prediction of events in a Poisson process with an unknown parameter.
ASSESSING PREDICTIVE DISTRIBUTIONS IN THE ABSENCE OF DATA 133 Note that if Fwere to be a beta distribution with parametersand, then (5.14) would simplify as
P X1=x1 X2=x2=++x1+x2−x1−x2+2
++2