Reliability and Risk: A Bayesian Perspective

Preface

Over the past few years, there has been an increasing emphasis on what is commonly referred

to as ‘risk’, and how best to manage it. The management of risk calls for its quantification, and

this in turn entails the quantification of its two elements: the uncertainty of outcomes and the

consequences of each outcome. The outcomes of interest here are adverse events such as the

failure of an infrastructure element, e.g. a dam; or the failure of a complex system, e.g. a nuclear

power plant; or the failure of a biological entity, e.g. a human being. ‘Reliability’ pertains to

the quantification of the occurrence of adverse events in the context of engineering and physical

systems. In the biological context, the quantification of adverse outcomes is done under the label

of ‘survival analysis’. The mathematical underpinnings of both reliability and survival analysis

are the same; the methodologies could sometimes be very different. A quantification of the

consequences of adverse events is done under the aegis of what is known as utility theory.

The literature on reliability and survival analysis is diverse, scattered and plentiful. It ranges

over outlets in engineering, statistics (to include biostatistics), mathematics, philosophy, demography,

law and public policy. The literature on utility theory is also plentiful, but it is concentrated

in outlets that are of interest to decision theorists, economists and philosophers. However, despite

what seems like a required connection, there appears to be a dearth of material describing a

linkage between reliability (and survival analysis) and utility. One of the aims of this book is

to help start the process of bridging this gap. This is done in two ways. The first is to develop

material in reliability with the view that the ultimate goal of doing a reliability analysis is to

appreciate the nature of the underlying risk and to propose strategies for managing it. The second

is to introduce the notion of the ‘utility of reliability’, and to describe how this notion can be cast

within a decision theoretic framework for managing risk. But in order to do the latter, we need

to make a distinction between reliability as an objective chance or propensity, and survivability

as the expected value of one’s subjective probability about this propensity. In other words, from

the point of view of this book

reliability is a chance not a probability

and that probability, which describes one’s uncertainty about reliability, is personal. It is this

de Finettian-style perspective of an objective propensity and a subjective probability that distinguishes

the material here from that of its several competitors. Hopefully, it changes the way in

which one looks at the subject of reliability and survival analysis; a minor shift in paradigm, if

you will.

With the above as a driving principle, the underlying methodology takes a Bayesian flavor,

and the second aim of this book is to summarize this methodology in its broader context. Much

of this Bayesian methodology is directed toward predicting lifetimes of surviving units. Here,

the probabilistic notion of ‘exchangeability’ plays a key role. Consequently, a chapter has been

devoted to this important topic, namely, Chapter 3.

The personalistic interpretation of probability is controversial, and unlike what is done by us,

not all Bayesians subscribe to it. Thus it is deemed important to trace the historical evolution

of personal probability, and to contrast it with the other interpretations. With that in mind, a

brief history of probability has been included here. This historical material is embedded in a

chapter that summarizes the quantification of uncertainty from a Bayesian perspective. Readers

whose exposure to the essentials of Bayesian inference is limited may find the material of this

chapter useful – so I hope. At the very least, Chapter 2 serves as the springboard in which the

terminology and notation used throughout the book are established.

The heart of the book starts with Chapter 4, on stochastic models of failure. Included herein

are mathematical concepts such as absolute continuity, singularity, the Lebesgue integral and

the Lebesgue–Stieltjes integral, notions that may appear to be distracting. However, these are

required for a better appreciation of some modern developments in reliability, in particular, the

treatment of interdependence. They are included here because most analysts who do reliability,

risk and survival analysis tend to be applied statisticians, engineers, and other physical and

biological scientists whose exposure to concepts in mathematical analysis may be limited.

As a final point to this preface, I feel obliged to be upfront about some limitations of this

book. For one, its current version does not have much by the way of examples, exercises or

data. I hope this limitation will be addressed in a subsequent edition. For now, the focus has

been on breadth of coverage and the spectrum of issues that the general topic of reliability and

survivability can spawn. An example is the use of concepts and notions in reliability theory that

are germane to econometrics and finance, in particular the assessment of income inequalities

and financial risk. Chapter 11 is devoted to these topics; hopefully, it gives this book an unusual

flavor among books in reliability and survival analysis. Another limitation of this book is that

in some instances, I may have merely mentioned an issue or a topic without dwelling on its

details in any substantive manner. This, I am aware, could frustate some readers. All the same,

I have chosen to do so in order to keep the material to a reasonable size and at the same time

maintain a focus on breadth and scope. As a compensatory measure, a large list of references has

been provided. On the matter of maintaining breadth and scope, I may have also ventured into

territory that some may consider unproven. My hope is that such excursions into the unknown

may provide a platform to others for new and additional research. Finally, given the size of this

book, and the amount of time it has taken to develop it, there are bound to be errors, typos,

inconsistencies and mistakes. For this I apologize to the readers and ask for their tolerance and

understanding. I would of course greatly value receiving comments pointing to any and all of

the above, and advice upon how to improvise upon the current text.

The material of this book can be most profitably used by practioners and research workers

in reliability and survivability as a source of information and open problems. It can also form

the basis of a graduate-level course in reliability and risk analysis for engineers, statisticians and

other mathematically orientated scientists, wherein the instructor supplements the material with

examples, exercises and real problems.