Understanding Uncertainty

Dennis V. Lindley

• Publisher: Wiley
• Year: 2013
• ISBN:
  9781118650233
• Edition: 2
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

Praise for the First Edition

"...a reference for everyone who is interested in knowing and handling uncertainty."
— Journal of Applied Statistics

The critically acclaimed First Edition of Understanding Uncertainty provided a study of uncertainty addressed to scholars in all fields, showing that uncertainty could be measured by probability, and that probability obeyed three basic rules that enabled uncertainty to be handled sensibly in everyday life. These ideas were extended to embrace the scientific method and to show how decisions, containing an uncertain element, could be rationally made.

Featuring new material, the Revised Edition remains the go-to guide for uncertainty and decision making, providing further applications at an accessible level including:

• A critical study of transitivity, a basic concept in probability
• A discussion of how the failure of the financial sector to use the proper approach to uncertainty may have contributed to the recent recession
• A consideration of betting, showing that a bookmaker's odds are not expressions of probability
• Applications of the book's thesis to statistics
• A demonstration that some techniques currently popular in statistics, like significance tests, may be unsound, even seriously misleading, because they violate the rules of probability

Understanding Uncertainty, Revised Edition is ideal for students studying probability or statistics and for anyone interested in one of the most fascinating and vibrant fields of study in contemporary science and mathematics.

Chapter 1

Uncertainty

1.1 Introduction

There are some statements that you know to be true, others that you know to be false, but with the majority of statements you do not know whether they are true or false; we say that, for you, these statements are uncertain. This book is about understanding uncertainty in this sense, about handling it, and, above all, about helping you to live comfortably with uncertainty so that you can better cope with it in your everyday life.

There are two comments that need to be made immediately. The first arises from the fact that the set of statements that you know to be true differs from my set, for you know things that I do not. Equally, things that are uncertain for you may be known to me; but there is more to it than that, for if we take a statement about which we are both uncertain, you may have more confidence that it is true than I do; we differ in our degrees of uncertainty. The upshot of these considerations is that uncertainty is a personal matter; it is not the uncertainty but your uncertainty. Admittedly, there are some situations where almost all agree on the uncertainty but these are rare and confined to special scenarios, for example, some aspects of gambling. Statements of uncertainty are personalistic; they belong to the person making them and express a relationship between that person and the real world about which a statement is being made. In particular, they are not objective in the sense that they express a property that is the same for all of us. It follows that throughout this book we will be referring to a person, conveniently called “you”, whose uncertainty is being discussed; it may sometimes be appropriate for you, the reader, to interpret it as referring to yourself but generally it applies to some unidentified person, or group of persons expressing a common opinion. You are uncertain about some aspect of the world and that uncertainty does not refer solely to you, or solely to the world, but describes a relationship between you and that world.

The second comment is to note that for any of us, for any “you”, the number of statements about which you are uncertain is vastly in excess of the number of statements for which their truth or falsity is known to you; thus all statements about the future are uncertain to some degree. Uncertainty is everywhere, so it is surprising that it is only in the twentieth century that the concept has been systematically studied and, as a result, better understood. Special types of uncertainty, like those arising in gambling, had been investigated earlier but the understanding of the broad notion, applicable to everyday life, is essentially a modern phenomenon. Because uncertainty is everywhere and affects everyone, a proper appreciation of it is vital for all persons, so this book is addressed to everyone who is prepared to listen to a reasoned argument about an ubiquitous concept. This book is for you, whoever you are. We begin with a collection of examples of uncertainty designed to demonstrate how varied, important, and numerous are statements where you genuinely do not know the truth.

1.2 Examples

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Example 1. It will Rain Tomorrow

For all of us who live in climates with changeable weather, this statement is uncertain. It has become almost a classic example of uncertainty because weather is of interest, even importance, to many of us; because meteorologists have seriously studied the question of how to make forecasts like this; and because it is a statement whose uncertainty will be removed after tomorrow has passed, so that it is possible to check on the quality of the statement, a feature of which meteorologists are very conscious and which will be discussed in §5.12. Notice too, that you can change the degree of your uncertainty about rain by looking out of the window, by consulting a barometer, or by switching on the TV, and we will see in Chapter 6 just how this change may be effected.

A careful discussion here would require clarification of what is meant by “rain”; will a trace suffice, or is at least 0.01 cm in the rain gauge needed before rain can be said to have fallen? Which place is being referred to and where will the gauge be placed? What is meant by “tomorrow”—from midnight to midnight, or 24 hours from 7 A.M., as might be administratively more convenient? In this chapter we deal with illustrative examples and can be casual, but later, when more precision is introduced, these matters will assume some importance, for example, when the skills of meteorologists in predicting the weather are being assessed, or when the quality of mercy in a court of law is described. Again we return to the point in §5.12.

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Example 2. The Capital of Liberia is Monrovia

The first example, being about the future, is uncertain for everyone living in a variable climate, but with Liberia the personal nature of uncertainty is immediately apparent, as many, but not all of us, are unsure about African politics. Your ignorance could easily be removed by consulting a reference source and, for this reason, such statements, commonly put in the form of a question, are termed almanac questions. The game of Trivial Pursuit is built around statements of this type and exploits the players' uncertainties.

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Example 3. The Defendant is Guilty

This is uncertainty in a court of law, and “guilt” here refers to what truly happened, not to the subsequent judgment of the court. Although Example 1 referred to the future and Example 2 to the present, this refers to the past. In the two earlier examples, the truth or falsity of the statement will ultimately be revealed; here it will usually remain forever uncertain, though the primary function of the court is, by the provision of evidence, to remove much of that uncertainty with the court's decision. The process of trial in a court of law will be discussed in §§6.6 and 10.14.

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Example 4. The Addition of Selenium to Your Diet will Reduce Your Chance of Getting Cancer

This is typical of many medical statements of interest today; in another example, selenium may be replaced by vitamin C and cancer by the common cold. Generally a treatment is held to affect a condition. Some medical statements you believe to be true because they are based on a large body of evidence, whereas others you may consider false and just quackery; but most are uncertain for you. They refer to topics that might come within the purview of science, where a scientist might rephrase the example in a less personal way as “selenium prevents cancer”. This last statement is a scientific hypothesis, is uncertain, and could be tested in a clinical trial, where the scientist would additionally be uncertain about the number of cancers that the trial will expose. Contrary to much popular belief, science is full of uncertainty and is discussed in Chapter 11. Scientific experiments and the legal trial of Example 3 are both methods for reducing uncertainty.

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Example 5. The Princes in the Tower were Murdered on the Orders of Richard III

Richard III was the king of England and mystery surrounds the deaths of two princes in the Tower of London during his reign. Much of what happened in history is uncertain and this statement is typical in that it deals with a specific incident whose truth is not completely known. The arguments to be presented in this book are often thought to be restricted to topics like gambling (Example 7), or perhaps science (Example 4), but not relevant to cultural matters like history, art (Example 6), or the law (Example 3). In fact, they have the potential to apply wherever uncertainty is present, which is everywhere. Admittedly historians are rarely explicit about their doubts but one historian, in accord with the thesis to be developed here, said that his probability, that the above statement about the princes was true, was 98%.

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Example 6. Many Eighteenth Century Painters Used Lenses and Mirrors

Until recently this was thought unlikely to be true but recent studies have produced evidence that strongly supports the idea. Science and art are not necessarily hostile; aside from optics and paint, they come together in the uncertainty that is present in them both.

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Example 7. A Card Drawn from a Well-Shuffled Pack will be an Ace

This example is typical of those that were discussed in the first systematic studies of uncertainty in the seventeenth century, in connection with gambling, and differs from the previous ones in that the degree of uncertainty has been measured and agreed by almost everyone. Because there are four aces in a pack of 52 cards, the chance of an ace is 4 divided by 52, or 1 in 13. Alternatively expressed, since there is one ace for every 12 cards of other denominations, the odds are 12 to 1 against an ace. (“Odds” and “chance” are here being used informally; their precise meaning will be discussed in §3.8.) It is usual to refer to the chance but, once you accept the common value, it becomes your chance. Some people associate personal luck with cards, so that for them, their chance may not be 1 in 13.

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Example 8. The Horse, High Street, will Win the 2:30 Race

Horse racing is an activity where the uncertainty is openly recognized and sometimes used to add to the excitement of the race by betting on the outcome. Notice that if High Street is quoted at odds of 12 to 1, so that a stake of 1 dollar will yield 12 if High Street wins, this largely reflects the amount of money placed on the horse, not any individual's uncertainty; certainly not the bookmaker's, who expects to make a profit. Your own odds will help you decide whether or not to bet at 12 to 1. The distinction between betting odds and odds as belief is explored in §3.8. Betting is discussed in §14.5.

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Example 9. Shares in Pharmaceutical Companies will Rise Over the Next Month

The buying and selling of stocks and shares are uncertain activities because you do not know whether they will rise or fall in value. In some ways, the stock exchange is like the race course (Example 8), but there is a difference in that the odds are clearly displayed for each horse, whereas the quantitative expression of doubt for the stock can only be inferred from its price now and how it has moved in the past, together with general information about the market. Gambling in the stock market differs from that at the casino (Example 7) because the chances at the latter are generally agreed whereas the existence of buyers and sellers of the same stock at the same time testifies to lack of agreement.

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Example 10. Inflation Next Year will be 3.7%

Statements of this type, with their emphatic “will be”, often appear in the media, or even in specialist publications, and are often called either predictions or forecasts (as with the weather, Example 1). They are surely uncertain but the confident nature of the statement tends to disguise this and makes the 3.7% appear firm, whereas everyone, were they to think about it, would realize that 3.8%, or even 4.5%, is a serious possibility. The assertion can be improved by inserting “about” before the figure, but this is still unsatisfactory because it does not indicate how much variation from 3.7% is anticipated. In general, predictions or forecasts should be avoided, because they have an air of spurious precision, and replaced by claims of the form “inflation next year will most likely be between 3.1% and 4.3%”, though even here “most likely” is imprecise. Exactly how uncertainty statements about a quantity, here an inflation index, should be made will be discussed in Chapter 9. Many people are reluctant to admit uncertainty, at least explicitly.

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Example 11. The Proportion of HIV Cases in the Population Currently Exceeds 10%

At first glance this example appears similar to the previous one but notice it is not an assertion about the future but one concerning the present, the uncertainty arising partly because not every member of the population will have been tested. It improves on Example 10 by making a claim about a range of values, above 10%, rather than a single value. People are often surprised by how little we know about the present, yet at the same time, do not want the uncertainty removed because the only method of doing so involves an invasion of privacy, here the testing for HIV. Uncertainty arising from an inability to question the whole population is considered in Chapter 9.

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Example 12. If an Election were to be Held Tomorrow, 48% Would Vote Democrat

There are two main causes for the uncertainty here, both of which are frequently commented upon and thought by many to make polls unsatisfactory. The first is the recognition that in reaching the 48% figure the pollsters only asked very few people, perhaps thousands in a population of millions; the second is caused by people either not telling the truth or changing their views between the question being posed and the action of voting. Methods for handling the first issue have been developed, and the polling firms are among the most sophisticated handlers of uncertainty in the world.

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Example 13. There will be a Serious Nuclear Accident in Britain Next Year

The uncertainty here is generally admitted and discussed. Two important features are the extreme seriousness of the statement if true, and the very small chance that it will be true. The balance between these two aspects is not easy to resolve and is of very real concern in a society where people are more comfortable with small risks of moderate chance like road accidents, than with accidents of a nuclear type. Methods are developed to handle this in §5.5.

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Example 14. Jesus was the Son of God

For at least some Christians, this statement is not uncertain, nor is it for atheists, whereas for agnostics, it is uncertain. It is included here because some people hold that the certainty felt by believers here is different in kind from the certainty they feel about Monrovia being the capital of Liberia (Example 2), at least after the almanac has been consulted, one being based on faith, the other on facts. This is a sensible distinction, for it is unsatisfactory to equate faith with checking an almanac. Nevertheless, some of the ideas to be considered in this book may be relevant to discussions concerning faiths.

Incidentally, it was ...