## Uncertainty about uncertainty

### Introduction

More often than not, managerial decisions are made on the basis of uncertain information. To lend some rigour to the process of decision making, it is sometimes assumed that uncertainties of interest can be quantified accurately using probabilities. As it turns out, this assumption can be incorrect in many situations because the probabilities themselves can be uncertain. In this post I discuss a couple of ways in which such *uncertainty about uncertainty* can manifest itself.

### The problem of vagueness

In a paper entitled, “Is Probability the Only Coherent Approach to Uncertainty?”, Mark Colyvan made a distinction between two types of uncertainty:

- Uncertainty about some underlying fact. For example, we might be uncertain about the cost of a project – that there will be a cost is a fact, but we are uncertain about what exactly it will be.
- Uncertainty about situations where there is no underlying fact. For example, we might be uncertain about whether customers will be satisfied with the outcome of a project. The problem here is the
*definition*of customer satisfaction. How do we measure it? What about customers who are neither satisfied nor dissatisfied? There is no clear-cut definition of what customer satisfaction actually is.

The first type of uncertainty refers to the lack of knowledge about something that we know exists. This is sometimes referred to as *epistemic uncertainty* – i.e. uncertainty pertaining to knowledge. Such uncertainty arises from imprecise measurements, changes in the object of interest etc. The key point is that we know for certain that the item of interest has well-defined properties, but we don’t know what they are and hence the uncertainty. Such uncertainty can be quantified accurately using probability.

Vagueness, on the other hand, arises from an imprecise use of language. Specifically, the term refers to the use of criteria that cannot distinguish between borderline cases. Let’s clarify this using the example discussed earlier. A popular way to measure customer satisfaction is through surveys. Such surveys may be able to tell us that customer A is more satisfied than customer B. However, they cannot distinguish between borderline cases because *any boundary between satisfied and not satisfied customers is arbitrary*. This problem becomes apparent when considering an indifferent customer. How should such a customer be classified – satisfied or not satisfied? Further, what about customers who choose not to respond? It is therefore clear that any numerical probability computed from such data cannot be considered accurate. In other words, the probability itself is uncertain.

### Ambiguity in classification

Although the distinction made by Colyvan is important, there is a deeper issue that can afflict uncertainties that appear to be quantifiable at first sight. To understand how this happens, we’ll first need to take a brief look at how probabilities are usually computed.

An operational definition of probability is that it is *the ratio of the number of times the event of interest occurs to the total number of events observed*. For example, if my manager notes my arrival times at work over 100 days and finds that I arrive before 8:00 am on 62 days then he could infer that the probability my arriving before 8:00 am is 0.62. Since the probability is assumed to equal the frequency of occurrence of the event of interest, this is sometimes called the *frequentist interpretation* of probability.

The above seems straightforward enough, so you might be asking: where’s the problem?

The problem is that *events can generally be classified in several different ways and the computed probability of an event occurring can depend on the way that it is classified*. This is called the reference class problem. In a paper entitled, “The Reference Class Problem is Your Problem Too”, Alan Hajek described the reference class problem as follows:

“The reference class problem arises when we want to assign a probability to a proposition (or sentence, or event) X, which may be classified in various ways, yet its probability can change depending on how it is classified.”

Consider the situation I mentioned earlier. My manager’s approach seems reasonable, but there is a problem with it: *all days are not the same as far as my arrival times are concerned*. For example, it is quite possible that my arrival time is affected by the weather: I may arrive later on rainy days than on sunny ones. So, to get a better estimate my manager should also factor in the weather. He would then end up with two probabilities, one for fine weather and the other for foul. However, that is not all: there are a number of other criteria that could affect my arrival times – for example, my state of health (I may call in sick and not come in to work at all), whether I worked late the previous day etc.

What seemed like a straightforward problem is no longer so because of the uncertainty regarding which reference class is the right one to use.

Before closing this section, I should mention that the reference class problem has implications for many professional disciplines. I have discussed its relevance to project management in my post entitled, “The reference class problem and its implications for project management”.

### To conclude

In this post we have looked at a couple of forms of uncertainty about uncertainty that have practical implications for decision makers. In particular, we have seen that probabilities used in managerial decision making can be uncertain because of vague definitions of events and/or ambiguities in their classification. The bottom line for those who use probabilities to support decision-making is to ensure that the criteria used to determine events of interest refer to* unambiguous facts* that are *appropriate to the situation at hand*. To sum up: decisions made on the basis of probabilities are only as good as the assumptions that go into them, and the assumptions themselves may be prone to uncertainties such as the ones described in this article.

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