3.1 Quantifying uncertainty

Quantifying uncertainty is a systematic way of assimilating and comparing uncertainties. This means that different personal uncertainties for the same object can be assessed consistently and also that personal uncertainties for different objects can be compared. This is because the framework of mathematics forces quantities to obey a coherent and consistent set of logical rules. The subset of mathematics which handles uncertainty is known as probability. The main benefit of using probability is the framework of logic that it enforces, rather than the quantification of uncertainty (although this is useful).

A probability is a number between 0 and 1 that describes the magnitude of uncertainty for the occurrence of an event. The probability must obey certain rules which we will show in subsequent examples in this chapter. A probability of 0 means that the event is impossible whilst a probability of 1 means that an event is certain. Uncertainty is described by probabilities which fall between 0 and 1. Probabilities of 0.5 describe an event whose occurrence is exactly as likely as its non-occurrence. Events whose occurrence is less likely than not should have a probability less than 0.5 on the scale, whilst events whose occurrence is more likely than not should have a probability greater than 0.5 on the scale. How close these probabilities are to 0 and 1 should reflect an individual’s magnitude of uncertainty. Since uncertainty is personal then it follows that probabilities are personal too. In forensic science, personal probabilities are generally interpreted as an individual’s degree of belief in the occurrence of an event. Not everyone agrees with this interpretation, but this historical debate is outside the scope of this book.

Constructing probabilities to describe uncertainty is often done by assuming a probabilistic model for how the uncertainty is expected to behave in reality. Since the process that is being modelled is uncertain, the expectations might not be exactly what is observed in practice. The most useful models can accurately align an individual’s magnitude of uncertainty to a quantitative probability, accepting that this can never be done perfectly.

We focus only on quantifying direct uncertainty, since this is what is done in practice when interpreting evidence. Verbal qualifiers are given for indirect uncertainties instead and are not covered here.