Marvin Rausand - Risk Assessment

(2.4)

that is a simple consequence of the multiplication rule for probabilities

The analyst's updated belief about , after she has access to the evidence , is called the posterior probability .

Thomas Bayes (1702–1761) was a British Presbyterian minister who has become famous for formulating the formula that bears his name – Bayes' formula (often written as Bayes formula). His derivation was published (posthumously) in 1763 in the paper “An essay toward solving a problem in the doctrine of chances” (Bayes 1763). The general version of the formula was developed in 1774 by the French mathematician Pierre-Simon Laplace (1749–1825).

An individual's belief in the occurrence of the event based on her prior belief and some additional evidence .

Initially, the analyst's belief about the event is given by her prior probability . After having obtained the evidence , her probability of is, from ( 2.4), seen to change by a factor of .

Bayes formula ( 2.4) can be used repetitively. Having obtained the evidence and her posterior probability , the analyst may consider this as her current prior probability. When additional evidence becomes available, she may update her current belief in the same way as previously and obtain her new posterior probability:

(2.5)

Further updating of her belief about can be done sequentially as she obtains more and more evidence.

By the posterior probability in ( 2.4), the analyst expresses her belief about the unknown state of nature when the evidence is given and known. The interpretation of in ( 2.4) may therefore be a bit confusing because is known. Instead, we should interpret as the likelihood that the (unknown) state of nature is , when we know that we have got the evidence .

In our daily language, likelihood is often used with the same meaning as probability , even though there is a clear distinction between the two concepts in statistical usage. In statistics, likelihood is used, for example, when we estimate parameters (maximum likelihood principle) and for testing hypotheses (likelihood ratio test).

In the first part of this chapter, we have used the word likelihood as a synonym for probability, as we often do in our daily parlance. The reason for this rather imprecise use of the word “likelihood” is that we wanted to avoid using the word “probability” until it was properly introduced – and because we wanted to present the main definitions of risk concepts with the same wording that is used in many standards and guidelines.

In risk analysis, it is not realistic to assume that the events are repeatable under essentially the same conditions. We cannot, for example, have the same explosion over and over again under the same conditions. This means that we need to use the Bayesian approach. Although most risk analysts agree on this view, there is an ongoing controversy about the interpretation of the subjective probability. There are two main schools:

1 (1) The first school claims that the subjective probability is subjective in a strict sense. Two individuals generally come up with two different numerical values of the subjective probability of an event, even if they have exactly the same knowledge. This view is, for example, advocated by Lindley (2007), who claims that individuals have different preferences and hence judge information in different ways.

2 (2) The second school claims that the subjective probability is dependent only on knowledge. Two individuals with exactly the same knowledge always give the same numerical value of the subjective probability of an event. This view is, for example, advocated by Jaynes (2003), who states:A probability assignment is “subjective” in the sense that it describes a state of knowledge rather than any property of the “real” world but is “objective” in the sense that it is independent of the personality of the user. Two rational human beings faced with the same total background of knowledge must assign the same probabilities [also quoted and supported by Garrick (2008)].