Risk Assessment. Marvin Rausand

Risk Assessment

a risk analysis, the word subjective may have a negative connotation. For this reason, some analysts prefer to use the word personal probability, because the probability is a personal judgment of an event that is based on the analyst's best knowledge and all the information she has available. The word judgmental probability is also sometimes used. To stress that the probability in the Bayesian approach is subjective (or personal or judgmental), we refer to the analyst's or her/his/your/my probability instead of the probability.

Example 2.9 (Your subjective probability)

Assume that you are going to do a job tomorrow at 10 o'clock and that it is very important that it is not raining when you do this job. You want to find your (subjective) probability of the event images : “rain tomorrow between 10:00 and 10:15.” This has no meaning in the frequentist (or classical) approach, because the “experiment” cannot be repeated. In the Bayesian approach, your probability images is a measure of your belief about the weather between 10:00 and 10:15. When you quantify this belief and, for example, say that images , this is a measure of your belief about images . To come up with this probability, you may have studied historical weather reports for this area, checked the weather forecasts, looked at the sky, and so on. Based on all the information you can get hold of, you believe that there is an 8% chance that event images occurs and that it will be raining between 10:00 and 10:15 tomorrow.

The Bayesian approach can also be used when we have repeatable experiments. If we flip a coin, and we know that the coin is symmetric, we believe that the probability of getting a head is images . In this case, the frequentist and the Bayesian approach give the same result.

An attractive feature of the Bayesian approach is the ability to update the subjective probability when more evidence becomes available. Assume that an analyst considers an event images and that her initial or prior belief about this event is given by her prior probability images :

Definition 2.25 (Prior probability)

An individual's belief in the occurrence of an event images prior to any additional collection of evidence related to images .

Later, the analyst gets access to the data images , which contains information about event images . She can now use Bayes formula to state her updated belief, in light of the evidence images , expressed by the conditional probability

(2.4) equation

that is a simple consequence of the multiplication rule for probabilities

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

Thomas Bayes

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).

Definition 2.26 (Posterior probability)

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

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

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

(2.5) equation

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

2.4.1.4 Likelihood

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