Marvin Rausand - Risk Assessment

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If we do a single experiment, we say that the probability of getting the outcome is and consider this probability a property of the experiment.

In a risk analysis, we almost never have a finite sample space of outcomes that occur with the same probability. The classical approach to probability is therefore not appropriate. Furthermore, to apply the frequentist approach, we must at least be able to imagine that experiments can be repeated a large number of times under nearly identical conditions. Because this is rarely possible, we are left with a final option, the Bayesian approach. In this approach, the probability is considered to be subjective and is defined as:

A numerical value in the interval representing an individual's degree of belief about whether or not an event will occur.

In the Bayesian approach, it is not necessary to delimit probability to outcomes of experiments that are repeatable under the same conditions. It is fully acceptable to give the probability of an event that can only happen once. It is also acceptable to talk about the probability of events that are not the outcomes of experiments, but rather are statements or propositions. This can be a statement about the value of a nonobservable parameter, often referred to as a state of nature . To avoid a too‐complicated terminology, we also use the word event for statements, saying that an event occurs when a statement is true.

The degree of belief about an event is not arbitrary but is the analyst's best guess based on her available knowledge about the event. The analyst's (subjective) probability of the event , given that her knowledge is , should therefore be expressed as

(2.3)

The knowledge may come from knowledge about the physical properties of the event, earlier experience with the same type of event, expert judgment, and many other information sources. For simplicity, we often suppress and simply write , but we should not forget that this is a conditional probability depending on .

In 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.

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 : “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 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 , this is a measure of your belief about . 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 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 . 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 and that her initial or prior belief about this event is given by her prior probability :

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

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