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 : “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 :
Definition 2.25 (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
(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
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 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.
2.4.1.4 Likelihood
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
