FLOOD FORECASTING MODELS AND UNCERTAINY REPRESENTATION - Modelling Uncertainty in Flood Forecasting Systems - page 18

Figure 2.2 . Venn diagram of (a) mutually exclusive events, and (b) intersection
of two events.
Subjective or Bayesian view of probability
In the subjective view, probability is used as a belief. An event is a statement and the
(subjective) probability of the event is a measure of the degree of belief that the subject
has in the truth of the statement. The basic idea in the application of this approach is to
assign a probability to any event on the basis of the current state of knowledge and to
update it in the light of the new information. The conventional procedure for updating a
prior probability in the light of new information is by using Baysian theorem.
Bayes' Theorem is named after Thomas Bayes, an 18 th century mathematician
(1702-1761) who derived a special case of this theorem. Bayes' theorem provides a rule
for updating the belief in a hypothesis H (i.e. the probability of H ) given additional
evidence E and background information (context) I, as
(2.6)
The left hand-side term, P(H|E,I), is called the posterior probability and it gives the
probability of the hypothesis H after considering the effect of evidence E in the context I .
The P(H|I) term is just the prior probability of H given I alone; that is, the belief in H
before the evidence E is considered. The term P(E|H,I) is called the likelihood, which
gives the probability of the evidence assuming the hypothesis H and background
information I is true. The denominator of the right-hand term, P(E|I), is the prior
probability of the evidence that can be regarded as a normalising or scaling constant. This
normalising constant is obtained by evaluating the exhaustive and exclusive set of
evidence scenarios (Hall, 1999):
(2.7)
The Bayesian theorem (Equation (2.6)) actually comes from a simple consequence of the
definition of conditional probability. The conditional probability of two sets provides the
dependency relationship between them. Given two sets A and B, the conditional
probability states that