FIGURE 5.2 Venn diagram illustrating the development of conditional probability

FIGURE 5.2 Venn diagram illustrating the development of conditional probability. 

For the Bayesian, the relationship is taken as an axiom, but its motivation reflects the real world with the foreshadowing of rules implied by the above frequentist treatment. Given the 2 events or propositions, A and B, then 

Application of Uncertainty Analysis to Ecological Risk of Pesticides 

Equation (5.10) is a statement of Bayes theorem. Since the theorem is proved using results or axioms valid for both frequentist and Bayesian views, its use is not limited to Bayesian applications. Note that it relates 2 conditional probabilities where the events A and B are interchanged. 

Bayesian interpretation and application of the theorem quantifies the development of information. Suppose that A is a statement or hypothesis, and let p A) stand for the degree of belief in the statement or hypothesis A, based on prior knowledge, it is called the prior probability. Let B represent a set of observations, then p(B A) is the probability that those observations occur given that A is true. This is called the likelihood of the data and is a function of the hypothesis. The left side, p(A B), is the new degree of belief in A, taking into account the observations B, it is called the posterior probability. Thus Bayes theorem tracks the effect that the observations have upon the changing knowledge about the hypothesis. The theorem can be expressed thus  

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