Revised Probabilites

In general, we start analyzing probabilistic situations by assigning probabilities of certain evetns according to our earlier experience under similar conditions, These probabilities may or may not be accurate. We can improve then by obtaining additional information through experiments or surveys. After a sufficient number of these improvements, we can reach a reasonable level of accuracy for the probability distribution of the outcomes. The methods for updating the belief in a hypothesis in the light of evidence obtained are referred to as Bayesian approaches. The Bayesian theorem provides a means of converting the degree of confidence probability that the decision maker had before any test data were obtained (prior probability) to the degree of confidence probability after the test data were obtained (posterior probability). It implies that the Bayesian approach combines experience with hard data to provide estimates similar to those obtained from the traditional statistical inference approach. It should be emphasized that the Bayesian approach requires extensive historical or subjective estimations for all the conditional probabilities and assumes that the hypotheses are independent. Bayesian probability inherently relies on the principle of the excluded middle; that is, some event is either true or false.

An extension of the Bayesian approach that removes the strong assumption of the principle of the excluded middle is the Dempster-Shafer theory of evidence (Shafer 1976; Klir and Yuan 1995). This theory makes a distinction between probability and ignorance. Rather than limiting belief in a hypothesis to a single number, Dempter-Shafter placers upper and lower bounds on belief. In their method, they separate belief in a hypothesis, belief in its negation, and ignorance about the hypothesis. It coincides probability theories when there is sufficient evidence for it to produce point estimations. However, because it is fundamentally very similar to probability theror, it suffers from the same need for large numbers probability assignments and from the need fro independence assumption. However, by considering ignorance, it avoids the inherent excluded middle principle ) probability theory’s treatment of randomness.

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