By Van Der Merwe A. J., Du Plessis J. L.

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**Extra resources for A Bayesian Approach to Selection and Ranking Procedures: The Unequal Variance Case**

**Example text**

G. A2 = Ac1 , the probabilities P(A1 ) and P(A2 ) are given by P(A1 ) = q1 , q1 + q2 P(A2 ) = q2 , q1 + q2 respectively. In the following theorem we generalize this result to more than two events. 2. Let A1 , A2 , . . e. P(Aj )/P(Ai ) = qj /qi . Then P(Ai ) = ✧ qi . 1. Consider an urn with balls of three colours. 50 % of the balls are red, 30 % black, and the remaining balls green. The experiment is to draw a ball from the urn. Clearly A1 , A2 , and A3 , deﬁned as the ball being red, black, or green, respectively, forms a partition.

For n = 2 this means that P(B1 ∩ B2 | Ai ) = P(B1 | Ai )P(B2 | Ai ). This property will be called conditional independence. 3. Let A1 , A2 , . . , Ak be a partition of S , and B1 , . . , Bn , . . a sequence of true statements (evidences). If the statements B are conditionally independent of Ai then the a posteriori odds after receiving the n th evidence qin = P(Bn | Ai )qin−1 , n = 1, 2, . . , where qi0 are the a priori odds. e. replaced by the posterior odds. This recursive estimation of the odds for Ai is correct only if the evidences B1 , B2 , .

Do these two approaches give diﬀerent heights for 100-year levels? We answer this question next. Consider a stationary stream, let t = 1 year and ucrt be chosen so that Pt (A) = 1/100. e. e. ucrt . However, if the stream is Poisson then Pt (A) = 1 − e−1/TA ≈ 1 TA and the diﬀerence is very small and not important in practice. The true advantage of the ﬁrst deﬁnition is that it can be used even for non-stationary The sign ≈ means that the value of the probability is estimated and hence uncertain.