Statistical tolerance limits for posterior distributions based on partially specified prior distributions  Page 26 
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Let a sample of size m be taken. This sample is of arbitreury size. (Optimal sample sizes will be discussed later. ) What is being proposed herein is that the concept of statistical tolerance limits be used to reflect the uncertainty in the posterior distribution caused by the finiteness of n. Hence^ the statement that is needed is of the following type: The probability is at least y that at least a proportion p of the posterior distribution is contained in the interval P m  mZo to pm + mZor . To this end, we may utilize the method of Proschan (u) who presented the theory of classical tolerance intervals. The problem there is to find a K such that the interval (x  k b , 5 + KA) (5) will contain at least a proportion p of the population with a probability of at least y when a sample of size M is taken from a population which is normally distributed with unknown mean A and known standard deviation B. Now, lOOy percent of the x's are contained within the interval (A  "(i7)/2 > * + ^(l_Y)/2 ) (6) where u^ is the standardized normal deviate such that the probability that it is exceeded is a. Note, lOOT percent of the x’s lie in the interval of Eq. (6), i.e., lOOy percent of the Eq. (5) 21
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Title  Statistical tolerance limits for posterior distributions based on partially specified prior distributions  Page 26 
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Full text  Let a sample of size m be taken. This sample is of arbitreury size. (Optimal sample sizes will be discussed later. ) What is being proposed herein is that the concept of statistical tolerance limits be used to reflect the uncertainty in the posterior distribution caused by the finiteness of n. Hence^ the statement that is needed is of the following type: The probability is at least y that at least a proportion p of the posterior distribution is contained in the interval P m  mZo to pm + mZor . To this end, we may utilize the method of Proschan (u) who presented the theory of classical tolerance intervals. The problem there is to find a K such that the interval (x  k b , 5 + KA) (5) will contain at least a proportion p of the population with a probability of at least y when a sample of size M is taken from a population which is normally distributed with unknown mean A and known standard deviation B. Now, lOOy percent of the x's are contained within the interval (A  "(i7)/2 > * + ^(l_Y)/2 ) (6) where u^ is the standardized normal deviate such that the probability that it is exceeded is a. Note, lOOT percent of the x’s lie in the interval of Eq. (6), i.e., lOOy percent of the Eq. (5) 21 