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**1 - 1**of**1**### Nonparametric Predictive Inference for Bernoulli quantities: two examples

"... Coolen (1998) presented Nonparametric Predictive Inference (NPI) for Bernoulli random quantities, based on a representation of Bernoulli data as outcomes of an experiment similar to that used by Bayes (1763), with Hill’s assumption A (n) (Hill 1968, 1988) used to derive direct predictive probabiliti ..."

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Coolen (1998) presented Nonparametric Predictive Inference (NPI) for Bernoulli random quantities, based on a representation of Bernoulli data as outcomes of an experiment similar to that used by Bayes (1763), with Hill’s assumption A (n) (Hill 1968, 1988) used to derive direct predictive probabilities for future observations based on available data. The lower and upper probabilities presented by Coolen (1998) have strong internal consistency properties in the theory of interval probability (Augustin and Coolen 2004, Weichselberger 2001). Due to the use of A (n) in deriving these lower and upper probabilities, they fit in a frequentist framework of statistics but can also be interpreted from Bayesian perspective (Hill 1988, Coolen 2006). NPI is also ‘perfectly calibrated ’ in the sense of Lawless and Fredette (2005). In this paper, we briefly give the main results from Coolen (1998), and we illustrate their use in two recently developed applications. Suppose that we have a sequence of n + m exchangeable Bernoulli trials, each with ‘success’ and ‘failure ’ as possible outcomes, and data consisting of s successes in n trials. Let Y n 1 denote the random number of successes in trials 1 to n, then a sufficient representation of the data for our inferences is Y n n+m 1 = s, due to the assumed exchangeability of all trials. Let Yn+1 denote the random number of successes in trials n + 1 to n + m. Let Rt = {r1,..., rt}, with 1 ≤ t ≤ m + 1 and 0 ≤ r1 < r2 <... < rt ≤ m, and, for ease of notation, let us define � � s+r0 s = 0. Then the NPI-based upper probability (Coolen 1998) for the event Y n+m n+1 ∈ Rt, given data Y n 1 = s, for s ∈ {0,..., n}, is P (Y n+m � �−1 � � � � � � � � n n + m �t s + rj s + rj−1 n − s + m − rj