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Extra info for A Bayesian Alternative to Parametric Hypothesis Testing

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Similarly, each random quantity is interpretable as a real‐valued function of the points Q:X = X(Q) is the value which X assumes if the true point is Q. The preceding case, E = E(Q), is simply the particular case which arises when the function can only take on the values 0 and 1. The same is true for random entities of any other kind: for example, a random vector is a vector which is a function of the point Q. 2. That all this can be useful and convenient as a form of representation is beyond question.

By saying that these are exhaustive (or, better, form an exhaustive family – but the phrase is cumbersome), we mean to assert that at least one of them must take place; that is, in the preceding notation, ⊢ Y ⩾ 1. This shows the relationship between the two conditions. e. 1. Partitions. A partition is a family of incompatible and exhaustive events – that is for which it is certain that one and only one event occurs. The coexistence of the conditions ⊢ Y ⩽ 1 and ⊢ Y ⩾ 1 means, in fact, ⊢ Y = 1. A partition can be finite or infinite: partitions (and, for the simplest conclusions, in particular finite partitions) have a fundamental importance in the calculus of probability (which, as already indicated, will consist in distributing a unit ‘mass’ of probability among the different events of each partition).

In the case of a single die, the ‘points’ are precisely 1, 2,…, 6, whereas for the two dice it is irrelevant whether we use 2, 3,…, 12, or 1, 2,…, 11. The colours, or results of the game, could similarly be coded numerically. In speaking of an m‐event we want, essentially, to emphasize the qualitative aspects of the alternatives. It is then appropriate to use the mathematical interpretation of them as unit vectors (1, 0,…, 0), (0, 1, 0,…,0),…, (0, 0, 0,…, 1) in an m‐dimensional space. In this way, writing Eh (h = 1, 2,…,m) for the events19 which consist in the occurrence of the hth alternative, an m‐event can be identified with the random vector (E1, E2,…, Em).