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Partly this is because (despite the fact that its density may seem somewhat barbaric at first sight) it is in many contexts the easiest distribution to work with, but this is not the whole story. The Central Limit Theorem says (roughly) that if a random variable can be expressed as a sum of a large number of components no one of which is likely to be much bigger than the others, these components being approximately independent, then this sum will be approximately normally distributed. Because of this theorem, an observation which has an error contributed to by many minor causes is likely to be normally distributed.

Thus, there is a 99% chance that he is guilty’. Alternatively, the defender may state: ‘This crime occurred in a city of 800,000 people. This blood type would be found in approximately 8000 people. ’ The first of these is known as the prosecutor’s fallacy or the fallacy of the transposed conditional and, as pointed out above, in essence it consists in quoting the probability P(E|I ) instead of P(I |E). The two are, however, equal if and only if the prior probability P(I ) happens to equal P(E), which will only rarely be the case.

Naturally, if no conditioning event is explicitly mentioned, the probabilities concerned are conditional on as defined above. 6 Some simple consequences of the axioms; Bayes’ Theorem We have already noted a few consequences of the axioms, but it is useful at this point to note a few more. We first note that it follows simply from P4 and P2 and the fact that H H = H that P(E|H ) = P(E H |H ) and in particular P(E) = P(E ). 8 BAYESIAN STATISTICS Next note that if, given H, E implies F, that is E H ⊂ F and so E F H = E H , then by P4 and the aforementioned equation P(E|F H ) P(F|H ) = P(E F|H ) = P(E F H |H ) = P(E H |H ) = P(E|H ).

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A Bayesian Alternative to Parametric Hypothesis Testing by Rueda R.


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