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By Le Bellac M.

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These computational difficulties are studied in Chapter 6, where we provide some general solutions. Still, they should not be considered a major drawback of the Bayesian approach. Indeed, Computational Statistics is currently undergoing such a rapid development that we can clearly reject the notion of a prior distribution chosen for its computational tractability, even though we may still rely on these particular distributions to present simpler and clearer examples in this book. On the contrary, it is stimulating to see that we are getting closer to the goal of providing more powerful and efficient statistical tools because of these new computational techniques, as they allow for the use of more complex prior distributions, which are in turn more representative of the available prior information.

If the first card drawn out of the urn is white, what is the probability that the proportion p of white cards is p0 ? , that p is uniformly distributed on {2/n, . . , (n − 1)/n}. The posterior distribution of p can then be derived using Bayes’s Theorem and p0 × 1/(n − 2) P (p = p0 | data ) = (n−1)/n p=2/n p × 1/(n − 2) n p0 . = n(n − 1)/2 − 1 9 It is also possible to picture a more Machiavellian Bayes who picked up this particular example in order to circumvent potential criticisms of the choice of the prior.

Berger (2000)) is that improper priors should be preferred to vague proper priors such as, a N (0, 1002 ) distribution say, because the later gives a false sense of safety owing to properness, while lacking robustness in terms of influence on the resulting inference. , Lindley (1965)), but the inclusion of improper distributions in the Bayesian paradigm allows for a closure of the inferential scope (figuratively as well as topologically). From a more practical perspective, the fact that the prior distribution is improper weakens the above symmetry between the observations and the parameters, but as long as the posterior distribution is defined, Bayesian methods apply as well.

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A short introduction to quantum information and quantum computation: solutions of exercises by Le Bellac M.


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