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Computer Age Statistical Inference by Trevor Hastie, Bradley Efron

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13 Objective Bayes Inference and Markov Chain Monte Carlo

From its very beginnings, Bayesian inference exerted a powerful influence on statistical thinking. The notion of a single coherent methodology employing only the rules of probability to go from assumption to conclusion was and is immensely attractive. For 200 years, however, two impediments stood between Bayesian theory’s philosophical attraction and its practical application.

1  In the absence of relevant past experience, the choice of a prior distribution introduces an unwanted subjective element into scientific inference.

2  Bayes’ rule (3.5) looks simple enough, but carrying out the numerical calculation of a posterior distribution often involves intricate higher-dimensional integrals. ...

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