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Bayesian Statistics: An Introduction, 4th Edition by Peter M. Lee

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10.1 Bayesian importance sampling

Importance sampling was first mentioned towards the start of Section 9.1. We said then that it is useful when we want to find a parameter θ which is defined as the expectation of a function f(x) with respect to a density q(x) but we cannot easily generate random variables with that density although we can generate variates xi with a density p(x) which is such that p(x) roughly approximates |f(x)|q(x) over the range of integration. Then

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The function p(x) is called an importance function. In the words of Wikipedia, ‘Importance sampling is a variance reduction technique that can be used in the Monte Carlo method. The idea behind importance sampling is that certain values of the input random variables in a simulation have more impact on the parameter being estimated than others. If these ‘important’ values are emphasized by sampling more frequently, then the estimator variance can be reduced.’

A case where this technique is easily seen to be valuable occurs in evaluating tail areas of the normal density function (although this function is, of course, well-enough tabulated for it to be unnecessary in this case). If, for example, we wanted to find the probability  that a standard normal variate was greater than 4, so that

a naïve Monte Carlo method would ...

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