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Suppose we are given a prior distribution about an unknown parameter $\pi(\theta)$. Also we are given $f(x_{1}, \dots, x_n|\theta)$. We want to find $\pi(\theta|x_1, \dots, x_n)$. Now $$\pi(\theta|x_1, \dots, x_n) = \frac{f(x_{1}, \dots, x_n|\theta) \ \pi(\theta)}{\int_{0}^{\infty} f(x_{1}, \dots, x_n|\theta) \ \pi(\theta) \ d \theta}$$

The denominator, is in general, hard to compute. But why do many people use Gibbs sampling to approximate the denominator as opposed to the Metropolis/Metropolis Hastings Algorithm where $\alpha \neq 1$?

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What's $\alpha$? – joriki May 6 '12 at 20:57

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