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| ====== mcmc ====== | ====== mcmc ====== | ||
| + | http:// | ||
| + | $ E_\pi[T(X)] = \int T(x)\pi(x) dx. $ | ||
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| + | In Bayesian inference, we are interested in posterior mean $E(\theta|y)$ or posterior variance $Var(\theta|y)$. | ||
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| + | One solution is to draw independent samples $ ( X^{(1)}, X^{(2)}, \cdots, X^{(N)} )$ from $\pi(x)$, then we can approximate | ||
| + | $ E_\pi[T(X)] \approx \frac{1}{N} \sum_{t=1}^N T( X^{(t) }) $ | ||
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| + | Law of large numbers -> 위 근사는 adoptable | ||
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| + | it is known that above approximation is still possible if we sample using a Markov chain. This is the main idea of MCMC method. | ||