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I know that one can approximate expectations of a function with respect to a pdf as such $$ \mathbb{E}_{p(x)}[\phi(x)] = \int \phi(x) p(x) dx \approx \frac{1}{N}\sum_{i=1}^N \phi(x^{(i)}) \qquad\qquad \text{where } x^{(i)} \sim p(x) $$ How can I approximate a normalizing constant in a similar way using Monte Carlo? $$ \mathcal{Z} = \int \widetilde{p}(x) dx \approx \,\, ? $$

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I would follow the same scheme as in importance sampling https://en.wikipedia.org/wiki/Importance_sampling, i.e I'll find a distribution $f(x)$ that I can easily sample from with the same domain as $\tilde{p}(x)$ and I'll approximate the normalizing constant as

$$Z = \int \tilde{p}(x)\frac{f(x)}{f(x)}dx= \int \frac{\tilde{p}(x)}{f(x)}f(x)dx \approx \sum_{n=1}^{N}\frac{\tilde{p}(x^{(n)})}{f(x^{(n)})} \ , \ \ x^{(n)}\sim f(x)$$

Fiodor1234
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