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Recently, I read slides on Variational Inference. The Bayesian rule is

$$ P(\mathbf{z} | \mathbf{x} ; \mathbf{\theta}) = \frac{P(\mathbf{x}|\mathbf{z}; \mathbf{\theta})P(\mathbf{z}; \mathbf{\theta})}{ P(\mathbf{x}; \mathbf{\theta}) } $$

And we can calculate the integral of $P(\mathbf{x}; \mathbf{\theta})$, however, why we can't computing the integral of it analytically?

Does it mean find calculating the integral is hard? Or we can find such closed-form formulation to calculate the integral but the computation is not polynomial?

Many slides claim that it is hard to compute the integral of $P(x)$ in Bayesian Inference analytically. But few slides explain why.

GoingMyWay
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    It means that calculating the integral is intractable. I.e. there is no closed form solution (in general). There are some nice cases where the integral is available in closed form, but these are exceptions rather than the rule. – jcken Feb 19 '21 at 09:54
  • @jcken Thank you. I got the point. I think the probability density of $P(X)$ is available but deriving the integral is hard or even impossible. – GoingMyWay Feb 19 '21 at 10:31
  • The density is typically only available up to proportionality (i.e. we have the top of the fraction, not the bottom). However, this is often enough to proceed with inference – jcken Feb 19 '21 at 10:34
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    That computing integrals analytically is hard is nothing special for Bayesian inference ... – kjetil b halvorsen Feb 19 '21 at 12:13

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