## What is an intractable integral? How do you recognize one?

Greetings,
I've been using Bayesian methods and especially Gibbs sampling to work on models of various complexity, and I've always accepted the pretty much standard textbook explanation as a sensible justification for using sampling methods: sometimes the posterior distributions of Bayesian models may be intractable.

It occured to me at one point that none of the resources I have properly explain what they mean by that. Google scholar search made me even more confused, since many works seem to use intractable to refer to "resource intensive calculation". Analytically intractable seems to be a better term I guess, but what that means exactly is not still clear to me.

How do you know a (multiple) integral is analytically intractable? Other than the obvious situations where -∞ and ∞ bounds, I mean. (Would this be intractable by the way?)

I'm looking for a proper discussion of the topic, preferably in a Bayesian inference context. It is amazin how many textbooks simply say "this integral may be interactable etc etc.. so let's use sampling"

Any pointers would be much appreciated

Kind regards
Seref

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 Ok, I have to confess that I've been bitten by my lack of proper understanding of the word intractable. My native language is not English, and I've taken intractable as impossible to calculate, which was wrong. I would still appreciate your input though, is not this term a bit subjective?

 Tags bayes, bayesian inference, integrable, sampling