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I have a question in terms of Bayesian computation.

Let's contrast parameter estimation and model selection. When performing a Bayesian hypothesis test, this involves computing fully marginalized likelihoods. When performing (Bayesian) parameter estimation, one computes partially marginalized posteriors.

The latter is normally far more easy to actually compute than the former. Why exactly is that? How could it be so much easier to computer the marginalization of all of your parameters except one versus marginalizing all of the parameters?

Is there a computational intuition behind why this is?

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  • $\begingroup$ Can you add details about a specific problem or reference? $\endgroup$ – AdamO May 22 '17 at 16:51
  • $\begingroup$ @AdamO I will have to look for resources on Bayesian computation. The intractability/high-dimensional space when performing this computation is describing in several MCMC books. However, I need to think of a resource which quantifies the difference between computing FMLs versus parameter estimation $\endgroup$ – ShanZhengYang May 22 '17 at 16:59
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    $\begingroup$ Marginalising a full posterior is easy once you can simulate from that posterior. Marginalising a likelihood function can be done by importance sampling and other numerical methods but never from a simulation of the posterior itself (if you exclude the infamous harmonic mean estimator). $\endgroup$ – Xi'an May 28 '17 at 20:14

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