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I am wondering if it is possible to re-calculate the normalizing constant of the posterior distribution for example the following

$$\pi(\theta|\boldsymbol{Y}) = \frac{L(\boldsymbol{Y}|\theta)\pi(\theta)}{\int L(\boldsymbol{Y}|\theta)\pi(\theta)d\theta}$$

often in the models I deal with the normalizing constant has no closed form so we by pass it as we are interested in the posterior of $\theta$ via the usual proportionality

$$\pi(\theta|\boldsymbol{Y}) \propto L(\boldsymbol{Y}|\theta)\pi(\theta)$$

My question is for Bayes factors and model averaging we require the normalizing constant. Is it possible to back calculate or approximate ${\int L(\boldsymbol{Y}|\theta)\pi(\theta)d\theta}$ given the proportional posterior?

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So, probably should have done this before I posted here but I hadn't found the right resource, once I found the right resource it exploded into many papers. It turns out there are many ways to approximate the normalizing constant. I point future people to the paper Diciccio et al (1997), as they explain all the methods with specific references

Diciccio, T. & Kass, R. & Raftery, A. Wasserman, L. (1997) Computing Bayes Factors by Combining Simulation and Asymptotic Approximations, Journal of the American Statistical Association, 92:439, 903-915

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  • $\begingroup$ And there are many more ways that have been developed in the last 20 years. $\endgroup$ – jaradniemi Aug 20 '18 at 19:57
  • $\begingroup$ Would you be so kind as to list the methods, or references? Most of the stuff I was looking at was 80-00's era $\endgroup$ – Cyrillm_44 Aug 22 '18 at 21:37

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