For the model given by some independent pairs $(x_i,y_i)$ identically generated from a bivariate Gaussian distribution, there is the convenient semi-conjugate family of "Normal-Wishart" prior distributions. It is mainly convenient because the posterior distributions are easy to simulate, without resorting to MCMC techniques. In particular the Jeffreys non-informative prior is at the boundary of the semi-cojugate family and the Jeffreys posterior is easily simulated.

But for the case when we assume equal variance for $x_i$ and $y_i$, is there a posterior distribution which is easy to simulate and achieving a good "frequentist performance" such as the Jeffreys posterior ? (that is, roughly speaking, the frequentist coverage of the $95\%$ credibility intervals approximately is $95\%$, for usual parameters of interest).

  • $\begingroup$ Did you mean to say that we assume the variance of $x_i$ and $y_i$ are known? Otherwise, if we assume equal but unknown variances for $x_i$ and $y_i$ then we can still use the Normal-Wishart prior. $\endgroup$ – user25658 Sep 9 '13 at 22:08
  • $\begingroup$ @BabakP The common variance is unknown. How do you put a Wishart prior on a variance matrix with equal diagonal entries ? $\endgroup$ – Stéphane Laurent Sep 10 '13 at 11:43
  • $\begingroup$ If you're after frequentist intervals, go straight to modified profile likelihood, I say. It's almost the same thing as using a probability-matching prior, but with no detour through Bayes' theorem. (There's a close relationship between modified profile likelihood and so-called integrated likelihood.) $\endgroup$ – Cyan Sep 11 '13 at 1:30
  • $\begingroup$ @Cyan I want a good frequentist performance but my parameter of interest is $\theta=f(\mu_1,\mu_2,\sigma)$ for a complicated function $f$. $\endgroup$ – Stéphane Laurent Sep 11 '13 at 6:09
  • $\begingroup$ @StéphaneLaurent Is there a one-to-one transformation between, say, $(\mu_1,\mu_2,\theta)$ and $(\mu_1,\mu_2,\sigma)$? If so, then I still say modified profile likelihood. It'll take brain power, be computationally intensive, and you'll want to check coverage by simulation, but it will work. $\endgroup$ – Cyan Sep 12 '13 at 4:53

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