credible intervals for functions of hyperparameters If I have a statistic $\nu(x, y)$ which is a function of hyperparameters (say just two for ease of explanation) $x$ and $y$ of a distribution $F(t|x , y)$ and associated prior $g(x,y)$ which is the proper way to compute $\alpha \%$ credible interval for $\nu\left(x_m,y_m\right)$ around the "mode" MLE value of $g(x,y)$ which occurs at $\left(x_{m},y_{m}\right)$


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*Identify two curves/loops $y(x)$ which have sub-maximum likelihood values of $\alpha/2$ and $\left(1-\alpha/2\right)$ and for each compute the line-integral average value of $\nu_{\alpha/2}(x,y(x))$ and $\nu_{\left(1-\alpha/2\right)}(x,y(x))$

*Using the likelihood distribution $g(x,y)$ to compute the 1-D distribution $h\left(\nu\right)$ and then pick of values from the CDF $H(\nu=\alpha/2)$ and $H(\nu=1-\alpha/2)$. Will $\nu \left(x_m, y_m \right)$ necessarily occupy any special place in the distribution $h\left(\nu\right)$, like its mean, median, or mode?
which is correct? Or is something else the correct way?
 A: If $X,Y$ are random variables, then the ability to write $Y=f(X)$ implies maximum mutual information and then there is in fact only one hyperparameter. So the first method doesn't make sense given the statistical framework of the problem. The second method is the typical approach in Bayesian analysis in which the distribution of $\nu(X,Y)$ is approximated by sampling and the empirical CDF used to calculate credibility regions. Therefore the second method is correct. 
Whether the mode $(x_m,y_m)$ has a special significance in the distribution of $Z=\nu(X,Y)$ will depend on the nature of $g$, $\nu$ and of $X,Y$ as well but in general $z_m$ will not be the mode of the new distribution. For example, consider $X\sim \mathcal{N}(0,1)$ and $Z=1/X$.
Addressing the comment, I think I see what you meant: finding curves $(x(t),y(t))$ such that the regions they define contain some percent of the domain according to $p(z)$. This is theoretically possible but unnecessary as you would first need to know $p(z)$ and then can more easily define credibility regions in $z$-space anyway. Note that the first method you describe still doesn't make much sense (why linear integral average when you are interested in probably over region?, etc.).
