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$X$ is a multivariate Gaussian, whose confidence region I can derive.

$Y$ is a function of $X$, specifically $Y = (x_1, x_2 - b x_1^2 + 100b, x_3, \dots, x_n)$. I can use change-of-variable technique to derive the pdf of $Y$.

Is it possible to also derive the confidence region of $Y$ and check if a point falls into this region?

Context: I'm replicating Haario et al (1999) paper on adaptive Metropolis algorithm, which uses Metropolis algorithm to draw samples from $X$ and $Y$, where $X$ is multivariate normal

$$ \begin{align} X &\sim MVN(0, C) \text{ with } C = diag(100, 1, \dots, 1) \\ \end{align} $$

enter image description here

and $Y$ is a banana-shaped, "twisted" normal

$$Y = (x_1, x_2 - b x_1^2 + 100b, x_3, \dots, x_n)$$

enter image description here

To check if their samples approximate well the distribution of Y, Haario et al calculate the percentage of samples falling within the 68.3% confidence region of Y. I don't know how to derive the confidence region of Y and check if a point falls into this regon.

Heisenberg
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I am not sure what you mean by "the" confidence region. If I were to create a confidence region at a given confidence level $k_\alpha$, I would pick an HPD region$$\mathfrak{C}_\alpha=\{x;\ p(x)\ge k_\alpha\}$$

A first approach to create a confidence region on $Y=h(X)$ is to take the transform of $\mathfrak{C}_\alpha$ by $h$, i.e. $$\mathfrak{C}^h_\alpha=\{y;\ \exists x\ \text{such that }p(x)\ge k_\alpha\}$$which has at least coverage $\alpha$.

But this may not be the HPD (i.e. smallest) confidence region because of the change of variable formula, which includes a Jacobian. Hence another approach is to take the pdf of $Y$ $p^h$ and derive directly the corresponding HPD region $$\mathfrak{C}^\star_\alpha=\{y;\ p^h(y)\ge k^\star_\alpha\}$$ When running an MCMC algorithm as in Haario et al (1999) it is possible to derive directly the HPD region by looking at the $100\alpha$% largest densities and constructing the convex envelope of the associated simulations.

Xi'an
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  • Yes, I should've said the HPD region -- thank you for the clarification! My goal (also Haario et al's goal) is to check that our MCMC samples correctly approximate the banana-shaped, twisted normal. Does this likely mean that Haario et al took the third approach? In that case, could you explain or point me towards how to "construct the convex envelope of the associated simulations" (in R) and check if a point falls within this envelope? – Heisenberg Dec 29 '17 at 05:12
  • I presume that, in this special case of Haario et al. (1999) the level set can be drawn as on the picture given that $(y_1,y_2)$ has a known marginal density. By trial and error, the level corresponding to a given $\alpha$ can be derived, without requiring my third approach (which anyway requires the MCMC to have converged). – Xi'an Dec 29 '17 at 08:24