MCMC sampling with sum constraints

Question

I'm interested in sampling a collection of variables with a sum constraint on them. For a simplified example:

Prior:

$X \sim \mathcal{N}(0, 1)$

$Y \sim \mathcal{N}(0, 1)$

Observation:

$X + Y = 1$

Proposed approach:

Use augmented variables $S$, $x'$, and $y'$, defined as:

$X = x' / S$

$Y = y' / S$

$S = x' + y'$

MCMC Algorithm:

1) Sample $x'$ given previous value of $S$ as being from $\mathcal{N}(0, S^2)$.

2) Sample $y'$ given previous value of $S$ as being from $\mathcal{N}(0, S^2)$.

3) "Sample" $S$ by setting it to $x' + y'$.

This feels like it should work, but how can I prove that it is a valid data augmentation scheme? Specifically, when sampling $x'$ using Gibbs sampling, why are we not forced to choose $x' = S - y'$ since we are conditioning on $S$ and $y'$?

Answer 1

Sorry, I'm not going to answer your question directly. But, I think you are thinking about the problem in the wrong way.

If the sum of X and Y are 1, then it means that Y is completely determined by X. In that case, you can just sample X, and let Y to be 1-X. That will give you a random sample of X,Y with X+Y=1. The distribution of X completely determines the distribution of Y, so you only need to sample X.