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A normal random matrix $\mathbf{X} \sim \mathcal{MN}_{n\times p}(\mathbf{M}, \mathbf{U}, \mathbf{V})$ is transformed as $$\mathbf{DXC}\sim \mathcal{MN}_{r\times s}(\mathbf{DMC}, \mathbf{DUD}^T, \mathbf{C}^T\mathbf{VC})$$ for $\mathbf D, \mathbf C$ full rank. So with the "standard normal random matrix" $\mathbf Z \sim \mathcal N_{n\times p}(0, I_n, I_p)$ ...


I think a simple way is using $Plug-in$ estimation. 1) estimate parameters of $\mu$ and $\Sigma$ of 3-variate normal based on data. $$(\hat{\mu} , \, \hat{\Sigma})$$ 2) Calculate the conditional distribution of $X_1|X_2,X_3$ and $X_1|X_2+X_3$ that depend on $\mu$ and $\Sigma$. 3) using $Plug-in$ estimation for $X_1|X_2,X_3$ and $X_1|X_2+X_3$ by ...


First in Gibbs Sampling, you update one parameter at the time is not correct. Gibbs sampling is using a decomposition of the distribution of interest into conditional distributions for blocks of components of the vector $X$ to be simulated. These blocks can be of any dimension. Provided all components of $X$ belong to at least one of the blocks, and that ...


What @whuber was trying to say is that when you have a conditional probability p(Y|X=x) you treat X as a given i.e. a constant x (note small case x). So in that case, the only random variable there is, is the error~N(0, 1), so Y|x~N(x, 1). I hope that clarifies this

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