Suppose we had two multivariate Gaussian random variables, with one being an affine transformation of the first $$X\sim\mathcal{N}(\boldsymbol\mu,\mathbf{\Sigma}), \\Y=\mathbf{A}X+\mathbf{b},$$ with $\mathbf{A}\in\mathbb{R}^{m\times n},\mathbf{b}\in\mathbb{R}^m$. I understand that it's a well-established fact that we can compute the marginal distribution of $Y$ as $Y\sim\mathcal{N}(\mathbf{A}\boldsymbol\mu +\mathbf{b},\mathbf{A\Sigma A}^\top).$
My question is:
- What different ways are there to compute the conditional distribution $p(Y|X)?$ do we have to compute the joint distribution–computing the cross-covariance terms–and then apply the formula involving the Schur complement for the conditional? Is there no faster way given the relationship between the random variables?
Best Answer
Since $Y$ is measurable with respect to $X$, $p(Y|X)$ will be a $\delta$ distribution at $AX + b$. The only randomness is in $X$, so if you condition that out you get a non-random result.