Suppose we have two random variables $X$ and $Y$ related by the equation:
$Y=kX+Z,$
where $k$ is an unknown constant scalar, and $Z$ is a zero mean Gaussian random variable independent of $X$. The statistical properties of $X$ and $Y$ are unknown, as is the covariance of $Z$.
Given $N$ independent samples drawn from $(X,Y)$:
$(x_1,y_1),\cdots,(x_N,y_N),$
for a new $x^*$, are there any existing methods to generate the corresponding new sample $y^∗$, such that its distribution is consistent with the conditional distribution $Y|X=x^*$?
I may have a naive method to do so. Given $(x_i, y_i)$ and $x^*$, we can sample a vector $\alpha=[\alpha_1, \cdots, \alpha_N]$ with $\|\alpha\|_2=1$, such that
$x^*=\sum_{i=1}^N \alpha_i x_i.$
Then, by letting $y^*=\sum_{i=1}^N \alpha_i y_i$, we can verify that the mean and covariance of the resulting $y^*$ are consistent with $Y|X=x^*$.
Are there existing methods that take a similar approach? If so, are there any performance analyses for these methods?
