Deriving the joint distribution of multivariate normal transformed into Bernoulli

Question

Given a covariance matrix $\sum_{ij}$ and a mean vector $\mu$ I have sampled $N$ multivariate normal vectors $Z = (z_1,...z_n)$ My goal is to create a vector of Bernoulli random variables $Y = (y_1,...y_n)$ where

$y_i = 1 $ if $|z_i| \leq C$ and 0 otherwise. where C is a specified constant, most likely a $Z_{\alpha}$ value.

Is there a closed form for the PMF of $Y$ ?

Currently I am taking a sample of very large $N$ and finding the proportions of each possible combination of $Y$ e.g $[y_1 = 1, ..., y_n = 1]$ through $[y_1 = 0, ... , y_n=0]$ and deriving $Y$'s PMF that way but it seems inefficient when trying to compute expected values and covariances

Answer 1

I'm not exactly sure what you're asking, since your specified constant $C$ is not well-defined, but I'll take a stab that you're looking either for a "$Z_\alpha$" that's "the same" in every direction, or a set of $Z_\alpha$'s that may vary in each direction (in which case, $C$ also varies with respect to each coordinate).

For the former, I think you're interested in the multivariate analogue of the inverse of the cdf $F(x)=P(Z\leq z)=z_\alpha$. The analogue is $\chi^2_n(\alpha)$, the quantile function (in your case) of a chi-squared distribution on $n$ degrees of freedom with area $\alpha$. So your $Y$ should be multivariate Bernoulli with uniform probability $1-\alpha$ in each coordinate, since the probability of lying in an ellipsoid within your multivariate normal that has measure $1-\alpha$ is...$1-\alpha$.

There are closed form expressions for the multivariate Bernoulli given in Whittaker, 1990; Graphical Models in Applied Mathematical Multivariate Statistics, or in https://arxiv.org/pdf/1206.1874.pdf.

For the latter, I'm unsure how to answer this in true closed form, since I'm not sure what the set of $C$'s are, but one way of doing it is to project the sampled point $x_j$ onto each coordinate, have an indicator random variable for each dimension (say, $I_i(x_j)$), and your $y_i$ would be an overall indicator variable of if all of the $I_i(x_j)=1$. This wouldn't have a closed form solution in general.

Answer 2

I think this is a hard unsolved problem, though I'm far from expert on it. It strikes me as a harder cousin of the orthant problem, which is already quite hard. See this paper for references.