Orthant Probability of Multi-variate Normal Distribution.
Orthant Probability is defined as $P(\vec{X}<0)$ where $X \sim N(0, \Sigma)$. It has been shown that there is no general analytical form of arbitrary dimensional Orthant Probability.
There are a few properties of Orthant Probabilities.
Properties of orthant probabilities
Symmetry:
$P(\vec{X}<0) = P(\vec{X} > 0)$.
One can use the Jacobian matrix and the transformation of C1 diffeomorphism to prove.
Odd dimension Orthant Probability can always be constructed from lower dimensional marginal probabilities.
One can prove using the axioms of probability and symmetry properties.
Bi-variate Orthant probabilities.
Define $Y \sim N(0, I)$, $X = G \cdot Y$. It’s not easy to see that $X \sim N(0, G \cdot G^T)$.
Using the Normal cone formed by $G$, we can use easily obtain the bi-variate orthant probabilities.
Tri-variate Orthant probabilities.
Geometry based solution
We can use the solid angle to calculate the orthant probabilities. And there is a geometry formula for the area of polar spherical triangle.
Odd dimension construction based solution.
We can construct dimension three results directly from marginal two-dimension results.
Degenerated Orthant probabilities.
In practice, $\Sigma$ is not necessarily full rank. There are numerical solutions for general solutions to solve high-dimensional Gaussian integrals. And the complexity grows exponentially as the dimension. Geometry based solution can help to reduce the complexity dramatically.