In this note we consider the problem of computing a probability:

\[\mathbb{P}(F(\eta) \ge z)\]

This is defined with respect to a finite number of (standard) normal random variables \(\eta := [\eta_1, \eta_2, \ldots, \eta_d]^T\). \(z\) is a fixed constant, and \(F: \mathbb{R}^d\rightarrow\mathbb{R}\) is some (nonrandom) function that defines the event of interest:

\[E_z := \{\eta: F(\eta) > z\}\]

If we recall the density for multivariate Gaussian random vectors, we have:

\[p_{\eta}(s) = \frac{1}{(2\pi)^{d/2}}\exp\bigg( -\frac12||s||_2^2 \bigg)\]

Then we have the integral:

\[\mathbb{P}(E_z) = \frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^d}\mathbf{1}_{E_z}\cdot \exp\bigg( -\frac12||s||^2 \bigg)ds\]

where \(\mathbf{1}_{E_z}(s)\) is the indicator function that is 1 for any \(s\in E_z\).

The issues preventing us from evaluating this integral in general are that:

  • \(F\) potentially may be difficult to evaluate.

  • The boundaries of integration may not have simple geometry.

However, what we can notice is that the density function is rapidly decreasing with respect to \(\|s\|^2\). Forgetting about the set \(E_z\) for now and suppose we want to integrate with respect to the entire space (which should give us 1). The minimizer would certainly be the mode, \(s^* = [0, \ldots, 0]^T\), then we recover 1 as the final integral indeed.

For more complicated sets \(E_z\), it might be interesting to postulate that only points \(s\) within some \(\delta\) ball around a constrained minimizer, will matter in the integral.

That is:

\[\mathbb{P}(F(\eta) > z) \approx \frac{1}{(2\pi)^{d/2}}\int_{E_z}\exp\bigg( -\frac12||s^*||^2 \bigg)ds = \frac{\mu(E_z)}{(2\pi)^{d/2}}\exp(-\frac12||s^*||^2)\]

where \(\mu(\cdot)\) denotes the Lebesgue measure. And since \(E_z\) does not necessarily contain \(0\), we define:

\begin{equation} \label{eqn:constrained-optimization} s^* := \text{argmin}_{s\in E_z}\frac12||s||^2 \end{equation}

Assuming that the above works well, evaluating the desired probability numerically has been converted into two sub-problems:

  • Evaluate \(\mu(E_z)\).

  • Solve constrained optimization problem \eqref{eqn:constrained-optimization}.

In particular, the above says roughly

\[\log \mathbb{P}(F(\eta) > z) \sim -\frac12\|s^*\|^2\]