Approximating Gaussian Integrals
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:
\[p_{\eta}(s) = \frac{1}{(2\pi)^{d/2}}\exp\bigg( \frac12s_2^2 \bigg)\]Then we have the integral:
\[\mathbb{P}(E_z) = \frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^d}\mathbf{1}_{E_z}\cdot \exp\bigg( \frac12s^2 \bigg)ds\]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:
\[\mathbb{P}(F(\eta) > z) \approx \frac{1}{(2\pi)^{d/2}}\int_{E_z}\exp\bigg( \frac12s^*^2 \bigg)ds = \frac{\mu(E_z)}{(2\pi)^{d/2}}\exp(\frac12s^*^2)\]where \(\mu(\cdot)\) denotes the Lebesgue measure. And since \(E_z\) does not necessarily contain \(0\), we define:
\begin{equation} \label{eqn:constrainedoptimization} s^* := \text{argmin}_{s\in E_z}\frac12s^2 \end{equation}
Assuming that the above works well, evaluating the desired probability numerically has been converted into two subproblems:

Evaluate \(\mu(E_z)\).

Solve constrained optimization problem \eqref{eqn:constrainedoptimization}.
In particular, the above says roughly
\[\log \mathbb{P}(F(\eta) > z) \sim \frac12\s^*\^2\]Enjoy Reading This Article?
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