Importance Simulation

Let us note \cD_f = \{\ux \in \Rset^{n} | g(\ux,\underline{d}) \leq 0\}. The goal is to estimate the following probability:

\begin{aligned}
    P_f &=& \int_{\cD_f} f_{\uX}(\ux)d\ux\\
    &=& \int_{\Rset^{n}} \mathbf{1}_{\{g(\ux,\underline{d}) \:\leq 0\: \}}f_{\uX}(\ux)d\ux\\
    &=& \Prob {\{g(\uX,\underline{d}) \leq 0\}}
  \end{aligned}

This is a sampling-based method. The main idea of the Importance Sampling method is to replace the initial probability distribution of the input variables by a more “efficient” one. “Efficient” means that more events will be counted in the failure domain \cD_f and thus reduce the variance of the estimator of the probability of exceeding a threshold. Let \underline{Y} be a random vector such that its probability density function f_{\underline{Y}}(\underline{y}) > 0 almost everywhere in the domain \cD_f,

\begin{aligned}
    P_f &=& \int_{\Rset^{n}} \mathbf{1}_{\{g(\ux,\underline{d}) \leq 0 \}}f_{\uX}(\ux)d\ux\\
    &=& \int_{\Rset^{n}} \mathbf{1}_{\{g(\ux,\underline{d}) \leq 0 \}} \frac{f_{\uX}(\ux)}{f_{\underline{Y}}(\ux)}f_{\underline{Y}}(\ux)d\ux
  \end{aligned}

The estimator built by Importance Sampling method is:

\begin{aligned}
    \hat{P}_{f,IS}^N = \frac{1}{N}\sum_{i=1}^N \mathbf{1}_{\{g(\underline{Y}_{\:i}),\underline{d}) \leq 0 \}}\frac{f_{\uX}(\underline{Y}_{\:i})}{f_{\underline{Y}}(\underline{Y}_{\:i})}
  \end{aligned}

where:

  • N is the total number of computations,
  • the random vectors \{\underline{Y}_i, i=1\hdots N\} are independent, identically distributed and following the probability density function f_{\uY}

Confidence Intervals

With the notations,

\begin{aligned}
    \mu_N &=& \frac{1}{N}\sum_{i=1}^N \mathbf{1}_{\{g(\underline{y}_{\:i}),\underline{d}) \leq 0 \}}\frac{f_{\uX}(\underline{y}_{\:i})}{f_{\underline{Y}}(\underline{y}_{\:i})}\\
    \sigma_N^2 &=& \frac{1}{N}\sum_{i=1}^N (\mathbf{1}_{\{g(\underline{y}_i),\underline{d}) \leq 0 \}}\frac{f_{\uX}(\underline{y}_{\:i})}{f_{\underline{Y}}(\underline{y}_{\:i})} - \mu_N)^2
  \end{aligned}

The asymptotic confidence interval of order 1-\alpha associated to the estimator P_{f,IS}^N is

\begin{aligned}
    [ \mu_N - \frac{q_{1-\alpha / 2} . \sigma_N}{\sqrt{N}} \: ; \: \mu_N + \frac{q_{1-\alpha / 2} . \sigma_N}{\sqrt{N}} ]
  \end{aligned}

where q_{1-\alpha /2} is the 1-\alpha / 2 quantile from the standard distribution \cN(0,1).

This method could also be found under the name “Strategic Sampling”, “Ponderated Sampling” or “Biased Sampling” (even if this estimator is not biased as it gives exactly the same result).

References:

  • W.G. Cochran. Sampling Techniques. John Wiley and Sons, 1977.
  • M.H. Kalos et P.A. Monte Carlo Methods, volume I: Basics. John Wiley and Sons, 1986.
  • R.Y. Rubinstein. Simulation and the Monte Carlo Method. John Wiley and Sons, 1981.