Demo: Monte Carlo Methods
Estimate a normal tail probability by crude Monte Carlo and by importance sampling. This is a compact way to see why proposal choice matters for rare events.
Mathematical setup
The target probability is
\[p_\gamma=\Pr(Z>\gamma)=E_f[\mathbf 1\{Z>\gamma\}], \qquad Z\sim N(0,1).\]Crude Monte Carlo uses
\[\hat p_{\mathrm{crude}}=\frac{1}{M}\sum_{m=1}^M \mathbf 1\{Z_m>\gamma\}, \qquad Z_m\sim f.\]Importance sampling draws $Y_m\sim q=N(\delta,1)$ and weights by $w(Y_m)=f(Y_m)/q(Y_m)$:
\[\hat p_{\mathrm{IS}}=\frac{1}{M}\sum_{m=1}^M \mathbf 1\{Y_m>\gamma\}w(Y_m).\]This estimate is unbiased for the displayed setup because both $f$ and $q$ are normalized and known exactly. The effective sample size displayed by the widget is
\[\mathrm{ESS}=\frac{\left(\sum_m w_m\right)^2}{\sum_m w_m^2},\]a diagnostic for weight concentration, not a proof of accuracy.
What to try
- Set $\gamma$ near 1.5. Crude Monte Carlo usually works because the event is not too rare.
- Move $\gamma$ toward 3 or 4. Crude estimates can become unstable unless $M$ is large.
- Set the proposal shift near the threshold. If the shift is too small, few proposal samples hit the tail; if too large, weights can become unstable.
The importance estimate uses unnormalized $f/q$ weights and is unbiased here because both the target and proposal densities are known. The effective sample size summarizes weight degeneracy: large weights from a few draws reduce useful information.