Mean-Square Error of the Monte-Carlo Method

Mathematically, there holds:

$$\mathbb{E}\left[{\left|\mu -{\hat{\mu }}_K\right|}^2\right]=\mathbb{V}\left[{\hat{\mu }}_K\right]+\stackrel{0}{\overbrace{{\left(\mu -\mathbb{E}\left[{\hat{\mu }}_K\right]\right)}^2}}$$

where,

• $\mu$ → True but unknown Expected value
• $\mu -\mathbb{E}\left[{\hat{\mu }}_K\right]$ → is called the bias.

1. The Monte Carlo method is unbiased, i.e.,

   $$\mathbb{E}\left[{\hat{\mu }}_K\right]=\mu$$

2. If $\mathbb{V}[Y]<\infty$, the mean-square error (MSE) satisfies

   $$\mathbb{E}\left[{\left|\mu -{\hat{\mu }}_K\right|}^2\right]=\mathbb{V}\left[{\hat{\mu }}_K\right]=\frac{\mathbb{V}[Y]}{K}.$$

   Where,

   • $\mathbb{V}[Y]$ → is the Variance of the model’s output random variable $Y$
   • The MSE is independent of the number of uncertain input parameters.
   • There are different ways to reduce the error:
   • Increase sample size $K$ (can be very expensive if the model is complex)
   • Decrease variance (Quasi-Monte Carlo, control variate method, multilevel Monte Carlo)