Monte Carlo Method to Approximate Expectation

Monte Carlo approximation for an expected value is given as:

$$\boxed{\mu \approx {\tilde{\mu }}_K:=\frac{1}{k}\sum _{i=1}^K{y}^{(i)}=\frac{1}{k}\sum _{i=1}^K\mathcal{M}(x^{(i)})}$$

Where,

• $K$→ Total number of samples generated
• $x$ →Random sample input which are independent and identically distributed (i.i.d.) realizations of the input random variable $X$.
• $\mathcal{M}(x^{(i)})$ Model, $\mathcal{M}$, when evaluated at the $i-$th random input sample, $x^{(i)}$

Note: $y=\mathcal{M}(x)$

Multi-dimensional Case

$$\boxed{\mu =\mathbb{E}[\gamma ]\approx {\tilde{\mu }}_K=\frac{1}{K}\sum _{i=1}^K{y}^{(i)}=\frac{1}{K}\sum _{i=1}^K\mathcal{M}\left({\mathbf{x}}^{(i)}\right)}$$

The sample ${\left\{{\mathbf{x}}^{(i)}\right\}}_{i=1}^K$ now contains vectors.

1. If $X_1,\dots ,X_{n_{\text{in }}}$ are independent: Generate $x_1^{(i)},x_2^{(i)},\dots ,x_{n_{\text{in }}}^{(i)}$ independently of each other
2. If $X_1,\dots ,X_{n_{\text{in }}}$ are dependent: Draw $\mathbf{x}{}^{(i)}$ directly from the joint PDF $f_X$

Calculation Steps

1. Generate a sample of model inputs $\{x^{(i)}{\}}_{i=1}^K$.
2. Compute the corresponding model output sample $\{y^{(i)}=\mathcal{M}(x^{(i)}){\}}_{i=1}^K$.
3. Estimate the expected value using the mean of the output sample.