What should be the underlying distribution behind Monte Carlo simulation?

Question

When we are trying to use Monte Carlo simulation to solve a problem that does not have analytical solution, how do we decide what should be the underlying distribution from which we draw these random numbers?

The examples I find use normal distribution, but when should we use what type of distribution?

Answer 1

When one uses Monte Carlo methods to approximate an integral$$\mathfrak{I}=\int_\mathcal{X} H(x)\,\text{d}x$$there is a wide range of possible choices for the distribution used in the simulation. The reason is that $\mathfrak{I}$ can be represented in an infinite number of ways as$$\mathfrak{I}=\int_\mathcal{X} h(x)f(x)\,\text{d}x$$where $f$ is a probability density and $h$ is defined as $h(x)=H(x)/f(x)$. The requirements on $f$ are that

1. the distribution associated with $f$ can be easily simulated, i.e. that there exists an available simulation algorithm (which may rely on MCMC or SMC methods);
2. the support of $f$ contains the support of $H$;
3. the resulting Monte Carlo approximation$$\frac{1}{N}\sum_{i=1}^N h(x_i)$$ has a finite variance, which is equivalent to$$\int_\mathcal{X} h^2(x)f(x)\,\text{d}x=\int_\mathcal{X} \frac{H^2(x)}{f(x)}\,\text{d}x<\infty$$

This still leaves open a wide variety of choices, which may be compared through their variance corrected by the computing cost.