Evaluate an integral using importance sampling

Question

Estimate $\int^{1}_{0}e^{x} dx$ using importance sampling. Should I use beta distribution as proposal distribution and uniform distribution as target ?

Answer 1

As the range of the integral is $(0,1)$, it is better to take an importance density restricted to $(0,1)$ like the Uniform $\mathcal U(0,1)$ or a Beta $\mathcal Be(a,b)$, rather than an importance function with a larger support like Exponential or Normal. Intuitively, including a random number of zeroes in the Monte Carlo average is both a waste of computer time and an unnecessary source of randomness! Here is a comparison of the four (for 10³ simulations, replicated 10² times):

that seems to put a black mark on the Beta. However, the choice of a Beta $\mathcal B(2,1)$ is quite poor as $$\frac{\exp(-x)}{x^{2-1}(1-x)^{1-1}}$$ has infinite variance. Using instead a Beta $\mathcal B(1/2,1)$ leads to much better performances:

As a final experiment, restricting the simulations of both Exponential and Normal variates to $(0,1)$ leads to an obvious improvement

with a zero variance in the case of the truncated Exponential importance function, since it uses the probability of being between $0$ and $1$, which is the value of the integral itself.

Answer 2

This integral is easy using direct monte carlo integration, however if you want to use importance sampling, you can choose the target distribution as $f(x)=e^x$, i.e. harder to sample than uniform. And $h(x)=\Pi(x)$, i.e. $1$ within $[0,1]$ and zero otherwise. Choose normal ($g(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2})$ or beta if you want, sample from it and take the mean, i.e. $$I\approx \frac{1}{N}\sum_{x \sim g(x)} \frac{f(x)h(x)}{g(x)}$$