Imagine you are using a path tracer to integrate the illumination over each pixel in an image. Rather than use a fixed number of samples per pixel, you want to adaptively choose the number of samples based on the behavior of the integrand. In particular, you'd like to guarantee that the integration error $|\hat{I}_n - I|$ is below a user specified tolerance $\epsilon$. As discussed in class, it is in general very difficult (perhaps impossible!) to provide an absolute guarantee that the error is below the given tolerance. However, it is still possible to come up with a reasonable scheme that works well in practice.

Question: Describe an adaptive Monte Carlo sampling scheme that is appropriate for the kinds of integrals that appear in rendering. Think in terms of the expected value and variance of the collection of samples. What simplifying assumptions does your strategy implicitly make about the integrand? (Note: there is no one correct answer to this question! You should think creatively about a practical strategy that might give good results.)