Power heuristic in multiple importance sampling

Question

Both Mitsuba and Tungsten use the power heuristic to do multiple importance sampling (MIS).

Mitsuba:

inline Float miWeight(Float pdfA, Float pdfB) const {
    pdfA *= pdfA;
    pdfB *= pdfB;
    return pdfA / (pdfA + pdfB);
}

Tungsten:

static inline float powerHeuristic(float pdf0, float pdf1)
{
    return (pdf0*pdf0)/(pdf0*pdf0 + pdf1*pdf1);
}

Advanced Global Illumination gives the balance heuristic: $$F=\cfrac1N\sum_{i=1}^n\sum_{j=1}^{n_i}\cfrac{f(X_{i,j})}{{\sum\limits_{k=1}^n}\frac{c_k}Np_k(X_{i,j})}$$

The power heuristic may be like this: $$F=\cfrac1N\sum_{i=1}^n\sum_{j=1}^{n_i}\cfrac{f(X_{i,j})p_i(X_{i,j})}{{\sum\limits_{k=1}^n}\frac{n_k}Np_k^2(X_{i,j})}$$

I know they are all valid estimators.

What's the motivation to use the power heuristic?

What about the cube heuristic? $$F=\cfrac1N\sum_{i=1}^n\sum_{j=1}^{n_i}\cfrac{f(X_{i,j})p_i^2(X_{i,j})}{{\sum\limits_{k=1}^n}\frac{n_k}Np_k^3(X_{i,j})}$$

Answer 1

PBRT v3 Page 799:

Veach determined it empirically.