I'd like to estimate an integral of the following form using Monte Carlo method:

$$ \int_{t_1}^{t_2} g(t) \left[ \int_{- \infty}^{\infty} f(t, u) du \right] ^\gamma dt$$

In case of $\gamma$ being a positive integer (say, $2$) I can rewrite it as follows:

$$ \int_{t_1}^{t_2} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} g(t) f(t, u_1) f(t, u_2) du_1 du_2 dt$$

That is, I can simply take two independent unbiased estimates of the inner integral and multiply them. My question is: could something similar be done in case of fractional $\gamma$?

  • $\begingroup$ There are myriad ways to estimate integrals. How do you want to do it? Using what information, with what computing limitations, to what accuracy? $\endgroup$
    – whuber
    Apr 6, 2023 at 13:59
  • $\begingroup$ @whuber It's Monte Carlo integration in the context of real-time computer graphics. Usually there's budget for 1-4 samples per frame only, but for most pixels you can average results from previous frames (that's why I want an unbiased estimate). $\endgroup$ Apr 6, 2023 at 15:36
  • $\begingroup$ But Monte-Carlo integration of what, exactly? There are two distinct integrals here. If we suppose you can readily evaluate the inner integral, this is a generic MC problem. Or maybe you want to approximate the inner integral, take its power, and then compute the outer integral directly? You need to edit this post to provide enough context for us to find a reliable interpretation of what you're trying to ask. $\endgroup$
    – whuber
    Apr 6, 2023 at 15:42
  • $\begingroup$ @whuber I can compute neither of the integrals directly. I'm trying to approximate both of them using single MC method, rather than two nested ones. Does it make sense? $\endgroup$ Apr 6, 2023 at 15:58
  • 2
    $\begingroup$ There is an entire branch of mathematics devoted to estimating integrals. Indeed, much of physics is involved in that too. For instance, Feynman diagrams are a method to estimate otherwise intractable integrals. Thus, at a minimum, your question needs to specify you reject all such approaches and require MC methods. $\endgroup$
    – whuber
    Apr 7, 2023 at 12:27

1 Answer 1


There seems to be a Bernoulli factory for solving this problem, namely producing unbiased estimates of the powered inner integral. Let us assume

  1. $0<\gamma<1$,
  2. $0<\varrho=\int f(t,x)\,\text dx<1$ (removing the dependence on $t$ to simplify notations).

Simulating a Bernoulli variate with probability and expectation $\varrho^{\gamma}$ can then be achieved by the following scheme, provided by Peter Occil (and attributed to Mendo, 2019):

 Repeat the following process, until this algorithm returns a value x:
 i. Set k=1
 ii. Generate a Bernoulli B(ϱ) variate z; if z=1, return x=1
 iii. Else, with probability γ/k, return x=0
 iv. Else, set k=k+1 and return to ii.

An R rendering of the above


Each time $\varrho$ is called, it can furthermore be replaced with an unbiased and independent estimator.

  • $\begingroup$ Thanks for Bernoulli factories! They look promising. $\endgroup$ Apr 7, 2023 at 7:59

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