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I am aiming to get a numerical value for a five-dimensional integral using Monte Carlo Integration. I am getting good results using the Mean Value Method, but I would like to try to use Importance Sampling to get even better results.

My main source gives me an example of how to do this method using a one-dimensional integral, but it doesn't even mention how to replicate it for multivariable integrals. My main issue is generating a set of random numbers from a multivariable probability distribution function.

If I understand correctly the process of doing it for a single variable is setting up the equation:

$$\int_0^{x}dx^{\prime}\,p(x^\prime) = z$$

and then solving for $x$ ($z$ are the uniformly distributed random numbers generated).

How can I replicate this procedure for an arbitrary 5-dimensional pdf $p(x_1,x_2,x_3,x_4,x_5)$?

I could find no good sources on this, most of the ones that I found dealt specifically with n-dimensional Gaussian distributions.

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Let's recap what you want to do: You have some set $V \subset \mathbb{R}^5$ and want to approximate the integral of some function $f\colon V \to \mathbb{R}$: $$ \int_V f(x) \,\mathrm{d}x $$ The "mean value method" sounds like a Monte Carlo-type approximation of the form $$ \int_V f(x) \,\mathrm{d}x \approx \frac{\operatorname{vol}(V)}{N} \sum_{i=1}^N f(X_i), $$ where $X_i$ are iid. random variables on $V$ with a uniform distribution and $\operatorname{vol}(V)$ is the volume of $V$.

The problem with this vanilla Monte Carlo approach is that, depending on the function $f$, most samples will not contribute to the integral since $f(X_i) \approx 0$. In importance sampling, the idea is to sample from a distribution that is adapted to the shape of $f$ instead of uniformly sampling from $V$.

To this end, let $p\colon V \to [0, \infty)$ be the density of a probability distribution $P$ on $V$. We approximate $$ \int_V f(x) \,\mathrm{d}x = \int_V f(x) \frac{p(x)}{p(x)} \,\mathrm{d}x = \int_V \frac{f(x)}{p(x)} \,\mathrm{d}P(x) \approx \frac{1}{N} \sum_{i=1}^N \frac{f(Y_i)}{p(Y_i)}, $$ where $Y_i$ are iid. random variables drawn from $P$.

As you've mentioned, we can exactly sample $Y_i$ from $P$ in one dimension by using inversion sampling. This only works if we can invert the cumulative distribution function of the distribution.

In multiple dimensions, there are several ways to draw from a given distribution:

  • Rejection sampling,
  • Gibbs sampling (if we can efficiently draw from the one-dimensional conditionals),
  • Markov Chain Monte Carlo (only provides asymptotically correct samples in general, but could be applied to the integration problem directly without the importance sampling detour),
  • some more.

For selecting a method, we need more information on the actual distribution. Is there some structure that can be exploited (e.g., mixture of Gaussians)?

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  • $\begingroup$ I am doing trying to calculate four-body decays. The structure of the functions I am trying to integrate usually has the structure $f(x) \propto \dfrac{1}{(x^2-m^2)^2 + m^2 \Gamma^2}$, where $m$ and $\Gamma$ is the mass and decay width of intermediate particles, values that can be obtained numerically. Which method you think would best fit me? Do you have any sources I could draw from? $\endgroup$
    – pollux33
    Oct 13 '21 at 9:06
  • $\begingroup$ @pollux33, how is $x^2$ to be understood (e.g., $\lvert \vec{x} \rvert^2 = \vec{x} \cdot \vec{x}$)? Is the distribution on the full $\mathbb{R}^5$ or does it have finite support? Is the distribution isotropic? $\endgroup$
    – cos_theta
    Oct 13 '21 at 10:11
  • $\begingroup$ I am dealing with two different sets of integration variables, let's call one $S_i$ where and the other $\tau_i$, both are related to each other (i.e. $S_i = S_i (\vec{\tau})$). When I work with $S_i$, the $x$ in my previous comment could be a single $S$ or it could be a linear combination of up to three of them. How each $S$ depends on $\tau$ could be a simple or a complex equation. If you want the specific relations, you can check this paper (page 15): lib-extopc.kek.jp/preprints/PDF/1983/8311/8311072.pdf Which has been my main help in getting the integration limits. $\endgroup$
    – pollux33
    Oct 13 '21 at 10:43

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