# Importance Sampling for multidimensional integrals and random numbers from multivariable pdf's

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.

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)?

• 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? Oct 13 '21 at 9:06
• @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? Oct 13 '21 at 10:11
• 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. Oct 13 '21 at 10:43