Nested Uniform Distributions in Monte Carlo Integration

Question

In terms of importance sampling for numerical Monte Carlo integration we can proceed as follows:

\begin{align} \int_{\Omega} p(\mathbf{x}) d\mathbf{x} &= \int_{\Omega} p(\mathbf{x}) \frac{q(\mathbf{x})}{q(\mathbf{x})}d\mathbf{x} \\ & = E_{\mathbf{x} \sim q(\mathbf{x})}\left[\frac{p(\mathbf{x})}{q(\mathbf{x})}\right] \\ & \approx \frac{1}{N} \sum_{i=1}^N \left[\frac{p(\mathbf{x}_i)}{q(\mathbf{x}_i)}\right] \end{align}

Therefore we can empirically estimate the integral by sampling $\mathbf{x}$ from some distribution $q(\cdot)$.

Now in my problem I know for a fact that my $\mathbf{x}$ is a finite vector ($\mathbf{x}=[x_1,x_2,x_3,...x_n]$), all values are contained in the interval $[0,1]$ (so $\Omega = [0,1]^d$), and there is a particular ordering to my $\mathbf{x}$, i.e.:

$x_1 \geq x_2$

$x_2\geq x_3$

$x_2 \geq x_3$ etc....

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So given the structure of my problem (nested, all values in $[0,1]$), my questions are:

Question 1: Given this nested structure to my problem, is it better to choose certain $q(\cdot)$ functions over others?

I feel that because all my variables are in the interval $[0,1]$, and I have no reason to believe any particular weighting of the points (I am actually evaluating an integral to help find a volume), so should I choose a uniform distribution?

Question 2: If I opt for a uniform distribution for $q(\cdot)$, do I need to reflect the prior structure of my variables in the sampling procedure?

i.e. I cannot just sample $\mathbf{x} \sim U[0,1]^n$, as this will occasionally violate my required hierarchical/ordering/nested structure.

Question 3: In many importance sampling integration problems, they give the example of uniform 1D, in which case $q(\cdot) = \frac{1}{b-a}$, so this term can be removed from the expectation/sum to the front. How can I calculate this same volume, assuming I opt for $q(\cdot)$ to have a nested hierarchical structure? How should this scale and be dealt with

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Thus ultimately I am unable to understand how to properly include such a prior knowledge of the nested nature of my variables into MC integration.

Answer 1

Question 1: Given this nested structure to my problem, is it better to choose certain $q(⋅)$ functions over others?

A generic recommendation is to have the importance function $q(⋅)$ supported by the same support as the integrand $p(⋅)$. Therefore, if $\mathbf x$ belongs to the restricted unit hypercube, the function $q(⋅)$ should only be positive over this restricted unit hypercube.

Question 2: If I opt for a uniform distribution for $q(⋅)$, do I need to reflect the prior structure of my variables in the sampling procedure?

The constraint on the components of $\mathbf x\in(0,1)^n$ can be expressed by a change of variables: \begin{align*} x_1 &= y_1 &y_1\ge 0\\ x_2 &= x_1 + y_2 &y_2\ge 0\\ &\vdots \\ x_n &=x_{n-1}+y_n &y_n\ge0\\ &\qquad\qquad y_1+\ldots+y_n\le 1 \end{align*} which indicates that $\mathbf y=(y_1,\ldots,y_n)$ belongs to the simplex of $\mathbb R^n$ (or $\mathbb R^{n+1}$ depending on the convention). A natural family of distributions on the simplex of $\mathbb R^{n}$ is made of the Dirichlet $\mathcal D(\alpha_1,\ldots,\alpha_{n+1})$ distributions with density $$f(\mathbf y)=\dfrac{\Gamma(\alpha_1)\cdots\Gamma(\alpha_{n+1})}{\Gamma(\alpha_1+\cdots+\alpha_{n+1})}y_1^{\alpha_1-1}\cdots y_{n+1}^{\alpha_{n+1}-1}$$ over this simplex of $\mathbb R^{n}$. The Uniform case corresponds to $$\alpha_1=\cdots=\alpha_{n+1}=1$$

Question 3: In many importance sampling integration problems, they give the example of Uniform 1D, in which case $q(⋅)=1/(b−a)$, so this term can be removed from the expectation/sum to the front. How can I calculate this same volume, assuming I opt for $q(⋅)$ to have a nested hierarchical structure?

The volume of the simplex of $\mathbb R^{n}$ is given by the constant $$\dfrac{\Gamma(\alpha_1+\cdots+\alpha_{n+1})}{\Gamma(\alpha_1)\cdots\Gamma(\alpha_{n+1})}$$