Let us define a random variable $X$ with density function $p(x)$. We wish to calculate $\mathbb{E}[f(X)] = \int f(x)p(x)dx$. We can compute the expectation by Monte Carlo simulations as

$$\mathbb{E}[f(X)] =\frac{1}{N} \sum_{i=1}^{N} f(x_i)p(x_i)$$

where $x_i$ are sampled from $p(x)$.

Sometimes it is impossible to generate random samples from $p(x)$. In such cases we use importance sampling and estimate

$$\mathbb{E}[f(X)] =\frac{1}{N} \sum_{i=1}^{N} \frac{f(x_i)p(x_i)}{g(x_i)}$$

where $x_i$ are sampled from $g(x)$.

My question is that if we know that $\delta_1 \le \mathbb{E}[f(X)] \le \delta_2$, then how do we exploit this information in estimating $\mathbb{E}[f(X)]$ using importance sampling.

I've done a lot of google search looking for importance sampling with prior information etc. However, the closest that I have come to involve evaluating the $\mathbb{E}[f(X)]$ over some subset of the support of the random variable $X$.

Still it does not help me with utilizing the additional information that I have about $\mathbb{E}[f(X)]$ from the bounds.

I would like to know the following:

**Has such a thing been studied?**

**If yes, can somebody point me to a useful reference?**