Calculate a normal average from uniform samples Ok, this feels like it should be pretty basic, but maybe I'm missing the right terminology to google this.
Suppose I have a (non-analytical) function $f(x)$ that I have evaluated for a large number of uniform values $x_i \in [-1, 1]$. That is, I have uniform samples of points $(x_i, f_i=f(x_i))$.
Now I want to know what the expectation value $\bar{f}$ of $f$ would be if the values of $x$ were drawn from a Normal distribution of width $\sigma \ll 1$. Ultimately, I want to do this for different values of $\sigma$, so that I can generate a plot of $\bar{f}(\sigma)$.
My first intuition was to simply multiply the values $f_i$ with a weight $w_i$ where $w_i$ is the value of the probability density function (PDF) for the normal distribution with $\sigma$. I would then normalize with $\sum w_i$ and average the weighted and normalized $f_i$. However, that does not seem to produce correct results.
I've also tried rejection sampling (taking a simple average of the $f_i$, but randomly excluding $f_i$ if a random value $r_i \in [0, 1] $ is smaller than the normalized PDF at $x_i$). This gives me good results, but it's quite slow.
Is there a better method than rejection sampling for this? It still seems to me like I should be able to average my original $f_i$ with some appropriate weighting, but maybe I'm wrong.
There's lots of methods for generating normal-distributed $x_i$ from uniform $x_i$, of course, but that's not what I want to do: I don't want to evaluate $f(x)$ for any new values $x_i$, so all I have are the existing $f_i$. My $x_i$ in this case are in fact random, but in principle it should also work for $x_i$ on a dense regular grid.

On principle you simply cannot estimate$$\int_{-\infty}^{\infty} f(x) \varphi_\sigma(x)\,\text dx$$based on $f(x_i)$'s with $x_i\in(-1,1)$ since there is not information for $f(x)$ when $x\not\in(-1,1)$...
Now, if $\sigma$ is small enough for$$\int_{-1}^{1} \varphi_\sigma(x)\,\text dx\approx 1$$you can consider$$\frac{2}{n}\sum_{i=1}^nf(x_i)\varphi_\sigma(x_i)$$as a pseudo-importance sampler but, depending on the function $f$, the approximation can completely go south, i.e. be completely unreliable: take an extreme example when $f$ is very large on $(-2,-1)$ and very small or null over $(-1,1)$:
$$f(x)=\begin{cases}
1/\varphi_\sigma(x) &\text{when }x\in(-2,-1)\\
0 &\text{elsewhere}
\end{cases}
$$
Then
$$\int_{-\infty}^{\infty} f(x) \varphi_\sigma(x)\,\text dx= 1$$
while
$$\frac{1}{n}\sum_{i=1}^nf(x_i)\varphi_\sigma(x_i)=0$$
Obviously, if the support of $f$ is $(-1,1)$ then
$$\int_{-\infty}^{\infty} f(x) \varphi_\sigma(x)\,\text dx= \int_{-1}^{1} 2f(x) \varphi_\sigma(x)\,\frac{\text dx}{2}=2\mathbb E[f(U)\varphi_\sigma(U)]$$when $U\sim\mathcal U(-1,1)$.
