> For the complete documentation index, see [llms.txt](https://www.learnros2.com/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://www.learnros2.com/review-of-mathematics/probability.md).

# Probability

##

## Importance Sampling

According to wiki, importance sampling is a Monte Carlo method for evaluating properties of a particular distribution, while only having samples generated from a different distribution than the distribution of interest.

$$
\begin{align\*}
E\_p\[f(X)] & = \int{}f(x)p(x)dx \\
& = \int{}f(x)\frac{p(x)}{q(x)}q(x)dx \\
& = E\_q\[\frac{p(X)}{q(X)}f(X)]

\end{align\*}
$$

The term $$\frac{p(x)}{q(x)}$$is a weight factor. The left side of the equation is the expectation of $$f(X)$$ under distribution $$p(x)$$. Suppose we have samples of $$X$$under distribution $$q(x)$$, we can reconstruct the expectation of $$f(X)$$ under distribution $$p(x)$$ by weighting the samples (i.e. multiplying $$f(x)$$by $$\frac{p(x)}{q(x)}$$.

The distribution $$p(x)$$is sometimes called the target distribution and they are often not directly accessible.


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