# Importance Sampling

Importance sampling is a powerful and pervasive technique in statistics, machine learning and randomized algorithms.

## Basics

Importance sampling is a technique for estimating the expectation $\mu$ of a random variable $f(x)$ under distribution $p$ from samples of a different distribution $q.$

The key observation is that $\mu$ is can expressed as the expectation of a different random variable $f^*(x)=\frac{p(x)}{q(x)}\! \cdot\! f(x)$ under $q.$

$$\mathbb{E}_{q}\! \left[ f^*(x) \right] = \mathbb{E}_{q}\! \left[ \frac{p(x)}{q(x)} f(x) \right] = \sum_{x} q(x) \frac{p(x)}{q(x)} f(x) = \sum_{x} p(x) f(x) = \mathbb{E}_{p}\! \left[ f(x) \right] = \mu$$

Technical condition: $q$ must have support everywhere $p$ does, $p(x) > 0 \Rightarrow q(x) > 0.$ Without this condition, the equation is biased! Note: $q$ can support things that $p$ doesn't.

Terminology: The quantity $w(x) = \frac{p(x)}{q(x)}$ is often referred to as the "importance weight" or "importance correction". We often refer to $p$ as the target density and $q$ the proposal density.

Now, given samples $\{ x^{(i)} \}_{i=1}^{n}$ from $q,$ we can use the Monte Carlo estimate, $\hat{\mu} \approx \frac{1}{n} \sum_{i=1}^n f^{*}(x^{(i)}),$ as an unbiased estimator of $\mu.$

## Remarks

There are a few reasons we might want use importance sampling:

1. Convenience: It might be trickier to sample directly from $p.$

2. Bias-correction: Suppose, we're developing an algorithm which requires samples to satisfy some "safety" condition (e.g., a minimum support threshold) and be unbiased. Importance sampling can be used to remove bias, while satisfying the condition.

3. Variance reduction: It might be the case that sampling directly from $p$ would require more samples to estimate $\mu.$ Check out these great notes for more.

4. Off-policy evaluation and learning: We might want to collect some "exploratory data" from $q$ and evaluate different "policies", $p$ (e.g., to pick the best one). Here's a link to a future post on off-policy evaluation and counterfactual reasoning and some cool papers: counterfactual reasoning, reinforcement learning, contextual bandits, domain adaptation.

There are a few common cases for $q$ worth separate consideration:

1. Control over $q$: This is the case in experimental design, variance reduction, active learning and reinforcement learning. It's often difficult to design $q,$ which results in an estimator with "reasonable" variance. A very difficult case is in off-policy evaluation because it (essentially) requires a good exploratory distribution for every possible policy. (I have much more to say on this topic.)

2. Little to no control over $q$: For example, you're given some dataset (e.g., new articles) and you want to estimate performance on a different dataset (e.g., Twitter).

3. Unknown $q$: In this case, we want to estimate $q$ (typically referred to as the propensity score) and use it in the importance sampling estimator. This technique, as far as I can tell, is widely used to remove selection bias when estimating effects of different treatments.

Drawbacks: The main drawback of importance sampling is variance. A few bad samples with large weights can drastically throw off the estimator. Thus, it's often the case that a biased estimator is preferred, e.g., estimating the partition function, clipping weights, indirect importance sampling. A secondary drawback is that both densities must be normalized, which is often intractable.

What's next? I plan to cover "variance reduction" and off-policy evaluation in more detail in future posts.