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 $\mu$ can be 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, $f(x) p(x) > 0 \Rightarrow q(x) > 0.$ 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 $q$ as 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 to use importance sampling:

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

2. Bias-correction: Suppose we're developing an algorithm that 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 (https://timvieira.github.io/blog/counterfactual-reasoning-and-learning-from-logged-data/) 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. As far as I can tell, this technique is widely used to remove selection bias when estimating the 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.