Importance sampling integration is a variance reduction technique used to estimate integrals, especially when direct sampling from the integrand is difficult or inefficient. Instead of sampling uniformly, importance sampling concentrates samples in regions where the integrand contributes most to the integral's value. This leads to a more accurate estimate with fewer samples.
The Core Idea
The fundamental concept behind importance sampling is to evaluate the expected value of a function by generating random samples from a probability density function (PDF) that approximates the integrand. This carefully chosen PDF, called the importance function, allows us to reduce the variance of the estimator compared to using a uniform sampling strategy or a PDF unrelated to the integrand.
How it Works
Here's a breakdown of the process:
-
Identify the Integral: You want to compute an integral of the form:
∫ f(x) dx -
Choose an Importance Function: Select a probability density function g(x) that:
- Is similar in shape to |f(x)|. The better the approximation, the more effective the variance reduction.
- Is easy to sample from.
- Has the property that g(x) > 0 whenever f(x) ≠ 0. This ensures that we can sample from regions where the integrand is non-zero.
-
Rewriting the Integral: The integral can be rewritten as:
∫ f(x) dx = ∫ [f(x) / g(x)] * g(x) dx = Eg[f(x) / g(x)]
where Eg[ ] denotes the expected value with respect to the distribution g(x).
-
Sampling and Estimation: Generate N samples, x1, x2, ..., xN, from the importance function g(x). Estimate the integral using the following formula:
Integral ≈ (1/N) Σi=1N [f(xi) / g(xi)]
The term f(xi) / g(xi) is called the importance weight. It corrects for the fact that we are sampling from g(x) instead of a uniform distribution or the function f(x) itself.
Advantages of Importance Sampling
- Variance Reduction: The primary benefit is a lower variance in the estimate compared to Monte Carlo integration with uniform sampling, especially when the integrand has peaks or is highly variable.
- Efficiency: By focusing samples on important regions, it requires fewer samples to achieve a desired level of accuracy.
Disadvantages of Importance Sampling
- Choice of Importance Function: Selecting a good importance function can be challenging and crucial to the success of the method. A poorly chosen g(x) can actually increase the variance.
- Estimating the PDF: Knowing or accurately estimating the unnormalized PDF is required.
Example
Imagine estimating the integral of a function f(x) that has a sharp peak. Instead of sampling uniformly across the domain, you could choose a Gaussian distribution g(x) centered around the peak of f(x). This ensures that more samples are concentrated in the region where f(x) contributes most to the integral, leading to a more accurate estimate.
Applications
Importance sampling is widely used in:
- Computer Graphics: Rendering complex scenes, especially with global illumination.
- Finance: Pricing options and other financial derivatives.
- Particle Physics: Monte Carlo simulations of particle interactions.
- Bayesian Statistics: Estimating posterior distributions.
In summary, importance sampling integration is a powerful technique for estimating integrals by strategically sampling from a probability distribution that reflects the behavior of the integrand, thereby improving efficiency and reducing variance in the estimated result.
