Rejection Sampling

Definition

Rejection sampling is a statistical technique used to generate random samples from a target probability distribution by using samples from a proposal distribution. This method involves drawing samples from the proposal distribution and accepting or rejecting them based on a comparison of the target and proposal distributions, which helps ensure that the final samples reflect the desired distribution. It’s particularly useful when direct sampling from the target distribution is difficult or impossible.

5 Must Know Facts For Your Next Test

1. Rejection sampling requires a proposal distribution that is easy to sample from and that sufficiently overlaps with the target distribution.
2. The acceptance criterion involves comparing a uniform random number with a scaled version of the target density; if the sample falls below this curve, it is accepted.
3. The efficiency of rejection sampling is determined by the ratio of the area under the target distribution to the area under the proposal distribution, impacting how many samples are ultimately accepted.
4. This technique is widely used in Bayesian statistics for generating samples from posterior distributions, especially when these distributions are complex.
5. Rejection sampling can be less efficient than other methods, such as Markov Chain Monte Carlo (MCMC), particularly when the acceptance rate is low due to poor choice of proposal distribution.

Review Questions

• How does rejection sampling ensure that the generated samples accurately represent the target distribution?
  • Rejection sampling ensures accurate representation by accepting samples based on a criterion that compares their likelihood under both the target and proposal distributions. Specifically, samples drawn from the proposal distribution are only accepted if they fall below a threshold defined by scaling the target density. This process effectively filters out samples that do not align with the target distribution, allowing only representative samples to be retained.
• Evaluate the impact of selecting an inappropriate proposal distribution on the efficiency of rejection sampling.
  • Selecting an inappropriate proposal distribution can significantly hinder the efficiency of rejection sampling. If the proposal does not adequately cover the support of the target distribution or if it leads to a low acceptance rate, many proposed samples will be rejected, requiring more draws to achieve a satisfactory number of accepted samples. This inefficiency results in increased computation time and may negate some advantages of using rejection sampling over direct sampling methods.
• Critically analyze how rejection sampling compares to other sampling techniques like MCMC in terms of practicality and application in Bayesian statistics.
  • Rejection sampling and MCMC are both valuable techniques in Bayesian statistics, but they differ in practicality based on their application contexts. While rejection sampling is straightforward and conceptually simpler, it can become impractical for high-dimensional distributions due to poor acceptance rates. In contrast, MCMC methods are designed to navigate complex posterior landscapes more effectively and often yield better efficiency for high-dimensional problems. The choice between them largely depends on the characteristics of the target distribution and computational resources available.

Related terms

Proposal Distribution: A probability distribution from which samples are drawn in rejection sampling, typically chosen to be easy to sample from and to cover the support of the target distribution.

Acceptance Rate: The proportion of proposed samples that are accepted during the rejection sampling process; it can influence the efficiency of the method.

Monte Carlo Method: A broad class of computational algorithms that rely on repeated random sampling to obtain numerical results, of which rejection sampling is one specific technique.