Rejection Sampling

Definition

Rejection sampling is a technique used to generate random samples from a target probability distribution by using samples from a simpler, proposal distribution. The method involves sampling from the proposal distribution and then deciding whether to accept or reject each sample based on a criterion related to the target distribution. This approach is particularly useful when direct sampling from the target distribution is difficult or impossible.

5 Must Know Facts For Your Next Test

1. Rejection sampling works by selecting a proposal distribution that is easy to sample from and encompasses the target distribution over its support.
2. The acceptance criterion typically involves comparing a uniform random number against an appropriately scaled version of the target probability density function.
3. It is crucial for the proposal distribution to be chosen wisely; if it poorly approximates the target, many samples may be rejected, leading to inefficiency.
4. Rejection sampling can be extended to multi-dimensional distributions, but the complexity increases as dimensions grow due to the curse of dimensionality.
5. This method is commonly used in Bayesian statistics and machine learning for generating samples from posterior distributions.

Review Questions

• How does rejection sampling utilize both a proposal distribution and an acceptance criterion to generate samples from a target distribution?
  • Rejection sampling begins by selecting a proposal distribution that is easier to sample from compared to the target distribution. Samples are drawn from this proposal distribution, and each sample is then evaluated against an acceptance criterion, which typically compares a uniform random number against a scaled version of the target's probability density function. If the sample meets this criterion, it is accepted; otherwise, it is rejected. This process allows for effective approximation of samples from more complex distributions.
• What are some strategies for choosing an effective proposal distribution in rejection sampling, and how do they impact the efficiency of the sampling process?
  • Choosing an effective proposal distribution is key in rejection sampling. Ideally, the proposal should closely resemble the shape of the target distribution while being easy to sample from. Strategies include using distributions that are known to approximate the target well or adjusting parameters of known distributions to fit. The efficiency directly depends on how many samples are accepted; if too many are rejected because the proposal does not cover the target well, this leads to wasted computation and time.
• Evaluate how rejection sampling can be integrated with other statistical methods to enhance sampling efficiency in complex models.
  • Rejection sampling can be integrated with techniques such as importance sampling or Markov Chain Monte Carlo (MCMC) methods to improve efficiency when working with complex models. By combining rejection sampling with importance sampling, one can adjust for cases where certain regions of the sample space are more relevant than others, thereby reducing variance in estimations. In MCMC methods, rejection sampling can serve as a stepping stone to ensure that proposed moves maintain adherence to the target distribution, ultimately leading to better convergence and more reliable estimations in Bayesian analysis.

Related terms

Proposal Distribution: A probability distribution from which samples are drawn in rejection sampling, chosen for its ease of sampling and ability to cover the target distribution.

Acceptance Ratio: The probability that a sample drawn from the proposal distribution will be accepted during rejection sampling, defined as the ratio of the area under the target distribution to the area under the proposal distribution.

Monte Carlo Method: A statistical technique that uses random sampling to estimate mathematical functions and simulate processes, with rejection sampling being one of its specific applications.