Independent Sampling (Standard Monte Carlo Methods)

Independence Sampling - Types

• Simple Sampling (Simple Monte Carlo) - similar to MLE
  • requires sampling from the actual distribution in order to make estimations
  • if one can’t sample from actual distribution, use rejection sampling, importance sampling, etc
• Rejection Sampling - as you scale to higher dimensions the inefficiency scales exponentially (requires evaluation of target distribution)
• Sampling in Bayesian Networks:
  • Logic Sampling - approximates 𝐏(𝑬=𝒆) queries, and some 𝐏(𝑸=𝒒|𝑬=𝒆) queries
  • Likelihood Weighting Sampling - approximates 𝐏(𝑸=𝒒|𝑬=𝒆) and 𝐏(𝑬=𝒆) queries
• Importance Sampling (IS) - for both types of importance sampling: hard to choose appropriate proposal/sampling distribution and as you scale to higher dimensions the variance of weight (i.e. 𝐏(𝑋)/𝐐(𝑋)) increases exponentially. Therefore, not very useful for high dimensional distributions unless we could decompose it into several smaller distributions. If it can’t be done, consider using MCMC
  • Un-Normalized Importance Sampling (UNIS):
    • solves the problem in Simple Sampling by sampling from a different distribution and prescribe weights (importance) to each sample. This would simulate sampling from the actual distribution. However, in order to compute weights it requires evaluation of target distribution
    • if one can’t evaluate the target distribution, use normalized importance sampling
  • Normalized Importance Sampling (NIS) - solves 𝐏(𝑸=𝒒|𝑬=𝒆) more accurately than Un-normalized Importance Sampling
• Particle Filtering Sampling - Sequential Monte Carlo - generates all the samples for one variable before moving to the next variable