Estimating the posterior predictive distribution by sampling

This video explains how to approximate the posterior predictive distribution using sampling in a Bayesian framework. The presenter begins by motivating the need for the posterior predictive distribution, highlighting its use in prediction and model checking via posterior predictive checks. He then outlines the exact equation for the posterior predictive distribution and notes that it often involves intractable calculations, necessitating a sampling-based approximation. The method involves two steps: first, sample a parameter value from the posterior distribution; second, sample a new data point from the likelihood conditional on that parameter. Repeating this process many times and histogramming the sampled data points yields an approximation to the posterior predictive distribution. The video illustrates this with a simple example: estimating the proportion of six-year-old children who can read, using a binomial likelihood and a beta prior. Two scenarios are compared: one with a small sample (10 children, 2 readers) and one with a larger sample (1000 children, 200 readers). The resulting posterior predictive distributions are similar, demonstrating that the sampling distribution’s uncertainty dominates over parameter uncertainty in this case. The video concludes by summarizing the two sources of uncertainty captured by the posterior predictive distribution: parameter uncertainty and sampling variability.