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20.8: Appendix:

  • Page ID
    8824
  • 20.8.1 Rejection sampling

    We will generate samples from our posterior distribution using a simple algorithm known as rejection sampling. The idea is that we choose a random value of x (in this case prespondp_{respond}) and a random value of y (in this case, the posterior probability of prespondp_{respond}) each from a uniform distribution. We then only accept the sample if y<f(x)y < f(x) - in this case, if the randomly selected value of y is less than the actual posterior probability of y. Figure 20.6 shows an example of a histogram of samples using rejection sampling, along with the 95% credible intervals obtained using this method.

    # Compute credible intervals for example
    
    nsamples <- 100000
    
    # create random uniform variates for x and y
    x <- runif(nsamples)
    y <- runif(nsamples)
    
    # create f(x)
    fx <- dbinom(x = nResponders, size = 100, prob = x)
    
    # accept samples where y < f(x)
    accept <- which(y < fx)
    accepted_samples <- x[accept]
    
    credible_interval <- quantile(x = accepted_samples, 
                                  probs = c(0.025, 0.975))
    kable(credible_interval)
      x
    2.5% 0.54
    98% 0.73
    Rejection sampling example.The black line shows the density of all possible values of p(respond); the blue lines show the 2.5th and 97.5th percentiles of the distribution, which represent the 95 percent credible interval for the estimate of p(respond).
    Figure 20.6: Rejection sampling example.The black line shows the density of all possible values of p(respond); the blue lines show the 2.5th and 97.5th percentiles of the distribution, which represent the 95 percent credible interval for the estimate of p(respond).