6.6 Beta-Binomial MCMC

# define model in stan language
bb_model <- "
  data {
    int<lower = 0, upper = 10> Y;
  }
  parameters {
    real<lower = 0, upper = 1> pi;
  }
  model {
    Y ~ binomial(10, pi);
    pi ~ beta(2, 2);
  }
"
# https://github.com/stan-dev/rstan/wiki/Configuring-C---Toolchain-for-Windows#r-42
# use stan to simulate posterior
bb_sim <- rstan::stan(model_code = bb_model, data = list(Y = 9), 
               chains = 4, iter = 5000*2, seed = 84735)

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• Uses 4 chains and 10000 samples of which 1/2 are discarded by default for burn-in

• Result is a stanfit object, which can be used to extract the samples

# for examining using view
chains <- as.data.frame(as.array(bb_sim, pars = "pi"))
# look at a zoom in of the sample trace
mcmc_trace(bb_sim, pars = "pi", window = c(50,100),size =0.1)

• Trace shows the samples exploring the parameter space but also illustrates non-zero autocorrelation.

• We can also plot the resulting distribution of samples (book shows that this is close to beta-binomial expected)

mcmc_dens(bb_sim, pars = "pi") + 
  yaxis_text(TRUE) + 
  ylab("density")