Multilevel example with blocks

Purpose: to compare drjacoby infered posterior distribution against the exact analytical solution in a multi-level model implemented via likelihood blocks.

Model

There are g groups with means mu_1 to mu_g. These component means are drawn from a normal distribution with mean 0 and sd 1.0. Within each group there are n observations drawn around each component mean from normal disribution with sd 1.0.

g <- 5
mu <- rnorm(g)

n <- 5
data_list <- list()
for (i in 1:g) {
  data_list[[i]] <- rnorm(n, mean = mu[i])
}
names(data_list) <- sprintf("group%s", 1:g)

Likelihood and prior:

Parameters dataframe:

L <- 5
df_params <- define_params(name = "mu_1", min = -L, max = L, block = c(1, 6),
                           name = "mu_2", min = -L, max = L, block = c(2, 6),
                           name = "mu_3", min = -L, max = L, block = c(3, 6),
                           name = "mu_4", min = -L, max = L, block = c(4, 6),
                           name = "mu_5", min = -L, max = L, block = c(5, 6))

MCMC

mcmc <- run_mcmc(data = data_list,
                 df_params = df_params,
                 loglike = "loglike",
                 logprior = "logprior",
                 burnin = 1e3,
                 samples = 1e5,
                 chains = 10,
                 silent = TRUE)

Plots

Black = posterior draws

Red = analytical solution to multi-level model

Green = analytical solution assuming independent groups (no second level)

# extract sampling draws
output_sub <- subset(mcmc$output, phase == "sampling")

for (i in 1:5) {
  # get posterior draws
  mu_draws <- output_sub[[sprintf("mu_%s", i)]]
  
  # get analytical solution for this group
  x <- seq(-L, L, l = 1001)
  m <- mean(data_list[[i]])
  fx <- dnorm(x, mean = m * n/(n + 1), sd = sqrt(1/(n + 1)))
  
  # get analytical solution if no multi-level model
  fx2 <- dnorm(x, mean = m, sd = sqrt(1/n))
  
  # overlay plots
  hist(mu_draws, breaks = seq(-L, L, l = 1001), probability = TRUE, col = "black",
       main = sprintf("mu_%s", i))
  lines(x, fx, col = 2, lwd = 4)
  lines(x, fx2, col = 3, lwd = 4)
}

Bob Verity and Pete Winskill

2021-12-16

Model

MCMC

Plots