5 September 1

5.1 Announcements

  • Office hours 9:30 - 10:30 today or by appointment
    • Lots of good questions are being asked and answered!
  • Poll
  • Questions about assignment 2?

5.2 Bayesian hierarchical models

  • Review of the Bayesian hierarchical modeling framework

\[\text{Data model:} \;\;[\mathbf{z}|\mathbf{y},\boldsymbol{\theta}_{D}]\] \[\text{Process model:} \;\;[\mathbf{y}|\boldsymbol{\theta}_{P}]\] \[\text{Parameter model:} \;\;[\boldsymbol{\theta}]\]

5.2.2 Simulating data from the prior predictive distribution

  • The prior predictive distribution is capable of providing predictions/forecasts without the use of any data
    • Other fields call this “simulation modeling” or a “sensitivity analysis”
    • It is basically data free statistics (i.e., prediction, forecasts, and inference is 100% assumption driven)
    • Used as a form of model (assumption) checking in Bayesian statistics
    • Super easy to do and helps us “prototype” our statistical model before we put in any more work
  • Live demonstration in R for the whooping crane data example
  • Warning about using the data twice

5.2.3 Model fitting

  • Class discussion
  • Markov chain Monte Carlo
    • Algorithm 4.2 (see pg. 169 in Wikle et al. 2019)
    • Next time in class
  • Rejection sampling and Approximate Bayesian computation
    • Live demonstration in R for the whooping crane data example
##   Winter  N
## 1   1950 31
## 2   1951 25
## 3   1952 21
## 4   1953 24
## 5   1954 21
## 6   1955 28
# Required functions
calc.lambda <- function(t, t0, gamma, lambda0) {
    lambda0 * exp(gamma * (t - t0))
}
rdunif <- function(n, min, max) {
    sample(min:max, size = n, replace = TRUE)
}

# Algorithm settings
K.tries <- 10^6  # Number of simulated data sets to make
diff <- matrix(, K.tries, 1)  # Vector to save the measure of discrepancy between simulated data and real data
error <- 1000  # Allowable difference between simulated data and real data

# Known random variables and parameters
t0 <- 1949
t <- df1$Winter
z <- df1$N
n <- length(z)

# Unkown random variables and parameters
posterior.samp.parameters <- matrix(, K.tries, 3)  # Matrix to samples of save unknown parameters
colnames(posterior.samp.parameters) <- c("gamma", "lambda0", "p")

y <- matrix(, K.tries, n)  # Matrix to samples of save unknown number of whooping cranes

for (k in 1:K.tries) {
    # Simulate from the prior predictive distribution
    gamma.try <- runif(1, 0, 0.1)  # Parameter model
    lambda0.try <- rdunif(1, 2, 50)  # Parameter model 
    lambda.try <- calc.lambda(t = t, t0 = t0, gamma = gamma.try, lambda0 = lambda0.try)  # Mathematical equation for process model
    y.try <- rpois(n, lambda.try)  # Process model
    p.try <- runif(1, 0, 1)  # Prior or data model
    z.try <- rbinom(n, y.try, p.try)  # Data model
    
    # Record difference between draw of z from prior predictive distribution and
    # observed data
    diff[k, ] <- sum(abs(z - z.try))
    
    # Save unkown random variables and parameters
    y[k, ] <- y.try
    posterior.samp.parameters[k, ] <- c(gamma.try, lambda0.try, p.try)
}

# Calculate acceptance rate
length(which(diff < error))/K.tries
## [1] 0.015898

5.3 Summary and comments

  • Bayesian hierarchical modeling framework is incredibly flexible
    • Motivated by a data set or practical problem, you can build your own “custom” statistical models
    • You can always “turn the Bayesian crank” for whatever model you develop (with some warnings of course!)
  • Practice the process of model specification (what we did today in class) at home on a different data set