STAT 535, Homework 5, Spring 2024:

1) Do Problem 12.3 in the Bayes Rules! book.

2) Do Problem 12.4 in the Bayes Rules! book.

3) Read Exercise 12.5 in the Bayes Rules! book about the bald_eagles data.  
[Part 1 of Problem -- Required for everyone:] 
Fit a Poisson regression model relating Y to X1 and X2.  Use vague/uninformative priors.
Comment on whether X1 and/or X2 are apparently important or unimportant in explaining the count of eagles sightings.
In whatever way you choose, assess the fit of the Poisson model and comment on it.

[Part 2 of Problem -- Required for grad students, extra credit for undergrads:] 
Fit a model that includes X1, X2, and the interaction X1*X2.  Use whatever model selection technique you like to 
choose between the interaction model and the no-interaction model.

# Some possibly helpful code for reading in the bald_eagles data set.
# If you are using stan_glm in the rstanarm package, much of this is not necessary ...
# you can just use the variables in the bald_eagles data frame directly,
# and could use year:hours to specify the interaction.

# UPDATE:  If you are using base R and NOT 'stan_glm' to fit the Poisson regression, then it seems like you
# encounter numerical issues when fitting the interaction model.  To correct this (only necessary if using base R)
# just redefine the 'Year' predictor (x1) to be 'Years since 2000' ... 
# this cuts down on the huge values of x1 and x1x2.
# The modified code is below:

library(bayesrules)
data(bald_eagles)
y <- bald_eagles$count
x1.init <- bald_eagles$year
x1 <- x1.init-2000
x2 <- bald_eagles$hours
x1x2 <- x1*x2


4) Do Problem 13.1 in the Bayes Rules! book.

5) Do Problem 13.4(b,c,d) in the Bayes Rules! book.

6) Do Problem 13.5 in the Bayes Rules! book.

7) Do Problem 13.7 in the Bayes Rules! book.  
[For part (a), the pp_check thing is optional.  
For part (b), just report the estimated model on the probability scale.]

# Some possibly helpful code for reading in the hotel_bookings data set.
# If you are using stan_glm in the rstanarm package, much of this is not necessary ...
# you can just use the variables in the hotel_bookings data frame directly.

library(bayesrules)
data(hotel_bookings)
hotel_bookings
y <- hotel_bookings$is_canceled
y<-as.numeric(y)-1   # subtracting 1 converts the 1's and 2's to 0's and 1's.
x1 <- hotel_bookings$lead_time
x2 <- hotel_bookings$previous_cancellations
x3 <- hotel_bookings$is_repeated_guest
x4 <- hotel_bookings$average_daily_rate