discrete-continuous-AIC.R

#Create an explicit binomially distributed set of numbers
n = 1000
frac = 0.9
x = rep(c(1,0),times = c(n*frac, n*(1-frac)))

#Fit a Gaussian model and a binomial model to the same data
gauss_mod = glm(x~1,family = gaussian)
binom_mod = glm(x~1, family= binomial)

#Compare AIC
AIC(gauss_mod, binom_mod)

Overall result: the binomial model would be thrown out based on AIC comparisons, even though it's the actual model for the data:

	df	AIC
gauss_mod	2	425
binom_mod	1	648

As for why that's true: the likelihood for continuous distributions are based on probability density functions, which can range from 0 to infinity, meaning the log-Likelihood (LL) can range from -infinity -> +infinity. Discrete distributions use probability mass functions, with a maximum value of 1, or a maximum LL of 0.