rstanarm+loo - add baseline elpd

Summary:

To improve the interpretability of elpd, we can give a baseline elpd result without any covariates just by fitting the given distribution by family argument. We can do that without need to run Stan for the baseline model. We could print the baseline when printing the loo result for a single model.

Description:

Here is an example with Bernoulli. The example case is at https://cdn.rawgit.com/avehtari/BDA_R_demos/5964b1b7/demos_rstan/diabetes.html

compute baseline

m<-sum(2-as.numeric(diabetes$outcome)) # number of successes
ab=2 # beta prior weight for intercept, 2 is equal to uniform prior
elpd0<-log((m-1+ab/2)/(n-1+ab))m+log((n-m-1+ab/2)/(n-1+ab))(n-m) # -1 comes from leaving one out
elpd02<-log((m-1+ab/2)/(n-1+ab))^2m+log((n-m-1+ab/2)/(n-1+ab))^2(n-m)
sdelpd0=sqrt((c/(n-1)-(b/(n-1))^2)*n)
sprintf("The baseline by guessing the larger class elpd_loo %.1f sd %.1f",elpd0,sdelpd0)

alternatively if we want to use pointwise for making also the pairwise comparison

elpd0_pointwise=matrix(0,1,n)
mi=2-as.numeric(diabetes$outcome)
elpd0_pointwise[mi==1]=log((m-1+ab/2)/(n-1+ab))
elpd0_pointwise[mi==0]=log((n-m-1+ab/2)/(n-1+ab))
elpd0=sum(elpd0_pointwise)
sdelpd0=sd(elpd0_pointwise)*sqrt(n)
sprintf("The baseline by guessing the larger class elpd_loo %.1f sd %.1f",elpd0,sdelpd0)

Start by reviewing how a single-model loo result is printed and how the supplied family argument is handled. Compare the Bernoulli baseline formulas and the pointwise alternative in the issue, then determine how baseline elpd should be calculated and displayed without running Stan. Done means the single-model loo output includes the baseline result with its uncertainty.