R/Bayes: Why is a wildly different prior not affecting the posterior in bayesglm{arm}?

Question

I am trying to see how different priors affect the posterior estimates in bayesglm{arm}. But no matter what the prior is, why is the posterior hardly changing?

library(Zelig); data(turnout) #Using the "turnout" data from the "Zelig" package

library(arm) #Loading the "arm" package
model.1 = bayesglm(vote ~ race + age, family=binomial(link=logit),
                   prior.df=1, prior.scale=2.5,
                   data=turnout) #This is the default specificication with a weakly informative prior

summary(model.1) yields:

Coefficients:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept) 0.042320   0.176567   0.240  0.81058    
racewhite   0.641959   0.134136   4.786  1.7e-06 ***
age         0.011222   0.003046   3.683  0.00023 ***

Now compare this to:

model.2 = bayesglm(vote ~ race + age, family=binomial(link=logit),
                   prior.df=1, prior.scale=2.5, prior.mean=c(-100,0), #Specifying a crazy prior
                   data=turnout)

Summary(model.2) yields:

Coefficients:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept) 0.039891   0.176720   0.226 0.821410    
racewhite   0.645076   0.134465   4.797 1.61e-06 ***
age         0.011219   0.003047   3.682 0.000231 ***

What's going on here? Surely a prior which is so different would influence the maximum likelihood estimates enough to sharply change the posterior?

Answer 1

Your second prior is not quite so crazy as you think. A t-distribution with one degree of freedom has very wide tails. In this with location = -100 and scale = 2.5 you still have a density of > 1e-5 for 0.645 and 0.0112. This is still more than 1/2000 of the maximum density of your prior. This is easily overwhelmed by the likelihood function for your amount of data. BTW the 1/2000 is not intended to create a link to your sample size, i just want to made concrete how wide the tails of your prior are.