MAP estimators (cont’d)

In connection with Anthony’s comments, here are the details for the normal example. I am using a flat prior on when . The MAP estimator of is then . If I consider the change of variable , the posterior distribution on is

and the MAP in is then obtained numerically. For instance, the R code

f=function(x,mea) dnorm(log(x/(1-x)),mean=mea)/(x*(1-x))
g=function(x){ a=optimise(f,int=c(0,1),maximum=TRUE,mea=x)$max;log(a/(1-a))}
plot(seq(0,4,.01),apply(as.matrix(seq(0,4,.01)),1,g),type="l",col="sienna",lwd=2)
abline(a=0,b=1,col="tomato2",lwd=2)

shows the divergence between the MAP estimator and the reverse transform of the MAP estimator of the transform… The second estimator is asymptotically (in ) equivalent to .

An example I like very much in The Bayesian Choice is Example 4.1.2, when observing with a double exponential prior on . The MAP is then always !

The dependence of the MAP estimator on the dominating measure is also studied in a BA paper by Pierre Druihlet and Jean-Michel Marin, who propose a solution that relies on Jeffreys’ prior as the reference measure.