## MAP estimators (cont’d)

**I**n 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 .

**A**n 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 !

**T**he 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.

April 25, 2011 at 12:14 am

[...] estimators, and loss functions. I posted a while ago my perspective on MAP estimators, followed by several comments on the Bayesian nature of those estimators, hence will not reproduce them here, but the core of the [...]

September 13, 2009 at 10:32 pm

From Jerymn (2005) [http://dx.doi.org/10.1214/009053604000001273] the answer to my question is apparently that this is indeed the unique way to obtain parametrization-invariant MAP estimates.

September 13, 2009 at 5:31 pm

Druilhet and Marin’s paper is very illuminating. Does this imply that whenever one wants a MAP estimate, they should compute it wrt a (Bernardo) reference prior dominating measure? I’m pretty sure this is not prevalent in machine learning, a literature in which MAP estimation is used fairly often. I think parametrization invariance is pretty crucial for meaningful inference. Do there exist other (non-trivial and statistically reasonable) candidates for dominating measures that will give identical MAP estimates under reparametrization?

September 13, 2009 at 5:42 pm

Formally, you could define an arbitrary prior/reference measure on an arbitrary parameterisation and then impose invariance by reparameterisation. Of course this would be mostly meaningless…