4

I learned why MAP suffers from being reparametrization invariance while MLE not from this answer, but I don't know why reparametrization invariance even matters? What is the non-linear mapping concretely and why we do the non-linear mapping depicted below?

I wonder if I am wrong: After we obtain the posterior we can provide a point estimate summary, such as mean, median, and mode if necessary. In which scenarios do we apply non-linear mapping to the posterior? For the sake of what?

enter image description here

I learned from this section in this book that:

We will now show that different parameterizations can pick different points in this interval arbitrarily. enter image description here

But it is unclear why we apply the parameterization to the MAP. Any materials(tutorials) would also be very appreciated.

Lerner Zhang
  • 5,017
  • 1
  • 31
  • 52
  • 1
    This lack of invariance shows that the MAP is not a stable estimator. When considering the estimated density $f(\cdot|\theta)$ relying on the plugged-in MAP, this estimated density will vary according to the choice of parameterisation. – Xi'an May 29 '21 at 08:29
  • @Xi'an Thanks, professor. Could you please refer me to some materials on the `plugged-in MAP`? – Lerner Zhang May 29 '21 at 14:02

1 Answers1

5

If $X\sim\mathcal N(\mu,1)$ and $\mu\sim\mathcal N(0,10)$, the MAP of $\theta$ is $$\theta^\text{MAP}(x)=\dfrac{10}{11}x$$while the MAP of $\alpha=\alpha(\theta)=\exp\{\theta\}$ is $$\alpha^\text{MAP}(x)=\exp\left\{\dfrac{10}{11}x-\frac{\sqrt{10}}{\sqrt{11}}\right\}$$ The plot of the plugged-in density estimates $\varphi(\cdot;\theta^\text{MAP}(x),1)$ versus $\varphi(\cdot;\log\{\alpha^\text{MAP}(x)\})$ shows a clear difference between these two densities.

enter image description here

However the distinction decreases with the sample size increasing.

Xi'an
  • 90,397
  • 9
  • 157
  • 575
  • I think I have got the idea of the plugged-in MAP, but I still wonder when we need to do the plugged-in MAP? And for what purpose? I mean, I know that we provide a mean or mode of a posterior distribution for the purpose of summarizing the posterior using just one number. Then why the plugged-in MAP? – Lerner Zhang May 30 '21 at 01:01
  • You should expand on the reasons you want a point estimate and on its ultimate purpose... – Xi'an May 30 '21 at 07:57
  • I learned from [here](https://statswithr.github.io/book/losses-and-decision-making.html) that it can be used for decision making. – Lerner Zhang May 30 '21 at 08:26