Decision theoretic point estimation
Outline
Topics
- Deriving a point estimate from decision theory.
Rationale
We have seen in week 2 some examples of point estimates (posterior mean, posterior mode).
These are actually special cases of decision theory with specific choices of loss functions.
This page provides a general framework to answer the question: “how to summarize a posterior distribution with one point?”
Setup
Example
Assume: a square loss, \(L(a, p) = (a - p)^2\), where \(a \in A = \mathbb{R}\).
Some initial simplification: on the objective function…
\[ \begin{aligned} \delta_{\text{B}}(Y) &= \operatorname{arg\,min}\{ \mathbb{E}[L(a, X) | Y] : a \in A \} \\ &= \operatorname{arg\,min}\{ \mathbb{E}[(X - a)^2 | Y] : a \in A \} \\ &= \operatorname{arg\,min}\{ \mathbb{E}[X^2 | Y] - 2a\mathbb{E}[X | Y]] + a^2 : a \in A \} \\ &= \operatorname{arg\,min}\{ - 2a\mathbb{E}[X | Y]] + a^2 : a \in A \} \end{aligned} \]
Question: under a square loss, \(\delta_{\text{B}}\) can be simplified to…