I want to show that $g_{\tau}(\mathbf{x}) = \mathbf{x}^T\hat{\beta}$ where $\hat{\beta} = \mathbf{X}^+\mathbf{y}$ and $\tau$ denotes a fixed training set is an unbiased learner, in the sense that: $$\newcommand{\Tau}{\mathrm{T}}E g_\Tau(\mathbf{x}) = g^*(\mathbf{x})$$

where $g^*(\mathbf{x})=\mathbf{x}^T\beta $ is the optimal prediction function, where T denotes a random training set.

I think it's possible to use the Tower property: $EE(X|Y)=E(X)$ but I'm not sure how to proceed.

Thanks in advance!

  • $\begingroup$ I guess you are referring to linear regressions, am I correct? $\endgroup$ – Nicg Oct 18 at 8:06

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