Suppose that in the polynomial regression Example 2.

1 we select the linear class of functions \(\mathscr{G}_{p}\) with \(p \geqslant 4\). Then, \(g^{*} \in \mathscr{G}_{p}\) and the approximation error is zero, because \(g^{\mathscr{G}_{p}}(\boldsymbol{x})=g^{*}(\boldsymbol{x})=\boldsymbol{x}^{\top} \boldsymbol{\beta}\), where \(\boldsymbol{\beta}=[10,-1400,400,-250,0, \ldots, 0]^{\top} \in \mathbb{R}^{p}\). Use the tower property to show that the learner \(g_{\tau}(\boldsymbol{x})=x^{\top} \widehat{\boldsymbol{\beta}}\) with \(\widehat{\boldsymbol{\beta}}=\mathbf{X}^{+} \boldsymbol{y}\), assuming \(\operatorname{rank}(\mathbf{X}) \geqslant 4\), is unbiased:

\[ \mathbb{E} g_{\mathscr{T}}(\boldsymbol{x})=g^{*}(\boldsymbol{x}) \]