PRML-公式推导 - 3.63

考虑\(y(x),y(x')\)间的协方差

\[\begin{eqnarray} cov[y(x),y(x')] &=& cov[\phi(x)^Tw,w^T\phi(x')] \ &=& \phi(x)^TS_N\phi(x') = \beta^{-1}k(x,x') \tag{3.63} \end{eqnarray} \]

推导

\(cov[\phi(x)^Tw,w^T\phi(x')]=\mathbb{E}[\phi^T(x) w w^T\phi(x')]-\phi^T(x)m_Nm_N^T\phi(x')\)
\(=\phi^T(x)\mathbb{E}[ww^T]\phi(x')-\phi^T(x)m_Nm_N^T\phi(x')\)
\(=\phi^T(x)[cov(w)+\mathbb{E}[w]\mathbb{E}^T[w]]\phi(x')-\phi^T(x)m_Nm_N^T\phi(x')\)
\(因为有\mathbb{E}[w]\mathbb{E}^T[w]=m_Nm_N^T,所以和后面那一项抵消了\)
\(=\phi^T(x)S_N\phi^T(x')\)
\(=\beta^{-1}k(x,x')\)