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Relevance Vector Machine 2

feng posted @ Mon, 01 Oct 2012 19:39:44 +0800 in PRML , 1997 readers



在获得训练数据之前。$\mathbf{w}$是0均值的,那么给定一个$\mathbf{x}$之后,期望的输出应该是0。现在得到训练数据之后,我们可能发现输出的$\mathbf{t} = [t_1, t_2, ..., t_N]^T$并不完全是0。这个时候,在给定的训练数据之后,我们可以计算出来$\mathbf{x}$的后验概率。也就是计算

\[p(\mathbf{w}|\mathbf{t}, \mathbf{X}, \mathbf{\alpha}, \beta)\]。


\[p(\mathbf{w}|\mathbf{t}, \mathbf{X}, \mathbf{\alpha}, \beta) = \frac{p(\mathbf{w}|\mathbf{\alpha})p(\mathbf{t}|\mathbf{X}, \mathbf{\alpha}, \beta)}{p(t|\mathbf{X}, \mathbf{\alpha}, \beta)}\]。




\[-\frac{1}{2}\mathbf{w}^T A \mathbf{w} - \frac{1}{2}\sum_{i = 1}^{N}{\beta(t_i - \mathbf{w}^T \phi(\mathbf{x}_i))^2}\]

这里的$A = diag(\alpha_i)$。接下来进行化简,化简的过程中,我们可以尽情的扔掉与$\mathbf{w}$无关的项。最后得到的结果为

\[-\frac{1}{2}(\mathbf{w}^T(A + \beta \Phi^T\Phi)\mathbf{w} - 2\beta \mathbf{w}^T\Phi^T\mathbf{t})\]


\[-\frac{1}{2}(\mathbf{w} - \mathbf{m})^T\Sigma^{-1}(\mathbf{x} - \mathbf{m}) = -\frac{1}{2}(\mathbf{w}^T\Sigma^{-1}\mathbf{w} - 2 \mathbf{w}^T\Sigma^{-1}\mathbf{m})+const\]


\[\Sigma = (A + \beta\Phi^T\Phi)^{-1}\]

\[\mathbf{m} = \beta\Sigma\Phi^{T}\mathbf{t}\]






\[\mathcal{N}(\mathbf{x}|\mathbf{m}, \mathbf{\Sigma}) = \frac{1}{(2\pi)^{D/2}|\Sigma|^{1/2}} exp{-\frac{1}{2}(\mathbf{x} - \mathbf{m})^T \Sigma^{-1} (\mathbf{x} - \mathbf{m})}\]

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