3.3.

Lemma 15

\[\begin{eqnarray} \mathrm{E}_{w} \left[ \phi(w \mid X^{n}) \right] & = & \mathrm{E}_{w} \left[ \frac{ 1 }{ Z_{n}(\beta) } \phi(w) \prod_{i=1}^{n} p(X_{i} \mid w)^{\beta} \right] \nonumber \\ & = & \frac{ \int_{W} \left( \right) \exp \left( - \frac{n \beta}{2} \norm{ J^{1/2} \left( w - w_{0} - \frac{ \hat{\xi}_{n} }{ \sqrt{n} } \right) }^{2} \right) \ dw }{ \int_{W} \exp \left( - \frac{n \beta}{2} \norm{ J^{1/2} \left( w - w_{0} - \frac{ \hat{\xi}_{n} }{ \sqrt{n} } \right) }^{2} \right) \ dw } \left( 1 + o_{p}(1) \right) \nonumber \end{eqnarray}\]

proof

From (3.9),

\[\begin{eqnarray} & & Z_{n}^{(1)} = \int_{W} \exp \left( - \frac{ n \beta }{ 2 } \norm{ J(w_{0})^{1/2} \left( w - w_{0} - \frac{ \xi_{n} }{ \sqrt{n} } \right) }^{2} \right) \ dw \exp \left( \frac{\beta}{2} \norm{ \xi_{n}}^{2} \right) \phi(w_{0}) (1 + o_{p}(1)) \nonumber \\ & \Leftrightarrow & \phi(w_{0}) = \frac{ Z_{n}^{(1)} }{ \exp \left( - \frac{\beta}{2} \norm{ \xi_{n}}^{2} \right) \int_{W} \exp \left( - \frac{ n \beta }{ 2 } \norm{ J(w_{0})^{1/2} \left( w - w_{0} - \frac{ \xi_{n} }{ \sqrt{n} } \right) }^{2} \right) \ dw (1 + o_{p}(1)) } \nonumber \\ & \Leftrightarrow & \phi(w_{0}) = \frac{ \int_{K(w) < \epsilon} \exp \left( - n \beta K(w) \right) \phi(w) \ dw }{ \exp \left( - \frac{\beta}{2} \norm{ \xi_{n}}^{2} \right) \int_{W} \exp \left( - \frac{ n \beta }{ 2 } \norm{ J(w_{0})^{1/2} \left( w - w_{0} - \frac{ \xi_{n} }{ \sqrt{n} } \right) }^{2} \right) \ dw (1 + o_{p}(1)) } \end{eqnarray}\]

$\Box$

Lemma 16

\[\begin{eqnarray} \mathrm{E}_{w} \left[ w \right] & = & w_{0} + \frac{1}{\sqrt{n}} \hat{\xi}_{n} + o_{p}(\frac{1}{\sqrt{n}}) \label{equation_03_10} \\ \mathrm{E}_{w} \left[ (w - w_{0}) (w - w_{0})^{\mathrm{T}} \right] & = & \frac{ J^{-1} }{ n \beta } + \frac{ \hat{\xi}_{n} \hat{\xi}_{n}^{\mathrm{T}} }{ n } + o_{p}(\frac{1}{n}) \label{equation_03_11} \\ \mathrm{E}_{w} \left[ f(x, w) \right] & = & \frac{ J^{-1} }{ n \beta } + \frac{ \hat{\xi}_{n} \hat{\xi}_{n}^{\mathrm{T}} }{ n } + o_{p}(\frac{1}{n}) \label{equation_03_12} \end{eqnarray}\]

Moreover,

\[\begin{eqnarray} \mathrm{E}_{w} \left[ f(x, w)^{2} \right] - \mathrm{E}_{w} \left[ f(x, w) \right]^{2} & = & \frac{1}{n \beta} \mathrm{tr} \left( J^{-1} \left( \nabla f(x, w_{0}) \right) \left( \nabla f(x, w_{0}) \right)^{\mathrm{T}} \right) + o_{p}(\frac{1}{n}) . \label{equation_03_13} \end{eqnarray}\]

proof

\[\begin{eqnarray} f(x, w) & = & f(x, w_{0}) + (w - w_{0})^{\mathrm{T}} \nabla f(x, w_{0} + w^{+}) \nonumber \\ & = & (w - w_{0})^{\mathrm{T}} \nabla f(x, w_{0} + w^{+}) \nonumber \end{eqnarray}\]

By taking,

\[\begin{eqnarray} \mathrm{E}_{w} \left[ f(x, w) \right] & = & \mathrm{E}_{w} \left[ (w - w_{0})^{\mathrm{T}} \nabla f(x, w_{0} + w^{+}) \right] \nonumber \\ & = & \end{eqnarray}\]

$\Box$

Lemma 17

• $k \ge 2$,

\[\begin{eqnarray} \abs{ \left( \frac{\partial }{\partial \alpha} \right)^{k} \mathcal{G}_{n}(\alpha) } & = & O_{p}(\frac{1}{n^{k/2}}) \nonumber \\ \abs{ \left( \frac{\partial }{\partial \alpha} \right)^{k} \mathcal{T}_{n}(\alpha) } & = & O_{p}(\frac{1}{n^{k/2}}) \nonumber \end{eqnarray}\]

proof

By taylor’s theorem, there exists

\[\begin{eqnarray} f(x, w)^{k} & = & f(x, w_{0}) + \left( (w - w_{0})^{\mathrm{T}} \nabla f(x, w^{+}) \right)^{k} \nonumber \\ & = & \left( (w - w_{0})^{\mathrm{T}} \nabla f(x, w^{+}) \right)^{k} . \nonumber \end{eqnarray}\]

Hence

\[\begin{eqnarray} \abs{f(x, w)}^{k} & \le & \norm{(w - w_{0})}^{k} \norm{ \nabla f(x, w^{+}) }^{k} . \nonumber \end{eqnarray}\]

By Lemma 15,

\[\begin{eqnarray} \mathrm{E}_{w} \left[ \norm{ w - w_{0} - \frac{ \hat{\xi}_{n} }{ \sqrt{n} } }^{k} \right] & = & O_{p} \left( \frac{1}{n^{k/2}} \right) . \nonumber \end{eqnarray}\]

Since

\[\begin{eqnarray} \abs{a}^{k} & \le & 2^{k} \left( \abs{a - b}^{k} + \abs{b}^{k} \right), \nonumber \end{eqnarray}\] \[\begin{eqnarray} \mathrm{E}_{w} \left[ \norm{ w - w_{0} - \frac{ \hat{\xi}_{n} }{ \sqrt{n} } }^{k} \right] & = & \mathrm{E}_{w} \left[ \norm{ (w - w_{0}) - \frac{1}{\sqrt{n}} \hat{\xi}_{n} }^{2} \right] \nonumber \\ & = & \mathrm{E}_{w} \left[ \norm{ (w - w_{0}) }^{2} - 2 (w - w_{0})^{\mathrm{T}} \frac{1}{\sqrt{n}} \hat{\xi}_{n} - \norm{ \frac{1}{\sqrt{n}} \hat{\xi}_{n} }^{2} \right] \nonumber \\ & = & \mathrm{E}_{w} \left[ \norm{ (w - w_{0}) }^{2} - 2 (w - w_{0})^{\mathrm{T}} \frac{1}{\sqrt{n}} \hat{\xi}_{n} - J^{-1} \frac{1}{n} \norm{ \xi_{n} }^{2} \right] \nonumber \\ & = & \mathrm{E}_{w} \left[ \norm{ (w - w_{0}) }^{2} - 2 (w - w_{0})^{\mathrm{T}} \frac{1}{\sqrt{n}} \hat{\xi}_{n} - J^{-1} \frac{1}{n} \norm{ \xi_{n} }^{2} \right] & = & O_{p} \left( \frac{1}{n^{k/2}} \right) . \nonumber \end{eqnarray}\]

$\Box$