Algorithm 3: Confidence interval construction via nodewise regression

$$\begin{array}{cc}\hfill & \mathrm{Input}:\text{target data}({\mathit{X}}^{(0)},{\mathit{y}}^{(0)}),\text{source data}\{({\mathit{X}}^{(k)},{\mathit{y}}^{(k)}){\}}_{k=1}^K,\text{penalty parameters}\{{\lambda }_j{\}}_{j=1}^p\hfill \\ \hfill & \text{and}\{{\tilde{\lambda }}_j{\}}_{j=1}^p,\text{transferring set}𝒜,\text{confidence level}(1-\alpha )\hfill \\ \hfill & \mathrm{Output}:\text{Level-}(1-\alpha )\text{confidence interval}{I}_j\text{for}{\beta }_j\text{with}j=1,\dots ,p\hfill \\ \hfill & 1\text{Compute}\mathit{\beta }\text{via Algorithm 1}\hfill \\ \hfill & 2{\text{Compute}}^{{\widehat{\mathit{\gamma }}}_j^{𝒜}\leftarrow {\text{arg min}}_{\mathit{\gamma }}\left\{-\frac{1}{2(n_{𝒜}+n_0)}\sum _{k\in \{0\}{\cup }^{𝒜}}\Vert {\mathit{X}}_{\mathit{\beta },j}^{(k)}-{\mathit{X}}_{\mathit{\beta },-j}^{(k)}\mathit{\gamma }{\Vert }_2^2+{\lambda }_j\Vert \mathit{\gamma }{\Vert }_1\right\}}\text{for}j=1,\dots ,p\hfill \\ \hfill & 3\text{Compute}{\mathit{\varrho }}_j\leftarrow {\text{arg min}}_{\mathit{\varrho }}\left\{-\frac{1}{2n_0}{\Vert {\mathit{X}}_{\mathit{\beta },j}^{(0)}-{\mathit{X}}_{\mathit{\beta },-j}^{(0)}({\widehat{\mathit{\gamma }}}_j^{𝒜}+\mathit{\varrho })\Vert }_2+{\tilde{\lambda }}_j{\Vert \mathit{\varrho }\Vert }_1\right\}\hfill \\ \hfill & 4\text{Compute}{\widehat{\mathit{\gamma }}}_j^{(0)}\leftarrow {\widehat{\mathit{\gamma }}}_j^{𝒜}+{\mathit{\varrho }}_j,{\mathbf{\Sigma }}_{\mathit{\beta }}\leftarrow \sum _{k\in \{0\}{\cup }^{𝒜}}\frac{n_k}{n_{𝒜}+n_0}{\mathbf{\Sigma }}_{\mathit{\beta }}^{(k)},{\widehat{\tau }}_j^2={\mathbf{\Sigma }}_{\mathit{\beta },j,j}-{\mathbf{\Sigma }}_{\mathit{\beta },j,-j}{\widehat{\mathit{\gamma }}}_j\text{and calculate}\hfill \\ \hfill & \mathbf{\Theta }\text{via}(6),\text{where}{\widehat{\mathit{\gamma }}}_j^{(0)}=({\widehat{\gamma }}_{j,1}^{(0)},\dots ,{\widehat{\gamma }}_{j,j-1}^{(0)},{\widehat{\gamma }}_{j,j+1}^{(0)},\dots ,{\widehat{\gamma }}_{j,p}^{(0)}{)}^T.\hfill \\ \hfill & 5\text{Compute}{I}_j\leftarrow [{\widehat{b}}_j-{\mathbf{\Theta }}_j^T{\mathbf{\Sigma }}_{\mathit{\beta }}{\mathbf{\Theta }}_j{q}_{\alpha ∕2}∕\sqrt{n_0},{\widehat{b}}_j+{\mathbf{\Theta }}_j^T{\mathbf{\Sigma }}_{\mathit{\beta }}{\mathbf{\Theta }}_j{q}_{\alpha ∕2}∕\sqrt{n_0}]\text{for}j=1,\dots ,p,\text{where}\hfill \\ \hfill & {\widehat{b}}_j\text{is the}j-\text{th component of}\mathit{b}\text{in}(7),\text{and}{q}_{\alpha ∕2}{\text{is the}}^{\alpha ∕2}-{\text{left tail quantile of}}^{N(0,1)}\hfill \\ \hfill & 6\text{Output}{\{I_j\}}_{j=1}^p\hfill \end{array}$$