Question: How are different domains weighted in WDIRL?
Evidence: According to the above analysis, we proposed a weighted version of DIRL to address the problem caused by the shift of $\rm {P}(\rm {Y})$ to DIRL. The key idea of this framework is to first align $\rm {P}(\rm {Y})$ across domains before performing domain-invariant learning, and then take account the shift of $\rm {P}(\rm {Y})$ in the label prediction procedure. Specifically, it introduces a class weight $\mathbf {w}$ to weigh source domain examples by class. Based on the weighted source domain, the domain shift problem is resolved in two steps. In the first step, it applies DIRL on the target domain and the weighted source domain, aiming to alleviate the influence of the shift of $\rm {P}(\rm {Y})$ during the alignment of $\rm {P}(\rm {X}|\rm {Y})$. In the second step, it uses $\mathbf {w}$ to reweigh the supervised classifier $\rm {P}_S(\rm {Y}|\rm {X})$ obtained in the first step for target domain label prediction. We detail these two steps in §SECREF10 and §SECREF14, respectively.
The motivation behind this practice is to adjust data distribution of the source domain or the target domain to alleviate the shift of $\rm {P}(\rm {Y})$ across domains before applying DIRL. Consider that we only have labels of source domain data, we choose to adjust data distribution of the source domain. To achieve this purpose, we introduce a trainable class weight $\mathbf {w}$ to reweigh source domain examples by class when performing DIRL, with $\mathbf {w}_i > 0$. Specifically, we hope that:
and we denote $\mathbf {w}^*$ the value of $\mathbf {w}$ that makes this equation hold. We shall see that when $\mathbf {w}=\mathbf {w}^*$, DIRL is to align $\rm {P}_S(G(\rm {X})|\rm {Y})$ with $\rm {P}_T(G(\rm {X})|\rm {Y})$ without the shift of $\rm {P}(\rm {Y})$. According to our analysis, we know that due to the shift of $\rm {P}(\rm {Y})$, there is a conflict between the training objects of the supervised learning $\mathcal {L}_{sup}$ and the domain-invariant learning $\mathcal {L}_{inv}$. And the conflict degree will decrease as $\rm {P}_S(\rm {Y})$ getting close to $\rm {P}_T(\rm {Y})$. Therefore, during model training, $\mathbf {w}$ is expected to be optimized toward $\mathbf {w}^*$ since it will make $\rm {P}(\rm {Y})$ of the weighted source domain close to $\rm {P}_T(\rm {Y})$, so as to solve the conflict.
According to the above analysis
According to the above analysis, we proposed a weighted version of DIRL to address the problem caused by the shift of $\rm {P}(\rm {Y})$ to DIRL. The key idea of this framework is to first align $\rm {P}(\rm {Y})$ across domains before performing domain-invariant learning, and then take account the shift of $\rm {P}(\rm {Y})$ in the label prediction procedure. Specifically, it introduces a class weight $\mathbf {w}$ to weigh source domain examples by class. Based on the weighted source domain, the domain shift problem is resolved in two steps. 
The motivation behind this practice is to adjust data distribution of the source domain or the target domain to alleviate the shift of $\rm {P}(\rm {Y})$ across domains before applying DIRL. Consider that we only have labels of source domain data, we choose to adjust data distribution of the source domain. To achieve this purpose, we introduce a trainable class weight $\mathbf {w}$ to reweigh source domain examples by class when performing DIRL, with $\mathbf {w}_i > 0$. Specifically, we hope that:

and we denote $\mathbf {w}^*$ the value of $\mathbf {w}$ that makes this equation hold. 
We shall see that when $\mathbf {w}=\mathbf {w}^*$, DIRL is to align $\rm {P}_S(G(\rm {X})|\rm {Y})$ with $\rm {P}_T(G(\rm {X})|\rm {Y})$ without the shift of $\rm {P}(\rm {Y})$. According to our analysis, we know that due to the shift of $\rm {P}(\rm {Y})$, there is a conflict between the training objects of the supervised learning $\mathcal {L}_{sup}$ and the domain-invariant learning $\mathcal {L}_{inv}$. And the conflict degree will decrease as $\rm {P}_S(\rm {Y})$ getting close to $\rm {P}_T(\rm {Y})$. Therefore, during model training, $\mathbf {w}$ is expected to be optimized toward $\mathbf {w}^*$ since it will make $\rm {P}(\rm {Y})$ of the weighted source domain close to $\rm {P}_T(\rm {Y})$, so as to solve the conflict.
Answer: The answers are shown as follows:
* To achieve this purpose, we introduce a trainable class weight $\mathbf {w}$ to reweigh source domain examples by class when performing DIRL, with $\mathbf {w}_i > 0$