## 2 Proposed Method
### Instance-Based Transfer Learning  
Instance-based transfer learning focuses on using proper parts of the source domain data for training in the target domain by re-weighting. Instance-based transfer learning approaches are mainly based on the instance weighting strategy. In transfer learning, the source and target domains are used as training data consequently. Both domains have the same conditional probability distributions but follow different marginal probability distributions:  
\[P_{S}(X)\
eq P_{T}(X),\ P_{S}(Y|X)=P_{T}(Y|X). \tag{1}\]  
The performance of transfer learning depends on the relationship between the source and target domain instances, which determines how much useful knowledge for diagnosing the target domain can be transferred from the source to target domain. The amount of transferred knowledge or distributional characteristics of the instances is crucial factor in determining the transfer learning performance either. However, some instances of the source domain data are not suitable for the target prediction model. Therefore, directly transferring all the large source domain data may not be useful for the task of the target domain. Thus, it is necessary to measure the dissimilarity and distribution difference between the target and source domain instances and use it as a weight that influences model training to improve the transfer learning performance. This is the main concept of instance-based transfer learning using instance weighting. The main objective of instance weighting strategy is to reduce dissimilarity between source and target domain instances. Our instance weighting approach is based on the following equation [36,37]:  
\[ \begin{split}\mathcal{R}[P_{T},\ \ell(x,y;f)]&= \mathbb{E}_{(x,y)\sim P_{T}}[\ell(x,y;f)]=\mathbb{E}_{(x,y)\sim P_{S}}\Big{[} \frac{P_{T}(x,y)}{P_{S}(x,y)}\ell(x,y;f)\Big{]}\\ &=\mathbb{E}_{(x,y)\sim P_{S}}\Big{[}\frac{P_{T}(x)}{P_{S}(x)}\ell (x,y;f)\Big{]}=\mathbb{E}_{(x,y)\sim P_{S}}[\beta(x,y)\ell(x,y;f)],\end{split}  \tag{2}\]  
where \(\mathcal{R}\) is the regularized risk and \(\ell(x,y;f)\) is the loss function in the prediction model \(f\) using \(x\) and \(y\). \(\mathbb{E}\) is the expected risk of the loss function, \(\ell\), and \(\beta\) is instance weight for source domain. The prediction model \(f\) is trained to minimize the expected risk \(\mathbb{E}\) of the loss function. From the above Equation (2), theoretically, the value of the instance weight \(\beta\) is the same as \(P_{T}(x)\ /P_{S}(x)\). However, the exact ratio of \(P_{T}(x)\) to \(P_{S}(x)\) is usually not known [36]. Therefore, several methods to estimate the weight have been introduced. For example, kernel mean matching (KMM) procedure [37], which is presented by [PERSON] et al., deals with weight estimation by matching the means between source domain and target domain instances in a reproducing kernel Hilbert space (RKHS):  
\[\underset{\beta_{i}}{\text{argmin}}\ \parallel\ \frac{1}{n_{S}}\sum_{i=1}^{n_{S}} \beta_{i}\Phi\big{(}x_{S_{i}}\big{)}-\frac{1}{n_{T}}\sum_{j=1}^{n_{T}}\Phi \Big{(}x_{T_{j}}\Big{)}\ \parallel_{\mathcal{H}^{\prime}}^{2}, \tag{3}\]  
where \(\beta_{i}\) is a weighting parameter of \(i^{th}\) instance, and \(\Phi\) is the kernel mapping function. We can obtain instance weight \(\beta_{i}\) using KMM procedure and apply to model training process. In addition, [PERSON] et al. [38] proposed an approach called the Kullback-Leibler importance estimation procedure (KLIEP) to estimate the weight. It estimates the importance, which means the weight, through a variant of likelihood CV (cross validation) by minimizing the Kullback-Leibler divergence (KLD). These two methods were basically devised to solve the dataset shift problem. It is one of the important assumptions in machine learning that the training data and test data should be drawn from the same distribution. A situation in which the joint distribution of inputs and outputs is different in training and test data is referred to as a dataset shift.