## 3 Method
### Preliminaries
In addition, due to the inherent differences in data and models between the source and target domains, aligning the two domains in the absence of source domain data becomes even more challenging. The fixed source domain model leads to overconfident predictions, disregarding the discrepancies between domains and resulting in erroneous mappings of target domain data. To address this, we propose incorporating the uncertainty of neural network weights into the model predictions. This requires treating the model parameters \(W\) in a Bayesian framework, where the posterior distribution of model parameters is obtained by adjusting the data \(X\), i.e., \(p(W\mid X)=\frac{p(W)p(X\mid W)}{p(X)}\propto p(W)p(X\mid W)\). The prediction of the network \(g(\cdot)\) for an observation \(x\) is given by the predictive posterior distribution, i.e.,  
\[p(y_{c}\mid x,X)=\int_{W}\phi_{c}(g(x))p(W\mid X)\text{d}W \tag{4}\]  
where \(y_{c}\) is the probability belonging to \(c\)-th class. It is important to mention that the posterior \(p(W\mid X)\) in Equation (4) generally lacks a direct analytical solution and therefore requires approximation. In order to facilitate this, we engage a local approximation technique for the posterior. Specifically, we utilize the Laplace Approximation (LA) [29] for dealing with the model's uncertainty about weight parameters. Laplace Approximation (LA), a simple yet effective method for approximate Bayesian inference, approximates the posterior distribution of these parameters, thereby embodying the model's uncertainty over them and their impact on the predictions. LA estimates the actual posterior through the implementation of a multivariate Gaussian distribution, which is centered around a locally optimal point, and utilizes a covariance matrix derived from the inverse of the negative log posterior's Hessian \(\mathbf{H}\). Meaning, \(p(W\mid X)\approx\mathrm{N}\big{(}W\mid W_{\mathrm{MAP}},\mathbf{H}^{-1}\big{)}\), where \(\mathbf{H}:=-\,\
abla_{W}^{2}\log p(W\mid X)\big{|}_{W_{\mathrm{MAP}}}\). Here, \(W_{\mathrm{MAP}}\) signifies the maximum a posteriori probability (MAP) estimate of the parameters of the network. However, this technique has its potential limitations. Laplace approximation demands computing the inverse of the Hessian Matrix--used to comprehend the curvature characteristics of the distribution--implying computational difficulties and enhancing the associated computational cost when dealing with larger networks. Another significant factor would be requiring determining a point around which the approximation is to be set up; this is generally the mode (maximum) of the distribution, which can also be complex to compute for larger configurations. Since some large networks have a huge number of parameters, which makes calculating \(H\) very difficult, we also made another simplification by applying a Bayesian treatment only to hypothesis function \(g(\cdot)\), known as the last-layer Laplace approximation. \[p(y_{c}\mid h,X^{s})\approx\int_{w}\phi_{c}(g(h))\mathrm{N}\Big{(}W\mid W_{ \mathrm{MAP}},\mathbf{H}^{-1}\Big{)}\mathrm{d}W \tag{5}\]  
In this part, \(h\) denotes the potential representation of the target domain model. Although the last-layer LA significantly reduces the computational load for larger networks, calculating the Hessian can present challenges when the class count is high. To streamline computations, we hypothesize that \(\mathbf{H}\) can be Kronecker-factored as \(\mathbf{H}:=\mathbf{V}\otimes\mathbf{U}\). The ensuing approximation is known as the Kronecker-factored Laplace approximation. Subsequently, we utilize the estimates of uncertainty for target adaptation through the implementation of Monte Carlo (MC) integration. The approximate predictive posterior distributions are as follows:  
\[p(y_{c}\mid h,X^{s})\approx\frac{1}{M}\sum_{j=1}^{M}\phi_{c}(g(h)) \tag{6}\]  
Here, \(M\) stands for the total count of MC steps. To foster predictions with low entropy, we further scale the hypothesis outputs by a factor of \(1/\tau\), where \(W<\tau\leq 1\).