## 5 Discussion
### Future Work
\begin{tabular}{l l} \(\mathbf{X}:\{\mathbf{x},\mathbf{z}\}\) & matrix of covariates (including \(\mathbf{z}^{S},\mathbf{x}^{S}\)) for claim amount outcome (Severity). \\  & Focusing solely on severity, we omit the superscript \({}^{\prime}s^{\prime}\) for simplicity. \\ \(\mathbf{X}_{h}^{F}:\{x_{h}^{F},z_{h}^{F}\}\) & vector of covariates in policy \(h\) (Frequency). \\ \(\mathbf{x}^{F}\) & vector of continuous covariate (Frequency). \\ \(\mathbf{z}^{F}\) & vector of binary covariate (Frequency). \\ \(x_{h}^{F}\) & individual value of continuous covariate in policy \(h\) (Frequency). \\ \(z_{h}^{F}\) & individual value of binary covariate in policy \(h\) (Frequency). \\ \(\mathbf{X}_{h}^{S}:\{x_{h}^{S},z_{h}^{S}\}\) & vector of covariates in policy \(h\) (Severity). \\ \(\mathbf{x}^{S}\) & vector of continuous covariate, and \(\mathbf{x}^{*}\) indicates the mismeasurement (Severity). \\ \(\mathbf{z}^{S}\) & vector of binary covariate (Severity). \\ \(x_{h}^{S}\) & individual value of continuous covariate, and \(x_{h}^{*}\) indicates the mismeasurement in policy \(h\) (Severity). \\ \(z_{h}^{S}\) & individual value of binary covariate in policy \(h\) (Severity). \\ \(\mathbf{X}_{h}:\{x_{h},z_{h}\}\) & vector of covariates in policy \(h\). \\ \(\mathbf{x}\) & vector of continuous covariate, and \(\mathbf{x}^{*}\) indicates the mismeasurement. \\ \(\mathbf{z}\) & vector of binary covariate. \\ \(x_{h}\) & individual value of continuous covariate, and \(x_{h}^{*}\) indicates the mismeasurement in policy \(h\). \\ \(z_{h}\) & individual value of binary covariate in policy \(h\). \\ \(p_{0}(\cdot)\) & parameter model (for prior). \\ \(p(\cdot)\) & parameter model (for posterior). \\ \(f_{0}(\cdot)\) & data model (for continuous cluster). \\ \(f(\cdot)\) & data model (for discrete cluster). \\ \(E[\cdot\ ]\), \(V[\cdot\ ]\) & expectation and variance as point estimates. \\ \(\phi(\cdot)\) & probability density function of Standard Gaussian density. \\ \(\Phi(\cdot)\) & cumulative density function of Standard Gaussian density. \\ \(\theta_{j}\) & set of parameters-\(\beta,\sigma^{2}\)-associated with the outcome model \(f(Y|\mathbf{X})\) for \(j\) cluster (posterior sample: \(\theta_{j}^{(*)}\)). \\ \(\mathbf{w}_{j}\) & set of parameters-\(\pi,\mu,\lambda^{2}\)-associated with the covariate models \(f(\mathbf{X})\) for \(j\) cluster (posterior sample: \(\mathbf{w}_{j}^{(*)}\)). \\ \(\mathbf{w}_{j}\) & cluster weights (mixing coefficient) for \(j\) cluster (finalized sample: \(\mathbf{w}_{j}^{(*)}\)). \\ \(\mathbf{\beta}_{j}:\{\beta_{j0},\beta_{j1},\beta_{j2}\}\) & regression coefficient vector for a mean outcome estimation. \\ \(\mathbf{\beta}_{0},\Sigma_{\beta_{0}}\) & i.e., \(\tilde{\sigma}^{2}(\mathbf{X}^{T}\mathbf{X})^{-1}=\mathbf{X}^{T}\mathbf{X}( \Sigma Y-\Sigma\tilde{Y})^{T}(\Sigma Y-\Sigma\tilde{Y})/(n-p)\) obtained from the baseline multivariate Gamma regression of \(\Sigma\tilde{Y}>0\). \\ \(\sigma_{j}^{2}\) & cluster-wise variance or scale parameter for the outcome. \\ \(\pi_{j}\) & proportion parameter for Bernoulli covariate. \\ \(\mu_{j}\) & location parameter for Gaussian covariate \(\mathbf{x}\). \\ \(\lambda_{j}^{2}\) & dispersion parameter for Gaussian covariate \(\mathbf{x}\). \\ \(\mathbf{\kappa}_{j}:\{\kappa_{j0},\kappa_{j1}\}\) & regression coefficient vectors to explain the mean of the unobserved Gaussian covariate \(\mathbf{x}\backslash\mathbf{z}\). \\ \(\mathbf{\Sigma}_{j}^{2}\) & variance parameter for \(\mathbf{x}^{*}|\mathbf{x}\) to indicate the contamination level in the measurement model. \\ \(m_{0},\delta\) & hyperparameters of Multivariate Normal for \(\mathbf{\beta}_{0}^{S}\). \\ \(q_{0},\Lambda\) & hyperparameters of Inverse Wishart density for \(\Sigma_{\tilde{p}_{0}}^{S}\). \\ \(u_{0},v_{0}\) & hyperparameters of Inverse Gamma density for \(\sigma_{j}^{2}\). \\ \(\rho_{\alpha 1},\rho_{\alpha 2}\) & hyperparameters of Fink's function for \(u_{0}\). \\ \(\rho_{\alpha 1},\rho_{\alpha 2}\) & hyperparameters of Gamma density for \(v_{0}\). \\ \(\mu_{0},\lambda_{j}^{2}\) & hyperparameters of Gaussian density of \(\mu_{j}\). \\ \(c_{0},d_{0}\) & hyperparameters of Inverse Gamma density for \(\lambda_{j}^{2}\) \\ \(g_{0},h_{0}\) & hyperparameters of Beta density for \(\pi_{j}\). \\ \end{tabular}