## 7 Conclusions  
In this paper, we introduced a probabilistic machine learning approach using a Bayesian kernel machine regression model to assess the growth of indoor-cultured abalone, focusing on key environmental factors such as water temperature, dissolved oxygen, pH, salinity, and nutrient supply. The model employs a spike-and-slab prior to determine the influence of growth-related factors, effectively selecting significant predictors. By incor  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Training Set**} & \multicolumn{2}{c}{**Test Set**} \\ \cline{2-5}  & **WBKMR** & **BKMR** & **WBKMR** & **BKMR** \\ \hline Set 1 & 0.130 & 0.263 & 2.938 & 2.546 \\ Set 2 & 0.124 & 0.259 & 0.949 & 1.482 \\ Set 3 & 0.121 & 0.247 & 0.569 & 0.731 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Root mean square errors for comparison.  
porating heteroscedasticity and leveraging Gaussian processes, the proposed weighted Bayesian kernel machine regression (WBKMR) model with unequal variance provides a novel framework for capturing complex, non-linear relationships among the predictors.  
The Posterior Inclusion Probability (PIP) analysis highlighted that factors such as time, dissolved oxygen, and nutrient supply are critical in influencing abalone growth, while interaction analysis highlighted the combined effects of environmental variables, such as DO and nutrient supply. These findings offer actionable insights into the optimal management of environmental conditions for maximizing abalone growth in controlled aquaculture settings.  
The comparative analysis demonstrated that WBKMR consistently outperformed the baseline BKMR model, achieving lower root mean square errors (RMSE) across multiple datasets. This indicates the proposed model's superior predictive accuracy and its ability to provide meaningful interpretations of environmental dynamics affecting growth.  
While the WBKMR model represents a significant advancement, this study is not without limitations. The model is currently constrained to specific environmental factors and indoor farming conditions. Future work could extend the approach to include additional variables, such as interactions with other species or broader aquaculture practices. Incorporating real-time data collection and expanding the temporal scale could further enhance the model's predictive power and practical applicability, offering valuable tools for sustainable abalone farming.  
Conceptualization: H.-J.J., H.-S.J., H.-K.L. and S.J.; Data curation: S.-W.S., M.-J.C. and Y.-D.O.; Exploratory Data Analysis: G.C.; Analysis: G.C. and S.J.; Writing: S.J. and S.-W. All authors have read and agreed to the published version of the manuscript.  
This study was supported by Korea Institute of Marine Science & Technology Promotion(KIMST) funded by the Ministry of Oceans and Fisheries (RS-2022-KS221673, Big data-based aquaculture productivity improvement technology). [PERSON] was also supported by INHA UNIVERSITY research grant and the National Research Foundation of Korea (NRF) grant funded by the Korea government (RS-2023-00209229).  
The raw data supporting the conclusions of this article will be made available by the authors on request.  
Authors [PERSON] and [PERSON] were employed by the company Silicon Inc. The remaining authors declare that the re-search was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} BKMR & Bayesian Kernel Machine Regression \\ GP & Gaussian Process \\ MCMC & Markov chain Monte Carlo \\ KMR & Kernel Machine Regression \\ PIP & Posterior Inclusion Probability \\ WBKMR & Weighted Bayesian Kernel Machine Regression \\ \end{tabular}