# Bayesian Binary Search
## 2 Methodology
### Probabilistic Bisection Step
In scenarios where the underlying distribution is highly complex or rapidly changing, the chosen PDF estimation method may struggle to provide accurate probabilities, potentially leading to suboptimal performance. ### Future Research Directions

Our findings open up several exciting avenues for future research: \begin{table}
\begin{tabular}{l c c c} \hline \hline \(e\) & **Percent Decrease** & **Basic Mean Steps** & **BBS Mean Steps** \\ \hline
128 & 3.07\% & 14.26 \(\pm\) 2.26 & 13.82 \(\pm\) 2.35 \\
256 & 3.31\% & 13.26 \(\pm\) 2.26 & 12.82 \(\pm\) 2.35 \\
512 & 3.57\% & 12.26 \(\pm\) 2.26 & 11.82 \(\pm\) 2.35 \\
1024 & 3.89\% & 11.26 \(\pm\) 2.26 & 10.82 \(\pm\) 2.35 \\
2048 & 4.27\% & 10.26 \(\pm\) 2.26 & 9.82 \(\pm\) 2.35 \\
4096 & 4.73\% & 9.26 \(\pm\) 2.26 & 8.82 \(\pm\) 2.35 \\
8192 & 5.03\% & 8.26 \(\pm\) 2.26 & 7.84 \(\pm\) 2.30 \\
16,384 & 5.73\% & 7.26 \(\pm\) 2.26 & 6.84 \(\pm\) 2.29 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Lightning Channel Probing Comparison. Figure 2: Lightning Probing Experiment: Basic vs. BBS Convergence Comparison. **Adaptive PDF Estimation:** Developing methods to dynamically adjust the PDF estimation technique based on the observed search space characteristics could further improve the robustness and efficiency of BBS. * **Theoretical Bounds:** While we provide empirical evidence of BBS's effectiveness, deriving tighter theoretical bounds on its performance under various distribution types would strengthen its theoretical foundation. * **Application to Other Domains:** Exploring the applicability of BBS to other areas such as database indexing, computational biology, or optimization algorithms could reveal new use cases and challenges. ### Conclusions

Bayesian Binary Search represents a significant step towards more adaptive and efficient search algorithms. By incorporating probabilistic information, BBS demonstrates the potential to outperform classical binary search in non-uniform distributions, as evidenced by our experiments in both simulated and real-world scenarios. In the context of the Lightning Network, BBS not only improves search efficiency but also contributes to reducing unnecessary network load, potentially enhancing the overall performance and scalability of the system. The promising results open up new possibilities for optimizing search processes in various domains. As we continue to navigate increasingly complex and data-rich environments, algorithms like BBS that can adapt to the underlying structure of the problem space will become increasingly valuable. Future work in this direction has the potential to not only improve search efficiency but also to deepen our understanding of how probabilistic approaches can enhance fundamental algorithms in computer science and their real-world applications. **Author Contributions:** [PERSON], [PERSON] and [PERSON] participated in writing the main manuscript text and execution of experiments. All authors [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON] and [PERSON] reviewed the manuscript and contributed to experimental design. All authors have read and agreed to the published version of the manuscript. **Funding:** The authors are grateful to be able to produce this work while being employed/funded for their roles at Stillmark and Amboss Technologies. **Data Availability Statement:** The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author. **Conflicts of Interest:** Author [PERSON] was employed by the company Stillmark. Author [PERSON] was indepedently affiliated but employed by Ontario at the time of working on the project. Author [PERSON] was employed by VantAI and an advisor to Amboss Technologies. Authors [PERSON], [PERSON] and [PERSON] were employed by the company Amboss Technologies. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.