# 贝叶斯搜索理论

## 流程

1. 提出所有关于船只失踪事件的假设。
2. 针对每一假设，构造船只位置的空间分布概率。
3. 针对每一位置，假设已知船只位于此处，计算能找到失踪船只的概率分布。在海洋中，这一般取决于水深：在浅水处找到失踪物的机会比在深水处大。
4. 结合上述两个概率分布，构造整体的搜索成功的概率分布。
5. 构造搜索路径：始于高概率区，经过居中概率区，最后搜索低概率区。
6. 在搜索过程中，持续更新上述概率分布。例如，如果在某处未能找到失踪物，那么船只位置分布于此的概率要被降低。这一更新过程需要用到贝叶斯定理

## 數學

$p' = \frac{p(1-q)}{(1-p)+p(1-q)} = p \frac{1-q}{1-pq} < p.$

$r' = r \frac{1}{1- pq} > r.$

## 參考資料

• Stone, Lawrence D., The Theory of Optimal Search, published by the Operations Research Society of America, 1975
• Iida, Koji., Studies on the Optimal Search Plan, Vol. 70, Lecture Notes in Statistics, Springer-Verlag, 1992.
• De Groot, Morris H., Optimal Statistical Decisions, Wiley Classics Library, 2004.
• Richardson, Henry R; and Stone, Lawrence D. Operations Analysis during the underwater search for Scorpion. Naval Research Logistics Quarterly, June 1971, Vol. 18, Number 2. Office of Naval Research.
• Stone, Lawrence D. Search for the SS Central America: Mathematical Treasure Hunting. Technical Report, Metron Inc. Reston, Virginia.
• Koopman, B.O. Search and Screening, Operations Research Evaluation Group Report 56, Center for Naval Analyses, Alexandria, Virginia. 1946.
• Richardson, Henry R; and Discenza, J.H. The United States Coast Guard computer-assisted search planning system (CASP). Naval Research Logistics Quarterly. Vol. 27 number 4. pp. 659–680. 1980.
• Ross, Sheldon M., An Introduction to Stochastic Dynamic Programming, Academic Press. 1983.