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Bayesian search theory is the application of Bayesian statistics to the search for lost objects. It has been used several times to find lost sea vessels, for example the USS Scorpion. The usual procedure is as follows:

1. Formulate a number of hypotheses about what happened to the vessel.
2. Corresponding to each hypothesis construct a probability distribution for the location of the vessel
3. Construct a probability distribution for actually finding an object in location X if it really is in location X. In an ocean search, this is usually a function of water depth - in shallow water your chances of finding an object are good if you are looking in the right place. In deep water your chances are reduced.
4. Combine the above information coherently to produce an overall probability distribution. (Usually this simply means multiplying the two distributions together.) This gives the probability of finding the vessel by looking in location X, for all possible locations X. (This is like a contour map of probability.)
5. Construct a search path which starts at the point of highest probability and 'scans' over high probability areas, then intermediate probabilities, then the low probability areas.
6. Revise all the probabilities continuously as you search, i.e. if you have searched location X then the probability that the vessel is there is greatly reduced (though not usually zero) and the probabilities of all other locations must be increased. The revision process is done using Bayes' theorem.

The advantages of the Bayesian method are that all information available is used coherently (i.e. in a leakproof manner) and the method automatically produces estimates of the cost, for a given success probability. That is, even before one starts searching, one can say something like "there is a 65% chance of finding it in a 5-day search. That probability will rise to 90% after a 10-day search and 97% after 15 days" or some such statement. Thus the financial viability of the search can be estimated beforehand.

Apart from the USS Scorpion, other vessels located by Bayesian search theory include the MV Derbyshire, the largest British vessel ever lost at sea, and the SS Central America. It also proved successful in the search for a lost hydrogen bomb following an accident in Palomares, Spain.

Bayesian search theory is incorporated into the CASP (Computer Assisted Search Program) mission planning software used by the United States Coast Guard for search and rescue.

• Bayesian inference

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• 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.