Bayesian search theory is the application of Bayesian statistics

Bayesian statistics

Bayesian statistics is that subset of the entire field of statistics in which the evidence about the true state of the world is expressed in terms of degrees of belief or, more specifically, Bayesian probabilities...

 to the search for lost objects. It has been used several times to find lost sea vessels, for example the USS Scorpion

USS Scorpion (SSN-589)

USS Scorpion was a Skipjack-class nuclear submarine of the United States Navy, and the sixth ship of the U.S. Navy to carry that name. Scorpion was declared lost on 5 June 1968 with 99 crew members dying in the incident. The USS Scorpion is one of two nuclear submarines the U.S...

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Procedure

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 chances of finding an object are good if the search is in the right place. In deep water 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, and finally low probability areas.
6. Revise all the probabilities continuously during the search. For example, if the search of location X has yielded nothing then the probability that the vessel is there is greatly reduced (though not usually to zero) and the probabilities of all other locations must be increased. The revision process is done using Bayes' theorem
   Bayes' theorem
   In probability theory and applications, Bayes' theorem relates the conditional probabilities P and P. It is commonly used in science and engineering. The theorem is named for Thomas Bayes ....
   
   .

The advantages of the Bayesian method are that all information available is used coherently (i.e. in a "leak-proof" manner) and the method automatically produces estimates of the cost for a given success probability. That is, even before the start of searching, one can say, hypothetically, "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 a similar statement. Thus the economic viability of the search can be estimated before committing resources to a search.

Apart from the USS Scorpion

USS Scorpion (SSN-589)

USS Scorpion was a Skipjack-class nuclear submarine of the United States Navy, and the sixth ship of the U.S. Navy to carry that name. Scorpion was declared lost on 5 June 1968 with 99 crew members dying in the incident. The USS Scorpion is one of two nuclear submarines the U.S...

, other vessels located by Bayesian search theory include the MV Derbyshire

MV Derbyshire

The MV Derbyshire was an ore-bulk-oil combination carrier built in 1976 by Swan Hunter, as the last in the series of the Bridge-class sextet. She was registered at Liverpool and owned by Bibby Line....

, the largest British vessel ever lost at sea, and the SS Central America

SS Central America

SS Central America, sometimes called the Ship of Gold, was a 280-foot sidewheel steamer that operated between Central America and the eastern coast of the United States during the 1850s. She was originally named the SS George Law, after Mr. George Law of New York...

. It also proved successful in the search for a lost hydrogen bomb following the 1966 Palomares B-52 crash in Spain.

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

United States Coast Guard

The United States Coast Guard is a branch of the United States Armed Forces and one of the seven U.S. uniformed services. The Coast Guard is a maritime, military, multi-mission service unique among the military branches for having a maritime law enforcement mission and a federal regulatory agency...

 for search and rescue

Search and rescue

Search and rescue is the search for and provision of aid to people who are in distress or imminent danger.The general field of search and rescue includes many specialty sub-fields, mostly based upon terrain considerations...

. This program was later adapted for inland search by adding terrain and ground cover factors for use by the United States Air Force

United States Air Force

The United States Air Force is the aerial warfare service branch of the United States Armed Forces and one of the American uniformed services. Initially part of the United States Army, the USAF was formed as a separate branch of the military on September 18, 1947 under the National Security Act of...

 and Civil Air Patrol

Civil Air Patrol

Civil Air Patrol is a Congressionally chartered, federally supported, non-profit corporation that serves as the official civilian auxiliary of the United States Air Force . CAP is a volunteer organization with an aviation-minded membership that includes people from all backgrounds, lifestyles, and...

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Mathematics

Suppose a grid square has a probability p of containing the wreck and that the probability of successfully detecting the wreck if it is there is q. If the square is searched and no wreck is found, then, by Bayes' theorem, the revised probability of the wreck being in the square is given by

For each other grid square, if its prior probability is r, its posterior probability is given by