Problem statement
The USS Scorpion was a nuclear submarine in the U.S. Navy. It disappeared on May 22, 1968, close to the Azores archipelago, when returning to its base in Norfolk from a mission. The reasons are as yet uncertain. There was speculation about an explosion, accidental activation of a foul torpedo, or even a Soviet attack.
After several days elapsed without contact, the search for the submarine started. The vicinity of the last known position of the ship was divided in 400 sectors. A probability of containing the remains of the ship, using available information and experts’ assessments, was ascribed to each such sector.
Probabilities ascribed to each sector
To simplify things, let’s assume we have far fewer sectors to search, say 25. Furthermore, let’s assume we have the following probabilities for where our submarine is:
rm(list=ls(all=TRUE))
set.seed(1)
#create probabilities for location
nloc=5
seq1=1:nloc
combo=expand.grid(x=seq1,y=seq1)
combo$prob=dnorm(combo$x,mean=3,sd=2)*dnorm(combo$y,mean=3,sd=2)
combo$prob=combo$prob/sum(combo$prob)
#visual depiction of probabilities
pi1=matrix(NA,nloc,nloc)
for (i in 1:nrow(combo)){
pi1[combo$x[i],combo$y[i]]=combo$prob[i]
}
par(mfrow=c(1,1),ask=T)
image(pi1,x=1:nloc,y=1:nloc,xlab='',ylab='')
text(x=combo$x,y=combo$y,round(combo$prob,3),pos=1)
Furthermore, say that our probability of detecting the submarine, given that it is present in the sector that we are looking at, is \(\theta=0.1\). This probability is low because this is a very deep part of the ocean and it is easy to miss the submarine at the bottom of the ocean even if it is present. We assume that this parameter is known (i.e., we are not estimating it).
Finally, we assume that there are no false detections. In other words, you cannot detect the submarine if it is not there.
How should we go about trying to find this submarine?
Here are some guiding questions:
What are potential strategies that we could use to solve this problem?
Can we write some equations describing the relevant probabilities in this problem?
Assignment:
Calculate the probability that the submarine is in each sector assuming that we searched sector 13 and did not find anything.
Update again the probability that the submarine is in each sector. This time, assume that we have searched sector 12 (after searching in sector 13) and did not find anything.
Obs.: the sector numbers were defined as shown below:
## [,1] [,2] [,3] [,4] [,5]
## [1,] 1 2 3 4 5
## [2,] 6 7 8 9 10
## [3,] 11 12 13 14 15
## [4,] 16 17 18 19 20
## [5,] 21 22 23 24 25
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