|Quoting KIAS (Reply 59):|
1) That statement is mathematically incorrect. Your chances of finding an object do not decrease in probability proportionally with the diminishment of a search area, whether the probability of finding it is less than 100% or not.
No, it isn't incorrect, which is why I've been banging on about Bayesian probabilities since I joined this thread!
If, and only if, you know for sure that an object is within a given search area, does your probability of finding it go up with every [subsection of that] area you reject as not containing it.
To simplify the question, let's assume that you have 101 possible search areas (e.g. solutions to the satellite data), and some other data (e.g. hearing possible black box pings) tells you that it has a 90% probability of being in the area at the top of your list, Area1, and 1% in each of the remaining 100areas (Area2-Area101) . So obviously you start with Area1. Let's say you divide it into 100 subdivisions, searching 1 each day. Let's compute the probability that the plane is in Area1, given that information. Let A1 stand for "plane in Area1" and NFY stand for "Not Found Yet".
P(NFY|A1)=1-proportion of A1 searched so far, i.e. the proportion of Area1 unsearched, so, if it's there, and 80% has been searched, the probability of not having found it yet is 20%.
P(A1) = 90% (prior probability)
P(¬A1)=10% (from prior probability)
P(NFY|¬A1) = 100%(Areas2-Area101 haven't been searched yet).
P(A1|NFY)=Proportion of Area1 unsearched *.9/[Proportion of Area1 unsearched *.9)+.1]
This means that the smaller the proportion of Area1 still unsearched, the smaller will the numerator be in relation to the denominator, and therefore the smaller the probability that the plane is in Area1.
Note also that the smaller the probability that the plane is anywhere other than Area1, the more slowly does the probability that it is in Area1 decrease as Area1 is searched, and vice versa.
The Bayesian take-home message here, it seems to me, is that given that there is so little hard data, and so many potential solutions to satellite data (not all of equal probability of course), and that even near-hard data may come with confidence limits (did the military Radar really track this plane, not another? Did the Chinese search ship really hear the box? Did Ocean Shield really hear the box?), every search that comes up blank alters the posterior probabilities that some other inference from other data is correct. Eventually, it may turn out that a northern route has higher posterior probabilities, given new data.
It's a more rigorous version of Sherlock Holmes's "When you have eliminated the impossible, whatever remains, however improbable, must be the truth." Or rather "As you eliminate the more probable, whatever remains, however initially improbable, becomes more likely".
[Edited 2014-04-24 02:58:18]