In two previous posts, [1,2] I’ve covered the fundamentals of Bayesian probability theory. The second post looked into how the air distaster investigation team might have proceeded, once they received Inmarsat’s satellite data, to assess the likelihood of the flight path taken by MH370.
The post builds upon the second post above, taking a look how Bayesian reasoning could be used to assign priorities to different and disjunct search areas.
Let’s assume we are looking for a missing person, ship or aircraft, and furthermore that we have identified three all inclusive and mutually exhausive areas, that is, the target must reside in one of these areas.
For each of these areas, we assign an initial (‘prior’) probability for the target to be located within that area. For each area, we also assign a specific search team, each such team having an effectiveness factor that indicates how thoroughly the team will be able to search their area. For instance, an aircraft doing only visual surveillance of the area is able to cover a lot of ground quickly, but its effectiveness, i.e. ability to locate the target with high certainty is low, compared to e.g. a team equipped with search dogs.
The table below shows how the search is initially planned: there are three fully inclusive, mutually exclusive search areas A, B, C, each with an a priori assigned probability for target being there, and each with an effectiveness factor of the search team:
Area A B C Prior P 5% 70% 25% Factor E 0 0.8 0.3 Multiplying the P with the E for each of the areas gives us the initial target detection probability for each area: P(detect) 0% 56% 8% Let's assume that we only have two search teams available, so area A is left without search activity in the first round. Now, either one of the two search teams will locate the target, and in that case, the job is done. If however the target remains undetected, the search coordinator, after talking to the teams, might assign other teams with other capabilities, and thus effectiveness factors to the different areas. Let's say that for the second round of search, the new areas are assigned teams as follows (the search of area B is intensified): Area A B C Prior P 5% 70% 25% Factor E 0 0.9 0.3 Now, let's assume that also for this second round, the search is unsuccessful. Now, by applying Bayesian reasoning, the search coordinator can start drawing some conclusions: let's take area B as an example: the prior for B is 70%, but we have already searched that area with a relatively high effectiveness factor of 0.9. That should indicate that the probability for the target to reside in B has decreased after the searches already performed. Similarily, for area C, we have only invested a modest search effectiveness, and although our initial prior for C was moderate, now, after the initial searches, we will revise our beliefs using Bayes: to do that, for each area we now multiply the prior with the complement effectiveness, that is, P x (1 - E), which gives us the probability for the target remaining in the unsearched parts of the area. As the final step, we the divide each such probability with the sum of all three probabilities, which gives us a revised probability for each area for the target to reside there. Applying the numbers above, gives us the following revised (posterior) target location probability: Area A B C Posterior P 16% 24% 60% Since our initial assumption was the three areas are fully inclusive (but mutually exclusive), the sum of the probabilities must be 100%. As can be seen, based on these numbers, the search coordinator should now switch priority from area B to area C for continued searches, assigning the most effective team to area C.