## Bayesian reasoning and SAR for flight MH370

I covered the basics of Bayesian reasoning in a previous post.

So, let’s apply Bayesian reasoning to the search & rescue operation of flight MH370. As we now know, Inmarsat’s identification of the most likely path taken by MH370, by means of Doppler analysis, proved significant in identifying the most likely flightpath taken.

In this post, I’m speculating on how the crash investigation team might have used Inmarsat’s data:

When MH370 disappeared from ATC radar, it could have taken any course from its latest known position. Although some courses would rationally be much more likely than others – e.g. based on an assumption that the aircraft would continue flying towards its destination – let’s for this exercise assume that each one degree course of the compass is equally probable, and that we for some reason settle on a southerly course as our assumption. There goes 360 degrees to a complete circle, so our prior probability for a southerly course is thus 1/360, about 0.28%.

Then, Inmarsat finds this ‘strange’ Doppler effect indicating that some aircraft has been flying a southerly course in the relevant window of time. Let’s assume that their analysis of a multitude of similar Doppler tracks for a multitude of different flights eventually reveal a probability of 95% of the Doppler track giving an accurate description of the actual flight track, and a 5 procent rate of ‘false positive’, that is, a Doppler track showing southerly course without an actual flight following that path.

Thus, from a Bayesian perspective, our “Prior”, that is, our initial best guess for the course taken by MH370 has a very low probability of 0.28%, but with the new evidence, provided by Inmarsat’s Doppler analysis, we are able to get a revised probability.

Plugging in these numbers into the Bayesian formula described in the previous post, all of sudden, with the new Doppler analysis evidence provided by Inmarsat, our assumption that the flight proceeded at a specific (180 deg) southerly course, has the probability of 51%!

That is, by applying Bayesian analysis to the problem, we were able (in this hypotetical example) to narrow down our S&R options from a quarter of a percentage to more than 50%, for a single degree of search sector.

Should we increase the width of the search sector, from its current 1 deg, we get the posterior probabilities as follows (everything else being constant):

```Search sector width in degrees:        Posterior probability:
1 degree                                    51%
2 degrees                                   68%
3 degrees                                   76%
4 degrees                                   81%
5 degrees                                   84%
```

Obviously, by increasing the search sector size, we get a larger area to search, but as can be seen from the numbers above, the probability for the target residing in the selected area increases rapidly with the width of the sector, even for these fairly narrow sectors.