-->

Sunday, 8 September 2013

The Effect Of Being Observed

Two of my favourite Twitter accounts had this conversation on Saturday evening.

I had been looking out for an example like this after reading that Bayesian probability means that two people can update their beliefs differently on the same information.

When This Is Anfield is correct is when we can observe the length of an injury. If a player is expected to be out for two weeks but then does not play in two weeks time, it is far more likely that the manager was being cautious than that he is now out for two months.

I have set up a fiddle http://pythonfiddle.com/modelling-injuries-bayesically for the case where Anfield Index is correct - the case where we have not observed the expected length of the injury. I have taken the following injury lengths in weeks [1,2,3,4,6,8,13,26,52] and given them probabilities of 40%^(array index + 1) - except for 52 which gets whatever is left over.

So what do we see?


player has been out less than 1 week(s)
pdf is: [(1, 0.40000000000000002), (2, 0.23999999999999999), (3, 0.14399999999999999), (4, 0.086400000000000005), (6, 0.051839999999999997), (8, 0.031104), (13, 0.018662399999999999), (26, 0.011197439999999999), (52, 0.016796159999999997)]
expected total time out is  3.62461696

player has been out less than 2 week(s)
pdf is: [(2, 0.40000000000000002), (3, 0.23999999999999999), (4, 0.14400000000000002), (6, 0.086400000000000005), (8, 0.051840000000000004), (13, 0.031104), (26, 0.018662399999999999), (52, 0.027993599999999997)]
expected total time out is  5.3743616

player has been out less than 3 week(s)
pdf is: [(3, 0.39999999999999991), (4, 0.23999999999999999), (6, 0.14399999999999999), (8, 0.086399999999999991), (13, 0.05183999999999999), (26, 0.031103999999999996), (52, 0.046655999999999989)]
expected total time out is  7.623936

player has been out less than 4 week(s)
pdf is: [(4, 0.40000000000000002), (6, 0.23999999999999999), (8, 0.14399999999999999), (13, 0.086399999999999991), (26, 0.051839999999999997), (52, 0.077759999999999996)]
expected total time out is  10.70656

player has been out less than 6 week(s)
pdf is: [(6, 0.40000000000000002), (8, 0.24000000000000002), (13, 0.14399999999999999), (26, 0.086400000000000005), (52, 0.12959999999999999)]
expected total time out is  15.1776

player has been out less than 8 week(s)
pdf is: [(8, 0.40000000000000002), (13, 0.23999999999999999), (26, 0.14399999999999999), (52, 0.21599999999999997)]
expected total time out is  21.296

player has been out less than 13 week(s)
pdf is: [(13, 0.40000000000000002), (26, 0.24000000000000002), (52, 0.35999999999999999)]
expected total time out is  30.16

player has been out less than 26 week(s)
pdf is: [(26, 0.40000000000000002), (52, 0.59999999999999998)]
expected total time out is  41.6

player has been out less than 52 week(s)
pdf is: [(52, 1.0)]
expected total time out is  52.0

We can see that now as each week passes, and less severe odds drop out of the pool, the odds of the injuries left climb and so as each week passes you actually expect them to be out for more time.

This was very quick and dirty. You would obviously wanted continuous weeks, probably with exponential decay, and add in a percentage chance that the manager was taking a precaution the first week the player was fit. 



Arrow Key Nav