The Denominator—Where “Due” Happens

After a weekend of listening to football broadcasters, I need to detox. I still feel this zinger from a few weeks ago during the Sunday Night Football broadcast:  the commentator noted that Adam Vinatieri, the kicker for the Indianapolis Colts, had a career record of 18 of 36 when attempting field goals over 50 yards, and a season record of 2 of 4 at that distance.  No matter how you look at it, he said, it’s a 50-50 proposition.  Then Vinatieri kicked the field goal, upping his record on the season to 3 out of 5 from long range.  Then the commentator quipped, “So I bet he misses the next one.”

It is ironic that while the sports world is finally embracing the scientific method of numerical analysis, the marketing department exhibits and even celebrates statistical stupidity.  The talking heads will tell us Peyton Manning’s Total QBR, Yasiel Puig’s WAR, and other advanced metrics, but then tell us Vinatieri will miss the next one, and then cut to a commercial telling us that “it’s only weird if it doesn’t work,” which translates to:  “Superstition is only stupid if you’re not drinking beer.”

If, like me, you cringe when you hear the commentator say that Vinatieri is “due” to miss the next one, then you probably don’t need to read the rest of this blog post, but you should anyway, because I may succeed in finally giving you the explanation for something that you already know to be true, but which no drunk sports fan can ever correctly explain.  And you may one day be that drunk sports fan.

First, we’re going to have to switch this discussion to coin flips.  (If we go with the statistician’s other favorite—balls in urns—then we would have to convince the commentator why we want to sample with replacement, and we know that NFL broadcasters will resist the use of replacements.)  The reason we have to use coin flips in our discussion is that in the case of Vinatieri, the broadcaster is assuming that Vinatieri’s probability of success is 0.5, when in fact Vinatieri’s success probability (denoted p) is an unknown parameter that we need to guess.  The broadcaster’s logic is that if we know that the success probability p=0.5, then a missed field goal on the next opportunity would restore the 3 of 5 record to a 3 of 6 record, i.e., a 50% success rate.

Forgetting about the “due theory” for a second, a glaring problem here is that the argument is not logically consistent.  The only reason he thinks Vinatieri is a 50% kicker from long range is from Vinatieri’s historical kicking data.  But now that we see Vinatieri is 3 of 5 for the season (60%) or 19 of 37 (51.35%) for his career, we should update our estimate of his success probability p.  His true probability of success is some unknown number, but we would now estimate it to be something between 51.35% and 60% (depending on whether we think recent results are a better measure of his current ability), in which case the broadcaster should announce that Vinatieri is a favorite to make the next one.  Anyway, this avenue leads us to philosophical debates about Bayesian statistics, prior distributions (esp. empirical Bayes priors), and other deep concepts that distract us from our simple mission: to show the simple reason that “due theory” is bogus, and reconcile that bogusness with other seemingly contradictory beliefs.

So let’s keep this simple with a fair coin, an object with a known success probability of p=0.5.  Let’s suppose we have flipped the coin 100 times, and it has come up 65 heads and 35 tails.  Present this data to a civilian, or even an AP, and let him even inspect the coin.  Ask him to describe his beliefs about the current situation.  He will state three beliefs, and we will stipulate that these are facts:

1. The coin is fair, giving a 0.5 probability of heads on any given flip.
2. Each flip of the coin has no effect on any other flip.  Because the flips are independent, the prior results do not help us predict the next flip.
3. If we flip the coin many times, the fraction of heads observed must converge to 0.5 (the success probability given in Fact #1).

Interestingly, these three commonplace notions form the foundation of much of statistical work, and hence have precise technical terminology and definitions.  The first two facts are what a statistician means when he says that the coin flips are i.i.d.—independent and identically distributed.  The third fact is what he means when he says that our estimator is consistent.  An “estimator” is just a rule or way of coming up with a guess.  In this case, if we had to guess Vinatieri’s success rate, or the coin’s heads rate, our guess is just the number of successes as a fraction of the total number of attempts.  So, our rule generates the estimate 65/100 = 0.65 for the heads probability of the coin, which we know from inspection has a 0.5 heads probability.  To say that our estimator or rule is “consistent” means that if we collect data from a large number of trials, our guess will get closer and closer to the correct answer.  So, consistency means that the fraction of heads is going to get closer and closer to 0.5 as we keep flipping that coin.

So now we have these three facts, and we also have a sample of 65 heads out of a 100 flips.  What will happen on the next 100 flips?  Most people, even APs, are unable to reconcile the meaningless of prior flips (Fact #2) with the necessity of convergence to a 50% fraction (Fact #3).  They say they don’t believe in “due,” but they also “know” that the fraction has to gravitate to ½, and if you actually watch their AP activity, you will see that their actions belie their true beliefs.  I have seen AP machine players who don’t believe in “due,” but who switch to the machine that has not hit a Royal Flush for a while.  I have seen card counters who are on a winning streak at a casino who become afraid to play there, because they feel they are due for a loss, even though they pay lip service to the AP Creed of not believing in “due.”  Though we have not eliminated superstition in the AP world, at least embarrassment over superstition is a start.

Though they say they know the coin is not due, in their heart of hearts they expect the coin to restore the 50% fraction in the following way:  Over the next 100 flips, if tails is supposedly due, then we fix the 65/100 fraction by updating it to (65+35)/(100+100) = 100/200 = 0.5.  Now of course, they realize that it is silly to expect 65 tails in the next 100 flips.  After all, that’s 15 extra tails in only 100 flips.  They may then reason that heads got that lucky in the first 100, so it’s not inconceivable for tails to run that lucky in the next 100.  Still, they usually feel guilty about worshipping the Due idol, and they also know that Fact #3 somehow invokes the “long run,” so then they decide that the better way to restore order to the universe is to make tails due over the next 10000 flips.  If we just get 5015 tails, or 15 extra, over the next 10000 flips, then our overall heads fraction is (65+4985)/(100+10000) = 0.5.  This way tails only needs to be slightly lucky for a while to restore order.

Either way, what these people are trying to do is to fix the heads fraction in the numerator.  They are trying to cancel out the 15 extra heads by inserting 15 extra tails into the numerator, so that the fraction becomes ½.  Here is the secret about Due:  the heads fraction corrects itself in the denominator, not the numerator.  While the coin showed 15 extra heads in the first 100 flips, what happens to our 0.65 fraction if the next 100 flips are even?  Then the fraction becomes (65+50)/(100+100) = 0.575.  Then it becomes (65+50+50)/(100+100+100) = 0.55 if we stay even in the next 100 flips.  After 10000 additional even flips after the initial hundred, we would arrive at (65+5000)/(100+10000) = 0.5015.   Look how close we are to 50%!

To rectify the fact that heads ran lucky at the beginning, we do not need tails to get lucky.  If in future flips, the coin just stays true to its nature, with tails getting only its fair share, then the original excess of 15 heads gets constantly diluted by the ever-expanding denominator.  The 0.65 fraction cannot be maintained; heads cannot keep getting extra lucky given Facts #1 and #2.  No fixed excess of heads can survive getting diluted down to the 50% success rate as the denominator grows.

You may not like the fact that as we get closer and closer to 50% in this example, we are always on the high side.  I have two answers.  First of all, this is just an example.  If you collected real data on millions of flips, you would see the fraction bounce around, but the bounces would get smaller and smaller and the growing denominator would still wipe out any excess of heads or tails in the numerator.  Facts #1 and #2 guarantee that the future balance of heads and tails will dilute any prior excess of heads or tails in the numerator as the denominator grows.  Second, statisticians’ main priority is having a guarantee that as the data sample size grows, our guess will get really close to the truth.  Whether the guess is always high or always low along the way is usually not that big a deal.  Who cares if we are too high if the overestimate is shrinking and shrinking, and we are getting superclose to the truth as the sample grows.  Too  high?  No problem—just get more data and the overestimate shrinks in magnitude.  Anyway, we’ll save a discussion of the relative merits of “unbiasedness” and “consistency” for another day (or not!).