You're out walking on a cold winter's evening, contemplating the weekend's upcoming matches, when you're approached by a behatted, shadowy figure who offers to sell you a couple of statistical models that tip AFL winners. You squint into the gloom and can just discern the outline of a pocket-protector on the man who is now blocking your path, and feel immediately that this is a person whose word you can trust.
He tells you that the models he is offering each use different pieces of data about a particular game and that neither of them use data about which is the home team. He adds - uninformatively you think - that the two models produce statistically independent predictions of the winning team. You ask how accurate the models are that he's selling and he frowns momentarily and then sighs before revealing that one of the models tips at 60% and the other at 64%. They're not that good, he acknowledges, sensing your disappointment, but he needs money to feed his Lotto habit. "Lotto wheels" , you ask. He nods, eyes downcast. Clearly he hasn't learned much about probability, you realise.
As a regular reader of this blog you already have a model for tipping winners, sophisticated though it is, which involves looking up which team is the home team - real or notional - and then tipping that team. This approach, you know, allows you to tip at about a 65% success rate.
What use to you then is a model - actually two, since he's offering them as a job lot - that can't out-predict your existing model? You tip at 65% and the best model he's offering tips only at 64%.
If you believe him, should you walk away? Or, phrased in more statistical terms, are you better off with a single model that tips at 65% or with three models that make independent predictions and that tip at 65%, 64% and 60% respectively?
By now your olfactory system is probably detecting a rodent and you've guessed that you're better off with the three models, unintuitive though that might seem.
Indeed, were you to use the three models and make your tip on the basis of a simple plurality of their opinions you could expect to lift your predictive accuracy to 68.9%, an increase of almost 4 percentage points. I think that's remarkable.
The pivotal requirement for the improvement is that the three predictions be statistically independent; if that's the case then, given the levels of predictive accuracy I've provided, the combined opinion of the three of them is better than the individual opinion of any one of them.
In fact, you also should have accepted the offer from your Lotto-addicted confrere had the models he'd been offering each only been able to tip at 58% though in that case their combination with your own model would have yielded an overall lift in predictive accuracy of only 0.3%. Very roughly speaking, for every 1% increase in the sum of the predictive accuracies of the two models you're being offered you can expected about a 0.45% increase in the predictive accuracy of the model you can form by combining them with your own home-team based model.
That's not to say that you should accept any two models you're offered that generate independent predictions. If the sum of the predictive accuracies of the two models you're offered is less than 116%, you're better off sticking to your home-team model.
The statistical result that I've described here has obvious implications for building Fund algorithms and, to some extent, has already been exploited by some of the existing Funds. The floating-window based models of HELP, LAMP and HAMP are also loosely inspired by this result, though the predictions of different floating-window models are unlikely to be statistically independent. A floating-window model that is based on the most recent 10 rounds of results, for example, shares much of the data that it uses with the floating-window model that is based on the most recent 15 rounds of results. This statistical dependence significantly reduces the predictive lift that can be achieved by combining such models.
Nonetheless, it's an interesting result I think and more generally highlights the statistical validity of the popular notion that "many heads are better than one", though, as we now know, this is only true if the owners of those heads are truly independent thinkers and if they're each individually reasonably astute.
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