Things to know about maths
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The JC was never very good at maths — still isn’t — yet found himself, to his own great surprise, conducting a long if unspectacular career in financial services. At last count, 30 years. Along the way he has learned some helpful, if unintuitive, things, so it seems only right to collect them.
Averages are misleading
Ergodicity
“Ensemble averages” payoffs across a collection of people who roll the same dice once, are very different from the payoffs of the same person rolling the same dice repeatedly. This is why the house never loses. This is called ergodicity.
Non-representative
The average of a collection of measurements taken from real people, in itself, represents absolutely nothing. Taking new measurements which change the average does not affect any of the existing measurements. Call this the “Techbro on the Clapham Omnibus” problem, known in modern times as the “making a big hire to massage the diversity and inclusion stats” problem.
Grouping data is problematic
An average or trend which seems to point in one direction can, when you group the data differently, often point in another direction. This plays particular havoc with social scientists when they try to draw causal inferences — or moral imperatives — from data they have gathered with the careful intention of illustrating their own precious hypothesis.
It is called Simpson’s paradox.
Probabilities are misleading
They don’t work in complex systems
Probabilities are only meaningful for independent events in a bounded system where you know all possible outcomes, and the events do not influence each other. Like rolling dice or playing poker. Events that happen in life, and markets, are neither independent of each other, nor bounded in that way. Hence, probabilities, and sophisticated mathematic modelling techniques based on probabilities, don’t work when human behaviour is involved.
The maddening thing is it often seems like they work, for most of the time, and then suddenly they don’t.