Monday, September 1, 2014

Why Can't We Be Friends: The Difference Between Bayesian And Frequentist Statistics In Two Paragraphs

Universes as plenty as blackberries

Bayesians think data is deterministic and parameters random. This means that looking all and only at the real data can teach you something about the data. We have probabilistic thoughts about what process could have made that one and only sample. There is a massive set of possible worlds and we don't know which we are in - but everything is deterministic within that world. That set of worlds might be real (David Lewis, David Deutsch) or it might be subjective (Jaynes, Savage), it doesn't make a difference mathematically. That's why we need priors, to help us regulate that set. Bootstrapping and other sampling distribution techniques are unnecessary noise makers - we're in the world we are in, don't make up new ones. The world we're most likely in is the one that is most like the data we really see (the likelihood principle).

A primitive binary classifier

Frequentists (aka error statisticians) think data is random and parameters are deterministic. We know what world we are in, but have to eliminate noise, error, etc. The world itself is random! We don't need priors, we need to get an idea of how our world would change if we started in the same place. Bootstrapping and other sampling distribution techniques are good for doing exactly this. Who cares if a hypothesis matches our data really closely, our data might be filled with noise and imperfection (no likelihood principle).

An unusually calm disagreement in the social science

Bayesian and Frequentist philosophies cannot be totally reconciled, because there exists tests maximally efficient in one and incoherent in another. They are like elliptic and hyperbolic geometry in this way. Use of one theory or another for a given situation is a philosophically deep choice. One doesn't have to dismiss one or the other just because they disagree if you look at different situations. Instead one has to be honest about the flaws (and, perhaps, to a lesser extent the strengths) that one's method of choice has for the problem at hand.

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