Are "All Models Wrong"?
Researcher in Machine Learning, Causality and Recommender Systems. On a career break to write a book on Reward Optimizing Recommender Systems. Previously led a team of researchers and engineers at Criteo.
The “All models are wrong, some models are useful” quote attributed to George Box is a favorite in statistical presentations, but is it true, and what does it really mean?
Let me make a probabilistic statement that I don’t think is wrong. If I throw a coin, the probability that it comes up heads is between 0 and 1. How could this statement be considered wrong?
Perhaps you don’t think it’s a model due to the imprecision. What if I say the probability of heads is 0.5, is that in any meaningful sense wrong? What if I say the probability of heads is one in a million or 10^-6?
Wrong might mean contradicted by the data, but if you take that view I think almost no models are wrong. What if I say the the probability of heads is 10^-6 i.e. one in a million, and then it comes up heads. The model is not contradicted by the data, the probability of the observed data is not zero. David Aldous makes a similar point, if the probability of a Soviet Union break up was assigned a probability of 5% (in the 80s) was the prediction right or wrong?
To a subjective Bayesian, a model might represent the beliefs of a coherent person, perhaps no person exists with those beliefs, but again I am not sure that would make the model “wrong”. To a frequentist it might mean something like the density of the model does not resemble a histogram of collected data if you collect a data (although as David Freedman reminds us, in the long run we are all dead, and you can never step in the same river twice).
My point with this brief note is the meaning of probability and hence model is ambiguous and consequently the meaning of wrong is similarly ambiguous.
