All models are wrong

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"All models are wrong" is a common aphorism in statistics; it is often expanded as "All models are wrong, but some are useful". It is usually considered to be applicable to not only statistical models, but to scientific models generally. The aphorism is generally attributed to the statistician George Box, although the underlying concept predates Box's writings.

Quotations of George Box[edit]

The first record of Box saying "all models are wrong" is in a 1976 paper published in the Journal of the American Statistical Association.[1] The 1976 paper contains the aphorism twice. The two sections of the paper that contain the aphorism are copied below.

2.3  Parsimony
Since all models are wrong the scientist cannot obtain a "correct" one by excessive elaboration. On the contrary following William of Occam he should seek an economical description of natural phenomena. Just as the ability to devise simple but evocative models is the signature of the great scientist so overelaboration and overparameterization is often the mark of mediocrity.

2.4  Worrying Selectively
Since all models are wrong the scientist must be alert to what is importantly wrong. It is inappropriate to be concerned about mice when there are tigers abroad.

Box repeated the aphorism in a paper that was published in the proceedings of a 1978 statistics workshop.[2] The paper contains a section entitled "All models are wrong but some are useful". The section is copied below.

Now it would be very remarkable if any system existing in the real world could be exactly represented by any simple model. However, cunningly chosen parsimonious models often do provide remarkably useful approximations. For example, the law PV = RT relating pressure P, volume V and temperature T of an "ideal" gas via a constant R is not exactly true for any real gas, but it frequently provides a useful approximation and furthermore its structure is informative since it springs from a physical view of the behavior of gas molecules.

For such a model there is no need to ask the question "Is the model true?". If "truth" is to be the "whole truth" the answer must be "No". The only question of interest is "Is the model illuminating and useful?".

Box repeated the aphorism twice more in his 1987 book, Empirical Model-Building and Response Surfaces (which was co-authored with Norman Draper).[3] The first repetition is on p. 74: "Remember that all models are wrong; the practical question is how wrong do they have to be to not be useful." The second repetition is on p. 424, which is excerpted below.

... all models are approximations. Essentially, all models are wrong, but some are useful. However, the approximate nature of the model must always be borne in mind....

A second edition of the book was published in 2007, under the title Response Surfaces, Mixtures, and Ridge Analyses. The second edition repeats the aphorism twice, in contexts identical with those of the first edition (on p. 63 and p. 414).[4]

Box's widely cited book Statistics for Experimenters (co-authored with William Hunter) does not include the aphorism in its first edition (published in 1978).[5] The second edition (published in 2005; co-authored with William Hunter and J. Stuart Hunter) includes the aphorism three times: on p. 208, p. 384, and p. 440.[6] On p. 440, the relevant sentence is this: "The most that can be expected from any model is that it can supply a useful approximation to reality: All models are wrong; some models are useful".

Discussions[edit]

There have been varied discussions about the aphorism. A selection from those discussions is presented below.

In 1983, the statisticians Peter McCullagh and John Nelder published their much-cited book on generalized linear models. The book includes a brief discussion of the aphorism (though without citing Box).[7] A second edition of the book, published in 1989, contains a very similar discussion of the aphorism.[8] The discussion from the first edition is as follows.

Modelling in science remains, partly at least, an art. Some principles do exist, however, to guide the modeller. The first is that all models are wrong; some, though, are better than others and we can search for the better ones. At the same time we must recognize that eternal truth is not within our grasp.

In 1995, the statistician Sir David Cox commented as follows.[9]

... it does not seem helpful just to say that all models are wrong. The very word model implies simplification and idealization. The idea that complex physical, biological or sociological systems can be exactly described by a few formulae is patently absurd. The construction of idealized representations that capture important stable aspects of such systems is, however, a vital part of general scientific analysis and statistical models, especially substantive ones, do not seem essentially different from other kinds of model.

In 1996, an Applied Statistician's Creed was proposed.[10] The Creed includes, in its core part, the aphorism.

In 2002, K.P. Burnham and D.R. Anderson published their much-cited book on statistical model selection. The book states the following.[11]

A model is a simplification or approximation of reality and hence will not reflect all of reality. ... Box noted that "all models are wrong, but some are useful." While a model can never be "truth," a model might be ranked from very useful, to useful, to somewhat useful to, finally, essentially useless.

The statistician J. Michael Steele has argued somewhat against the aphorism as follows.[12]

... there are wonderful models — like city maps....

If I say that a map is wrong, it means that a building is misnamed, or the direction of a one-way street is mislabeled. I never expected my map to recreate all of physical reality, and I only feel ripped off if my map does not correctly answer the questions that it claims to answer.

My maps of Philadelphia are useful. Moreover, except for a few that are out-of-date, they are not wrong.

In 2008, the statistician Andrew Gelman responded to that, saying in particular the following.[13]

I take his general point, which is that a street map could be exactly correct, to the resolution of the map.

... The saying, "all models are wrong," is helpful because it is not completely obvious....

This is a simple point, and I can see how Steele can be irritated by people making a big point about it. But, the trouble is, many people don't realize that all models are wrong.

In 2013, the philosopher of science Peter Truran published an essay related to the aphorism.[14] The essay notes, in particular, the following.

... seemingly incompatible models may be used to make predictions about the same phenomenon. ... For each model we may believe that its predictive power is an indication of its being at least approximately true. But if both models are successful in making predictions, and yet mutually inconsistent, how can they both be true? Let us consider a simple illustration. Two observers are looking at a physical object. One may report seeing a circular disc, and the other may report seeing a rectangle. Both will be correct, but one will be looking at the object (a cylindrical can) from above and the other will be observing from the side. The two models represent different aspects of the same reality.

Truran's essay further notes that Newton's theory of gravitation has been supplanted by Einstein's theory of relativity and yet Newton's theory remains generally "empirically adequate". Indeed, Newton's theory generally has excellent predictive power. Yet Newton's theory is not an approximation of Einstein's theory. For illustration, consider an apple falling down from a tree. Under Newton's theory, the apple falls because Earth exerts a force on the apple—what is called "the force of gravity". Under Einstein's theory, Earth does not exert any force on the apple.[15] Hence, Newton's theory might be regarded as being, in some sense, completely wrong but extremely useful. (The usefulness of Newton's theory comes partly from being vastly simpler, both mathematically and computationally, than Einstein's theory.)

In 2014, the statistician David Hand made the following statement.[16]

In general, when building statistical models, we must not forget that the aim is to understand something about the real world. Or predict, choose an action, make a decision, summarize evidence, and so on, but always about the real world, not an abstract mathematical world: our models are not the reality—a point well made by George Box in his oft-cited remark that "all models are wrong, but some are useful".

Additionally, in 2011, a workshop on model selection was held in The Netherlands. The name of the workshop was "All models are wrong...".[17]

Historical antecedents[edit]

Although the aphorism seems to have originated with George Box, the underlying idea goes back decades, perhaps centuries. For example, in 1960, Georg Rasch said the following.[18]

… no models are [true]—not even the Newtonian laws. When you construct a model you leave out all the details which you, with the knowledge at your disposal, consider inessential…. Models should not be true, but it is important that they are applicable, and whether they are applicable for any given purpose must of course be investigated. This also means that a model is never accepted finally, only on trial.

Similarly, in 1947, the mathematician John von Neumann said that "truth … is much too complicated to allow anything but approximations".[19]

Relatedly, in 1942, the French philosopher-poet Paul Valéry said the following.[20]

Ce qui est simple est toujours faux. Ce qui ne l’est pas est inutilisable.

What is simple is always wrong. What is not is unusable.[21]

Valéry, Paul (1942), Mauvaises pensées et autres, Paris: Éditions Gallimard

See also[edit]

Notes[edit]

  1. ^ Box, G. E. P. (1976), "Science and statistics" (PDF), Journal of the American Statistical Association, 71: 791–799, doi:10.1080/01621459.1976.10480949.
  2. ^ Box, G. E. P. (1979), "Robustness in the strategy of scientific model building", in Launer, R. L.; Wilkinson, G. N., Robustness in Statistics, Academic Press, pp. 201–236, doi:10.1016/B978-0-12-438150-6.50018-2.
  3. ^ Box, G. E. P.; Draper, N. R. (1987), Empirical Model-Building and Response Surfaces, John Wiley & Sons.
  4. ^ Box, G. E. P.; Draper, N. R. (2007), Response Surfaces, Mixtures, and Ridge Analyses, John Wiley & Sons.
  5. ^ Box, G. E. P.; Hunter, W. G. (1978), Statistics for Experimenters, John Wiley & Sons.
  6. ^ Box, G. E. P.; Hunter, J. S.; Hunter, W. G. (2005), Statistics for Experimenters (2nd ed.), John Wiley & Sons.
  7. ^ McCullagh, P.; Nelder, J. A. (1983), Generalized Linear Models, Chapman & Hall, §1.1.4.
  8. ^ McCullagh, P.; Nelder, J. A. (1989), Generalized Linear Models (second ed.), Chapman & Hall, §1.1.4.
  9. ^ Cox, D. R. (1995), "Comment on “Model uncertainty, data mining and statistical inference”", Journal of the Royal Statistical Society, Series A, 158: 455–456.
  10. ^ Nester, M. R. (1996), "An applied statistician's creed" (PDF), Journal of the Royal Statistical Society, Series C, 45: 401–410, doi:10.2307/2986064.
  11. ^ Burnham, K. P.; Anderson, D. R. (2002), Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach (2nd ed.), Springer-Verlag, §1.2.5. [As of February 2019, combined editions of this book have over 44000 citations on Google Scholar.]
  12. ^ Steele, J. M., "Models: Masterpieces and Lame Excuses".
  13. ^ Gelman, A. (12 June 2008), "Some thoughts on the saying, "All models are wrong, but some are useful"".
  14. ^ Truran, P. (2013), "Models: Useful but not true", Practical Applications of the Philosophy of Science, Springer, p. 61–67, doi:10.1007/978-3-319-00452-5_10.
  15. ^ Under Einstein's theory of relativity, the primary reason the apple falls down is that Earth warps time, so that clocks near the base of the tree run more slowly than clocks high up in the tree; there is also a secondary reason, which is that Earth warps space. The empirical evidence for Einstein's theory is extremely strong—e.g. GPS relies on Einstein's theory, and it would not work if it relied on Newton's theory (Ashby 2002).
  16. ^ Hand, D. J. (2014), "Wonderful examples, but let's not close our eyes", Statistical Science, 29: 98–100, arXiv:1405.4986, doi:10.1214/13-STS446.
  17. ^ Wit, E.; van den Heuvel, E.; Romeijn, J.-W. (2012), "'All models are wrong...': an introduction to model uncertainty" (PDF), Statistica Neerlandica, 66: 217–236, doi:10.1111/j.1467-9574.2012.00530.x. [See too the workshop web page: "All models are wrong...".]
  18. ^ Rasch, G. (1960), Probabilistic Models for Some Intelligence and Attainment Tests, Copenhagen: Danmarks Paedagogiske Institut, p. 37–38; republished in 1980 by University of Chicago Press.
  19. ^ von Neumann, J. (1947), "The mathematician", in Haywood, R. B., Works of the Mind, University of Chicago Press, p. 180–196; republished in 1995 by Bródy F., Vámos T. (editors), The Neumann Compendium, World Scientific, p. 618–626.
  20. ^ The relatedness of Valéry's quotation with the aphorism "all models are wrong" has been noted by various authors, e.g. Vankat (2013, §1.7).
  21. ^ Some authors have given different English translations, e.g. Valéry (1970, p. 466), Wolfson & Murphy (1998), and Vankat (2013, §1.7). The translation presented here was given by Google Translate; it has only one word different from the translation of Wolfson & Murphy: "What" instead of "Whatever" (both occurrences).

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