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Considering the scenarios of exploring data, predicting (in the range of predictors), extrapolating and explaining- for which would one need a model? When can one do without one? [Edit] By "model" I mean a probabilistic representation of the data generating process.

As an example I would say, that for the purpose of exploratory data analysis, one can do nicely without a model (exceptions do exist), and for the purpose of making predictions "out of data" (extrapolation), one will not get very far without one. What more can be said about the different scenarios?

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One note, I would say that predicting "out of data" (or more commonly 'out of sample') and "extrapolation" are not quite the same thing. Predictive models are always hoping to be able to accurately guess an unknown response value of a future observation for which the predictor values are known, but this is generally understood as applying to cases where the predictor values are within the range of values of the data on which the model was trained. OTOH, extrapolation refers to predicting a response value for an observation whose predictor values lie outside that range. – gung Jul 22 '12 at 16:18

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Can you give your definition of model? I would say even in EDA we almost always use models (say fit a Loess line to a scatterplot, or kernel density estimate the PDF of a distribution, or identify outliers via some criterion), although they may be more implicit than explicit.

In the process of data reduction one takes a larger set of data and reduces it down into smaller sets of information that can be easily processed or understood. These smaller sets of information can be estimates of model parameters, or they can be reduced sets of information in graphical displays (they aren't really different though, to reduce the information in essence takes a model).

So to directly answer your question, I would say anytime you can not tell everything you want to know from your data by simply examining scatterplots or histograms of the distributions, then some type of model is necessary. Which given that definition I have personally never come across such a situation in which I did not need some type of model.

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+1, I largely agree w/ this, but I would not have put it quite so strongly. – gung Jul 22 '12 at 16:19
Gung, I'd go even one step beyond Andy W in terms of strong (and simple) statements: "If you make any assumptions, you have a model." As you say, even something as apparently model-less as EDA may make assumptions. – Wayne Jul 22 '12 at 18:28
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As an approach, EDA doesn't particularly try to be non-parametric: boxplots, halfspace depth& company are not devoid of assumptions. It's not about not having a model, it's about showing the deviation from the model. The whole EDA philosophy is distilled in this Tukey quote: "Numerical quantities focus on expected values, graphical summaries on unexpected values.". – user603 Jul 22 '12 at 19:12
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Consider the sample mean. I can compute it and discuss it as a location measure of the sample. What have I assumed? Same goes for inspecting a boxplot etc. I would consider these EDA without any model. – JohnRos Jul 22 '12 at 19:23
@JohnRos, I would consider a calculation of the mean an explicit model of the data generating process under most circumstances (i.e. it is the expected value). Also box-plots are frequently used as an explicit formulation to identify outliers. Regardless, if this answer isn't satisfactory I think you need to be more explicit about what is a model. You don't have to interpret the mean as the probabilistic statement about the distribution, but unless you do the mean in and of itself does not tell you much. – Andy W Jul 22 '12 at 19:29
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