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You can state and prove theorems with linear models. You can do inference and testing. This means papers and academics naturally love them. And therefore, they
by richrichie 3y ago
You can state and prove theorems with linear models. You can do inference and testing. This means papers and academics naturally love them. And therefore, they are everywhere.
Not the case with non-linear models. We need to throw computers at them.
- whatshisface 3y agoI think that's an overstatement. You can consider the nonlinearity as a perturbation of the linear theory, study the topology of the solutions or their symmetries, or sometimes even find exact solutions, like exist for a handful of transistor circuits.
- richrichie 3y agoIf the model is y = a + b * x + e, then we can precisely state probability distributions of a and b given x and y for reasonably general assumptions about error e. If the model is y = f(x) + e, where for example f is a neural net, there is not much we can say about the “quality” of parameters of f. That is, we can’t attach confidence interval. We will have to resort to expensive simulations and for most practical applications this is not workable (size of dataset). What you say is useful in a different context. Local linearization is a very powerful idea.
- whatshisface 3y agoMaybe not if there are many more parameters in f than there are samples in the dataset, but if there are a handful of parameters in f then a typical step is to take the inverse of the derivative of goodness-of-fit with respect to each fit parameter, to determine the precision with which each parameter was determined by the fit.