# Assuming a probability density for MLE to do model selection

Motivation: I am trying to use Akaike Information Criterion to assess model ranking and over-fitting risk for a set of nonlinear models. I am an electrical engineer with no formal statistical training (for model selection, that is!), so please bluntly correct anything that is wrong.

As I understand it, I must compute the maximum likelihood estimator for each model. I could assume the residuals are Gaussian* and then the MLE is a least squares one. I am not convinced this is adequate; if my models are nonlinear, does that imply a more complex probability density? And, if my residuals aren't Gaussian and I don't have a good assumption for a probability density, how would I choose one?

The values for the parameters of these models are chosen via an exploratory algorithm that uses RMSE as its cost function. The models have between 10-40 terms. The datasets have between 300 and 2000 measured values.

*I can also check this by observing them, no?: (histogram of the difference between the model predicted value and the observed value) -- though this is just for one set of data, I do not necessarily know that the errors will always be normal for each model against any possible data set. If I check these before each analysis for each model, what would I do if they aren't normal?

Motivation: I am trying to use Akaike Information Criterion to assess model ranking and over-fitting risk for a set of nonlinear models.

As I understand it, I must compute the maximum likelihood estimator for each model.

If you want an AIC, yes, you would need MLE. But the AIC is not automatically ideal. [One thing you should be careful of is if you're selecting a model on the same data you're using to do inference, you'll have a variety of problems.]

I could assume the residuals are Gaussian* and then the MLE is a least squares one. I am not convinced this is adequate; if my models are nonlinear, does that imply a more complex probability density?

I don't think so. If your model is of the form $y=f(X,\beta)+\varepsilon$, I don't see why nonlinear $f$ would imply anything about $\varepsilon$.

Your actual distribution about $E(Y)$ is almost certainly more complex than a Gaussian/ (When are data exactly Gaussian? I'd have thought almost never.)

And, if my residuals aren't Gaussian and I don't have a good assumption for a probability density, how would I choose one?

If you don't have a good basis for one, choosing a model by looking at the same data you use to fit the model would bias your AICs anyway. [You might be able to get around that by sample-splitting, or cross-validation for example.]

I can also check this by observing them, no?

Your residuals are proxies for your errors, yes, so for example, skewed residuals might indicate non-normality.

If I check these before each analysis for each model, what would I do if they aren't normal?

• You might consider a class of models such as generalized nonlinear models (like GLMs but nonlinear in the parameters) and still use AIC.

• You might stick with least-squares even though it's not ML, and treat AIC simply as a (monotonic transformation of) a penalized-MSE.

• Depending on what you need your models to achieve, you might consider modifying the current model to one with a slightly heavier tail, such as a contaminated-normal model, perhaps still with ML.

• You might consider using some different criterion to do both model selection and model fitting.

But in any case your actual errors won't be exactly normal; the question is the degree to which that will impact your inference. (The AIC, for example, may perform reasonably well at trading off fit for model complexity whether or not the fitted model is exactly right.)

though this is just for one set of data, I do not necessarily know that the errors will always be normal for each model against any possible data set.

Note that the marginal distribution of residuals will only approximate the error distribution if the other (more important) assumptions hold up; you should check that the model for the mean and the variance is reasonable first before trying to worry much about normality.

• What other forms of a model are there than y=f(X,β)+ε ? Or does this form merely imply how error is assessed (difference between f and y)? I don't follow "you should check that the model for the mean and the variance is reasonable first" -- does this mean that the mean and variance of the model-generated data need to match the measured data (with some tolerance)? If they don't, does this suggest some fundamental error in the form of the model? Does it make sense to look at AIC weights for each fold of a K-fold CV? Finally, are the errors of the held-out-from-training data more interesting? – pixels Oct 29 '14 at 9:16