Residual Entropy
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We describe an approach to improving model fitting and model generalization that considers the entropy of distributions of modelling residuals. We use simple simulations to demonstrate the observational signatures of overfitting on ordered sequences of modelling residuals, via the autocorrelation and power spectral density. These results motivate the conclusion that, as commonly applied, the least squares method assumes too much when it assumes that residuals are uncorrelated for all possible models or values of the model parameters. We relax these too-stringent assumptions in favour of imposing an entropy prior on the (unknown, model-dependent, but potentially marginalizable) distribution function for residuals. We recommend a simple extension to the Mean Squared Error loss function that approximately incorporates this prior and can be used immediately for modelling applications where meaningfully-ordered sequences of observations or training data can be defined.
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