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I need to do a logistic regression that will likely have a lot of zeros. Can someone explain penalized logistic regression to me like I'm dumb?

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You add a penalty to control properties of the regression coefficients, beyond what the pure likelihood function (i.e. a measure of fit) does. So you optimizie

$$ Likelihood + Penalty $$

instead of just maximizing the likelihood.

The elastic net penalty penalizes both the absolute value of the coefficients (the “LASSO” penalty), which has advantage of performing automatic variable selection by shrinking irrelevant coefficients to zero, and the squared size of the coefficient (the “ridge” penalty), which has been shown to limit the impact of collinearity. The level of penalization is typically chosen by cross-validation. You can use the R package glmnet to fit elastic net penalized models.

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