I have been reading some papers and I understood that adaptive lasso has the Oracle properties which lasso lacks. Does that mean adaptive lasso always better than lasso (let's focus on the simple linear case)? Or this is only true under some circumstances?
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6$\begingroup$ For forecasting, oracle property is detrimental. But it is useful for finding the "correct" model. So it depends on what you want to do. This is analogous to BIC- vs. AIC-based model selection. BIC is a consistent selector but underperforms in forecasting, i.e. is inefficient. AIC is efficient but not a consistent selector. It has been proven that efficiency and consistency is not possible at the same time. $\endgroup$– Richard HardyOct 20, 2017 at 20:04
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1$\begingroup$ Thanks a lot for your good comment. Do you mind explaining a bit more on the following: why for predicting, oracle property is bad. Is this mean when prediction is the goal, lasso should be used, and when inference is the goal we should use adaptive lasso? And why it is that efficiency and consistency cannot be achieved at the same time? Thanks again $\endgroup$– MZ75Oct 20, 2017 at 20:13
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1$\begingroup$ As I mentioned on your other version of this question, adding 'adaptive' to something comes with some cost -- so what is better depends on what you're trying to do better at; perhaps it would be better to ask about what the tradeoffs are. $\endgroup$– Glen_bOct 20, 2017 at 23:12
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