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I am confused about the correct way to write the elastic net. After reading some research papers there seems to be three forms

1) $\exp\{-\lambda_1|\beta_k|-\lambda_2\beta_k^2\}$

2) $\exp\{-\frac{(\lambda_1|\beta_k|+\lambda_2\beta_k^2)}{\sqrt{\sigma^2}}\}$

3) $\exp\{-\frac{(\lambda_1|\beta_k|+\lambda_2\beta_k^2)}{2\sigma^2}\}$

I just don't understand the correct way to add $\sigma^2$. Is any of the above expressions correct?

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    $\begingroup$ If you want to find the optimal $\beta$ parameters, i.e. the ones that maximize the expression, then all three expressions are equivalent. Also, this is only the prior (or penalty) part of the elastic net expression. Obviously you need to add the linear regression part as well. $\endgroup$ – Maurits M Jun 3 at 6:08
  • $\begingroup$ In page 5 here, they provide the marginal prior they use for the Bayesian elastic net Gibbs sampler. $\endgroup$ – Greenparker Jun 12 at 15:32

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