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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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If you want to find the optimal β 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.

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  • $\begingroup$ I've copied this comment by @MauritsM as a community wiki answer because it is, more or less, an answer to this question. We have a dramatic gap between answers and questions. At least part of the problem is that some questions are answered in comments: if comments which answered the question were answers instead, we would have fewer unanswered questions. $\endgroup$ – mkt Sep 11 at 15:05

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