What is the correct way to write the elastic net?

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?

Answer

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.

Attribution
Source : Link , Question Author : gbd , Answer Author :
mkt

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