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
*