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Can anyone explain to me the difference between penalised likelihood and maximum a posteriori?

I read a paper where the likelihood function is

$$L(\theta_1, \theta_2,\theta_3 ; x)=f(x|\theta_1, \theta_2) f(\theta_2|\theta_1,\theta_3)f(\theta_3)$$ or $$\ell(\theta_1, \theta_2,\theta_3 ; x)=\log(f(x|\theta_1, \theta_2)) +\log(f(\theta_2|\theta_1,\theta_3))+\log(f(\theta_3))$$

and the authors say they use maximum penalised likelihood to find $\widehat\Theta$. I thought that would be maximum a posteriori and penalised likelihood would use a likelihood function of the form. $$\ell(\Theta ; x)=\log(f(x|\Theta)) - \lambda g(\Theta)$$

Any thoughts?

kjetil b halvorsen
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Ursulla
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  • I don't know if this can be of any help, but - for example - quadratic loglikelihood penalization (squared $L_2$ norm) corresponds to having independent normal priors on the model parameters. I don't know the deatils of the paper you're referring to, but it's definitely not impossible to "exploit" PL for that purpose - Or maybe I've misunderstood your question? – boscovich Sep 30 '13 at 11:02
  • Possible duplicate of [Frequentism and priors](https://stats.stackexchange.com/questions/29835/frequentism-and-priors) – kjetil b halvorsen Mar 28 '18 at 10:12
  • The function you produce is not a likelihood because of the extra functions of the parameters at the end. – Xi'an Mar 31 '18 at 08:32

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