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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?

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  • $\begingroup$ 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? $\endgroup$
    – boscovich
    Commented Sep 30, 2013 at 11:02
  • $\begingroup$ Possible duplicate of Frequentism and priors $\endgroup$
    – kjetil b halvorsen ♦
    Commented Mar 28, 2018 at 10:12
  • $\begingroup$ The function you produce is not a likelihood because of the extra functions of the parameters at the end. $\endgroup$
    – Xi'an
    Commented Mar 31, 2018 at 8:32

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