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The maximum likelihood estimation procedure for linear mixed model is described in this link. It seems to me that something is wrong there.

In their Restricted Maximum Likelihood section the first equation is:

$$P(\boldsymbol y|\boldsymbol\theta,\sigma^2)=\int P(\boldsymbol y|\boldsymbol\beta,\boldsymbol\theta,\sigma^2) P(\boldsymbol \beta)d\boldsymbol\beta=\int P(\boldsymbol y|\boldsymbol\beta,\boldsymbol\theta,\sigma^2)d\boldsymbol\beta.$$

How can be $\int P(\boldsymbol y|\boldsymbol\beta,\boldsymbol\theta,\sigma^2) P(\boldsymbol \beta)d\boldsymbol\beta=\int P(\boldsymbol y|\boldsymbol\beta,\boldsymbol\theta,\sigma^2)d\boldsymbol\beta $ ?

Where is the $P(\boldsymbol \beta)$ in the last integral ?

  • Also what is the posterior distribution $P(\boldsymbol\beta|\boldsymbol y,\hat{\boldsymbol\theta}_R,\hat\sigma^2_R)$ ?

What will be the expression of the posterior distribution ?

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