= Observed-data likelihood with a missing covariate
{title2=$L_i=\int f_\beta(y_i\mid x)g_\eta(x)\,dx$}
If a regression response $Y$ is observed but covariate $X$ is sometimes missing, <likelihood> inference requires a joint model: for example $f_\beta(y\mid x)g_\eta(x)$. An incomplete record contributes $\int f_\beta(y\mid x)g_\eta(x)\,dx$ to the <observed-data likelihood>. <Missing at random> with distinct parameters allows the missingness mechanism to be omitted; it does not remove the need to integrate over $X$.
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