Let $\{\mathcal{N}(y_i;\mu(\theta),\Sigma(\theta):\theta \in \Theta\}$ be an SEM, where $y_i$ is one observation and $\theta$ a parameter vector. Let's say there are a lot of missing values in our observed sample $y=[y_1,\ldots,y_N]$. We can still get a maximum likelihood estimate $\hat{\theta}$ using full information maximum likelihood. Lets now say that we partition every observation $y_i$ into all values that are missing $a_i$ and all values that are not missing $b_i$. Thus, $y_i^\top=[a_i,b_i]$. We can then get the conditional density $p_\hat{\theta}(a_i|b_i)$ implied by the SEM by getting the conditional distribution out of the model implied joint density $$p_\hat{\theta}([a_i,b_i])^\top=\mathcal{N}([a_i,b_i];\mu(\hat{\theta}),\Sigma(\hat{\theta}))$$ where $\hat{\theta}$ is the maximum likelihood estimate for the parameters $\theta$.

I have some experience using SEM, however I have never seen that somebody used the conditional distribution $p_\hat{\theta}(a_i|b_i)$. I am surprised by this as there are a couple of useful applications for the conditional distribution $p_\hat{\theta}(a_i|b_i)$ (e.g., prediction and imputation). So, my question is: Is the conditional distribution $p_\hat{\theta}(a_i|b_i)$ used in the SEM community? If yes, for what and under which name?


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