How does mice::mice work? The idea of multiple imputation seems to be based on the decomposition
$$
p(\theta \mid y_{\text{obs}}) = \int p(\theta \mid y_{\text{obs}}, y_{\text{mis}})p( y_{\text{mis}} \mid y_{\text{obs}}) \text{d}y_{\text{mis}}.
$$
One can 


*

*simulate draws $y_{\text{mis}}^i \sim p( y_{\text{mis}} \mid y_{\text{obs}})$, 

*estimate the posterior $p(\theta \mid y_{\text{obs}}, y_{\text{mis}}^i)$ for each draw, and then 

*average these results, appealing to the law of total expectation.
This is interesting because it's probably more common to sample from the posterior first, and then use those parameter samples to sample unobserved data.
Does the R function mice::mice estimate $p( y_{\text{mis}} \mid y_{\text{obs}}) $  without reference to the parameters of interest $\theta$? If it does not, and it is actually using the decomposition
$$
p( y_{\text{mis}} \mid y_{\text{obs}}) = \int p( y_{\text{mis}} \mid \theta)p(\theta \mid y_{\text{obs}}) \text{d}\theta,
$$
then it doesn't seem like I can use any method I wish to do step (2). I'm sure this has been asked before, but I've been having a hard time finding the answer because many search results are devoid of any notation.
 A: mice assumes at least MAR missing mechanism, under MAR (and MCAR) which is an ignorable missing-data mechanism a model for missingness is not necessary when the statistical inference aims at $\theta$ (theorem of ignorability).So since you are more interest in $\theta$ then for correct inference, the observed-data posterior distribution $f( \theta|y_{\text{mis}})$  has to be considered 
$$f(\theta|y_{\text{mis}}) = \int f(\theta, y_{\text{mis}}| y_{\text{obs}})dy_{\text{mis}} = $$
$$ \int f(\theta|y_{\text{mis}},y_{\text{obs}})f(y_{\text{mis}} |y_{\text{obs}})dy_{\text{mis}}$$
this part $ f(\theta|y_{\text{mis}},y_{\text{obs}})$ called the complete posterior (step2) and this part $f(y_{\text{mis}} |y_{\text{obs}})$ called posterior predictive distribution, which typically we cannot directly draw from, so that 
$$f( y_{\text{mis}} \mid y_{\text{obs}}) = \int f( y_{\text{mis}} \mid \theta , y_{\text{obs}})f(\theta \mid y_{\text{obs}}) \text{d}\theta
$$
this part $f(\theta \mid y_{\text{obs}})$ called posterior step and this part $f( y_{\text{mis}} \mid \theta , y_{\text{obs}})$ called  the imputation step.
