Working with integrals and matrices- beginner question

Question

In the book Analysis of Incomplete Multivariate Data (Schafer, 1997), the writer tells about the following model: $Y$ is our data with missing entries, divided to $Y_{mis}$ and $Y_{obs}$, the missing and observable entry set. $R$ is an indicator variable matrix indicating for each entry of $Y$ if it's missing or not. $Y$ and $R$ depend on hidden variables: $\theta$ is data generating hidden parameter(s), and $\xi$ is the missing data parameter(s).

Can someone explain the integrals? Specifically, I want to know:

1. What are the rules used between equalities?
2. Intuitive reasoning about differentiating on non-trivial variable, such as $Y_{mis}$? How should I think about it?

2.3 The observed-data likelihood and posterior

2.3.1 Observed-data likelihood
Following arguments given by Rubin (1976) and Little and Rubin (1987), it can be shown that under ignorability, we do not need to consider the model for $R$ nor the nuisance parameters $ξ$ when making likelihood-based or Bayesian inferences about $θ$.

Because the observed data truly consist not only of Y obs, but also of R, the probability distribution of the observed data is actually given by

\begin{align} P(R, Y_{obs}|θ, ξ) &= \int P(R, Y| θ, ξ) dY_{mis} \\ &= \int P(R|Y, ξ)P(Y|θ) dY_{mis}, \tag{2.3} \end{align}

where the integral is understood to mean summation for distributions that are discrete.

Answer 1

It is based on the defintion of a marginal and conditional distributions such that \begin{align*} p(y)=\int p(y,x)dx=\int p (y|x)p (x)dx \end{align*}

Marginal distributions interact with conditional distributions such that: \begin{align*} p(y|z)=\int p(y,x|z)dx=\int p (y|x,z)p (x|z)dx \end{align*}

In your case under ignorablity assumption (MAR assumption and distinct parameter space) $f (r|y; \xi ) = f (r |y_{obs} ;\xi )$ you can impute successfully and obtain correct likelihood or Bayes inferences about the parameters without modeling the missingness