3
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ What does your second question refer to here? I don't see anything being differentiated. And what does "non-trivial" mean? $\endgroup$ – The Laconic Apr 12 '18 at 12:05
3
$\begingroup$

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

$\endgroup$
  • $\begingroup$ Thank you. For future readers I'll say what helped me: write down the bayes rule for discrete summation, then move to integral case, and apply as needed. $\endgroup$ – Danny Apr 12 '18 at 13:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.