Joint Summarized by Marginal or Conditional?
I happened to read Yixuan’s blog about a question related to the course Statistical Inference, whether two marginal distributions can determine the joint distribution. The question is adopted from Exercise 4.47 of Casella and Berger (2002).
This exercise aims to illustrate that marginal normality does not imply bivariate normality.
This question recalled me related material in Gibbs sampler, where the conditional distributions contain sufficient information to produce a sample from the joint distribution. The Hammersley-Clifford theorem (Robert and Casella (2004)) demonstrates this feature,
In a word, the full conditional distributions perfectly summarize the joint density, but the set of marginal distributions obviously fails to do so.
It is necessary to note that this theorem makes the implicit assumption that the joint density $f(x,y)$ exists. The assumption can be not satisfied (Example 10.29 of Robert and Casella (2004)), suppose a pair of conditional densities
are well defined, but
which violates the requirement of the Hammersley-Clifford theorem.