While studying a proposition about global and local Markov properties from Grimmett's Probability on Graphs, I face difficulties with proving the equivalence of two equations.
Let $G=(V,E)$ be a finite graph, without loops or multiple edges. We shall restrict ourselves to measures on the sample space $\Sigma = \{0,1\}^V$. Let $\pi$ be a positive probability measure on $\Sigma$. The following statements are equivalent:
- $\pi$ satisfies the global Markov property
- $\pi$ satisfies the local Markov property
- for all $A \subseteq V$ and any pair $u,v \in V$ with $u \notin A, v \in A$ and $u \nsim v$, \begin{align} (*) \qquad \frac{\pi(A \cup u)}{\pi(A)} = \frac{\pi(A \cup u \setminus v)}{\pi(A\setminus v)}. \end{align}
Difficulties arises when I want to show that equation $(*)$ is equivalent to equation $(**)$. \begin{align} (**) \qquad \frac{\pi(A \cup u)}{\pi(A) + \pi(A \cup u)} = \frac{\pi(A \cup u \setminus v)}{\pi(A\setminus v)+\pi(A \cup u \setminus v)}. \end{align} First, I don't get why the global Markov property gives that \begin{align} \frac{\pi(A \cup u)}{\pi(A) + \pi(A \cup u)} = \pi(\sigma_u =1 | \sigma_{V \setminus u} = A). \end{align} The global Markov property holds when $\pi$ satisfies \begin{align} \pi(\sigma_W = s_W | \sigma_{V \setminus W}= s_{V \setminus W}) = \pi(\sigma_W = s_W | \sigma_{\Delta W}= s_{\Delta W}), \end{align} where for any $W \subseteq V$, the external boundary is defined by $\Delta W = \{v \in V: v \notin W, v \sim w\ \text{for some } w \in W\}$ and the vector $\sigma \in \Sigma$ may be placed in one-to-one correspondence with the subset $\nu(\sigma) = \{ v\in V: \sigma_v =1\}$ of $V$.
Secondly, I do not get why equation $(*)$ is equivalent to $(**)$.