I am studying about Gibbs distribution concept and I am confusing a one term in that concept that is normalization term. According to the Hammersley–Clifford theorem, an random $x$ can equivalently be characterized by a Gibbs distribution $$P(x)=Z^{-1}\exp(-U(x))$$ where $Z$ is a normalizing constant called the partition function $$Z=\sum_x \exp(-U(x))$$ and $U(x)$ is an energy function of the form $$U(x)=\sum_{c \in C}V_c(x)$$ which is a sum of clique potentials $V_c(x)$ over all possible cliques $C$. A clique $c$ is defined as a subset of sites in $S$ (are neighborhood system of $x$) in which every pair of distinct sites are neighbors, except for single-site cliques
To make clearly understand, let get simple example. Assume $x$ has two values {0,1}. The $U(x=0)$ can be computed as
$$ \left[ \begin{array}{cc|c} 0&8&-8\\ 0&-8&0\\ 3&-4&5 \end{array} \right] $$ $U(x=1)$ is $$ \left[ \begin{array}{cc|c} 0&-8&8\\ 4&-5&-8\\ 7&8&-9 \end{array} \right] $$
My question is that how to compute $Z$ and $P(x=0)$ as well as $P(x=1)$?
As my understand, $Z$ is normalization over all $x$. Then $Z$ is computed $$Z=\sum_x \exp(-U(x))=\exp(-U(x=0))+\exp(-U(x=1))$$ $$Z=10^3 \times \left[ \begin{array}{cc|c} 0.002&2.9810&2.981\\ 0.001&3.1294&2.9820\\ 0.0001&0.0546&8.1031 \end{array} \right]$$
Then $P(x=0)$ and $P(x=1)$ are computed $$P(x=0)=\frac {\exp(-U(x=0))}{Z}=\left[ \begin{array}{cc|c} 0.5&00&1\\ 0.9820&0.9526&0.0003\\ 0.9820&1&0 \end{array} \right]$$
$$P(x=1)=\frac {\exp(-U(x=1))}{Z}=\left[ \begin{array}{cc|c} 0.5&1&0\\ 0.018&0.0474&0.9997\\ 0.0180&0&1 \end{array} \right]$$
Is it correct? Reference document in here
