  We consider finite relational signatures $\tau \subseteq \sigma$, a sequence
of finite base $\tau$-structures $(\mathcal{B}_n : n \in \mathbb{N})$ the
cardinalities of which tend to infinity and such that there is a fixed finite
upper bound of the degree of (the Gaifman graph of) every $\mathcal{B}_n$. We
let $\mathbf{W}_n$ be the set of all expansions to $\sigma$ of $\mathcal{B}_n$
we consider a probabilistic graphical model, a concept used in machine learning
and artificial intelligence, to generate a probability distribution
$\mathbb{P}_n$ on $\mathbf{W}_n$ for all $n$. We use a many-valued
``probability logic'' with truth values in the unit interval to express
probabilities within probabilistic graphical models and to express queries on
$\mathbf{W}_n$. This logic uses aggregation functions (e.g. the average)
instead of quantifiers and it can express all queries (on finite structures)
that can be expressed with first-order logic since the aggregation functions
maximum and minimum can be used to express existential and universal
quantifications, respectively. The main results concern asymptotic elimination
of aggregation functions (the analogue of almost sure elimination of
quantifiers for two-valued logics with quantifiers) and the asymptotic
distribution of truth values of formulas (the analogue of logical convergence
results for two-valued logics).
