My problem is relatively simple to describe. I have a collection $C$ of $n$ items and an $n\times n$ matrix $P$ such that $P_{ij}$ is the probability that items $c_i$ and $c_j$ belong to the same cluster.
We want to find the highest-probability assignment $\mathbf{a}$ of items to clusters. Any number of clusters may be used. In other words, we want to maximize the function:
$$L(\mathbf{a})=\prod_{i\neq j}P_{ij}\mathbb{1}(a_i, a_j)+(1-P_{ij})(1-\mathbb{1}(a_i, a_j))$$
where $\mathbb{1}(a_i, a_j)$ is $1$ if $a_i = a_j$ (i.e., $i$ and $j$ are from the same cluster) and $0$ otherwise.
I recognize that this may be a very hard problem to solve optimally. Convex relaxations and the like are fine, too.