  This paper is about models for a vector of probabilities whose elements must
have a multiplicative structure and sum to 1 at the same time; in certain
applications, as basket analysis, these models may be seen as a constrained
version of quasi-independence. After reviewing the basic properties of these
models, their geometric features as a curved exponential family are
investigated. A new algorithm for computing maximum likelihood estimates is
presented and new insights are provided on the underlying geometry. The
asymptotic distribution of three statistics for hypothesis testing are derived
and a small simulation study is presented to investigate the accuracy of
asymptotic approximations.
