  We define a linearly reinforced process called the *-Edge-Reinforced Random
Walk (*-ERRW ) which can be seen as a Yaglom reversible, hence non-reversible,
extension of the Edge-Reinforced Random Walk (ERRW) introduced by Coppersmith
and Diaconis in 1986. This family of processes also generalizes the r-dependent
ERRW introduced by Bacallado (2009). Under some assumptions on the initial
weights, the *-ERRW is partially exchangeable in the sense of Diaconis and
Freedman (1980), and thus it is a random walk in a random environment. The main
result of the paper gives the explicit expression of the mixing law, hence
extending the "magic formula" of Coppersmith and Diaconis from the case of
mixtures of reversible Markov chains to the case of mixtures of Yaglom
reversible Markov chains.
