  Computing the partition function $Z$ of a discrete graphical model is a
fundamental inference challenge. Since this is computationally intractable,
variational approximations are often used in practice. Recently, so-called
gauge transformations were used to improve variational lower bounds on $Z$. In
this paper, we propose a new gauge-variational approach, termed WMBE-G, which
combines gauge transformations with the weighted mini-bucket elimination (WMBE)
method. WMBE-G can provide both upper and lower bounds on $Z$, and is easier to
optimize than the prior gauge-variational algorithm. We show that WMBE-G
strictly improves the earlier WMBE approximation for symmetric models including
Ising models with no magnetic field. Our experimental results demonstrate the
effectiveness of WMBE-G even for generic, nonsymmetric models.
