  The Gumbel trick is a method to sample from a discrete probability
distribution, or to estimate its normalizing partition function. The method
relies on repeatedly applying a random perturbation to the distribution in a
particular way, each time solving for the most likely configuration. We derive
an entire family of related methods, of which the Gumbel trick is one member,
and show that the new methods have superior properties in several settings with
minimal additional computational cost. In particular, for the Gumbel trick to
yield computational benefits for discrete graphical models, Gumbel
perturbations on all configurations are typically replaced with so-called
low-rank perturbations. We show how a subfamily of our new methods adapts to
this setting, proving new upper and lower bounds on the log partition function
and deriving a family of sequential samplers for the Gibbs distribution.
Finally, we balance the discussion by showing how the simpler analytical form
of the Gumbel trick enables additional theoretical results.
