  This paper investigates the construction of model space priors from an
alternative point of view to the usual indicators for inclusion of covariates
in a given model. Assumptions about indicator variables often lead to
Beta-Binomial priors on the model space, which do not appropriately penalize
for model complexity when the parameters are fixed. This can be alleviated by
changing the parameters of the prior to depend on the number of covariates,
though justification for this is lacking from a first-principles point of view.
We propose viewing the model space as a partially ordered set. When the number
of covariates increases, an isometry argument leads to the Poisson distribution
as the unique, natural limiting prior over model dimension. This limiting prior
is derived using two constructions that view an individual model as though it
is a "local" null hypothesis and compares its prior probability to the
probability of the alternatives that nest it. We show that this prior induces a
posterior that concentrates on a finite true model asymptotically.
Additionally, we provide a deterministic algorithm that takes advantage of the
nature of the prior and explores good models in polynomial time.
