  Parametric inference posits a statistical model that is a specified family of
probability distributions. Restricted inference, e.g., restricted likelihood
ratio testing, attempts to exploit the structure of a statistical submodel that
is a subset of the specified family. We consider the problem of testing a
simple hypothesis against alternatives from such a submodel. In the case of an
unknown submodel, it is not clear how to realize the benefits of restricted
inference. To do so, we first construct information tests that are locally
asymptotically equivalent to likelihood ratio tests. Information tests are
conceptually appealing but (in general) computationally intractable. However,
unlike restricted likelihood ratio tests, restricted information tests can be
approximated even when the statistical submodel is unknown. We construct
approximate information tests using manifold learning procedures to extract
information from samples of an unknown (or intractable) submodel, thereby
providing a roadmap for computational solutions to a class of previously
impenetrable problems in statistical inference. Examples illustrate the
efficacy of the proposed methodology.
