I will describe some new finiteness results in algebraic statistics which say that, up to symmetry, many statistical models have finite implicit descriptions as the number of states of random variables ``go to infinity''. While the focus of the talk will be on applications to Markov bases (that is, generating sets of toric ideals) the results can apply much morem widely. The results follow from finiteness results about the polynomial ring in infinitely many variables under the action of the infinite symmetric group, and may be of independent interest. This is joint work with Christopher Hillar.