Empirical data sets are algorithmically compressible: reply to McAllister?

James McAllister's 2003 article, 'Algorithmic randomness in empirical data', claims that empirical data sets are algorithmically random, and hence incompressible. We show that this claim is mistaken. We present theoretical arguments and empirical evidence for compressibility, and discuss the matter in the framework of Minimum Message Length (MML) inference.