This paper extends the Bounded Real Lemma of the H-infinity control theory to stochastic systems under random disturbances with imprecisely known probability distributions. The statistical uncertainty is measured in entropy theoretic terms using the mean anisotropy functional. The disturbance attenuation capabilities of the system are quantified by the anisotropic norm which is a stochastic counterpart of the H-infinity norm. We develop a state-space criterion for the anisotropic norm of a linear discrete time invariant system to be bounded by a given threshold value. The resulting Anisotropy-based Bounded Real Lemma involves an inequality on the determinant of a matrix associated with a parameter-dependent algebraic Riccati equation.