  While on the one hand, chaotic dynamical systems can be predicted for all
time given exact knowledge of an initial state, they are also in many cases
rapidly mixing, meaning that smooth probabilistic information (quantified by
measures) on the system's state has negligible value for predicting the
long-term future. However, an understanding of the long-term predictive value
of intermediate kinds of probabilistic information is necessary in various
physical problems, and largely remains lacking.
  Of particular interest in data assimilation and linear response theory are
the conditional measures of the SRB measure on zero sets of general smooth
functions of the phase space. In this paper we give rigorous and numerical
evidence that such measures generically converge back under the dynamics to the
full SRB measures, exponentially quickly. We call this property conditional
mixing. We will prove that conditional mixing holds in a class of generalised
baker's maps, and demonstrate it numerically in some non-Markovian piecewise
hyperbolic maps. Conditional mixing provides a natural limit on the
effectiveness of long-term forecasting of chaotic systems via partial
observations, and appears key to proving the existence of linear response
outside the setting of smooth uniform hyperbolicity.
