  In optimal prediction methods one estimates the future behavior of
underresolved systems by solving reduced systems of equations for expectations
conditioned by partial data; renormalization group methods reduce the number of
variables in complex systems through integration of unwanted scales. We
establish the relation between these methods for systems in thermal
equilibrium, and use this relation to find renormalization parameter flows and
the coefficients in reduced systems by expanding conditional expectations in
series and evaluating the coefficients by Monte-Carlo. We illustrate the
construction by finding parameter flows for simple spin systems and then using
the renormalized (=reduced) systems to calculate the critical temperature and
the magnetization.
