A gravitational wave background (GWB) of sufficient strength, characterized by Ω, the energy density per logarithmic frequency interval in units of the closure density, would introduce timing residuals in the most stable millisecond pulsars. For a description pertaining to the observations of PSR’s 1937+21 and 1855+09 see Kaspi, Taylor and Ryba (1994), hereafter KTR, and references therein. Thorsett and Dewey (1996, see also this volume) present a method for placing a statistical upper limit on Ω. Their method however, cannot correctly account for the presence of a known level of white measurement noise in the timing residuals. We use a Bayesian approach which can best account for this white noise along with our lack of previous knowledge on the parameter Ω (McHugh, Zalamansky, Vernotte and Lantz, submitted).