The technical problems involved in obtaining Bayesian model estimates for the regression parameters in m similar groups are studied. The available computer programs, BPREP (BASIC), and BAYREG, both written in FORTRAN, require an amount of computer processing that does not encourage regular use. These programs are analyzed so that the performance of algorithms might be improved to a point where inclusion in the CADA Monitor would be feasible. Slight modifications to the preparatory program BPREP, and major changes to BAYREG have resulted in a speeded-up algorithm, reducing computer time by a factor of two to four in four benchmark examples--involving grade point average as a predictor of American College Test scores in junior colleges--believed to be fairly representative examples of the type of multiple regression problems where Bayesian m-group regression might be used. However, further use of the speeded algorithm is recommended before the program is included in the CADA Monitor. This report includes descriptions of: (1) an update to Jones and Novick's input description for use of the modified programs; (2) deficiencies of the old programs BPREP and BAYREG, and improvements in the new program, MBREG; (3) detailed proposal for a small revision of BPREP into a new preparatory program BPREP adapted to the new main program MBREG; (4) "leaps" which constitute the major performance improvement of MBREG over BAYREG. (Author/GDC)