We intend to continue the work under Grant GM18770, on probablity estimation, one of the most important applications of which will eventually be to medical diagnosis. The problems are different for continuous and discontinuous distributions though some variables will be "mixed" in the medical application. For both classes of problem one can use the method of penalized likelihood. For continuous distributions we have developed this method in detail for univariate problem. We hope to extend this work to bivariate problems. Also we hope we shall be able to look for "bumps" in density curves with the aid of the graphic data tablet. For multidimensional contingency tables the "penalty" we subtract from the log-likelihood is proportional to negative entropy. We hope to apply the method penalized likehood also to parametric problems. Related to probability estimation is the Bayes/non-Bayes compromise. It has been applied strikingly to significance tests for multinomial distributions, and we have made some progress also with univariate continuous applications. We intend to apply this method to contingency tables and to multivariate continuous problems, and hope to find new significance criteria having asymptotic distributions reliable into their extreme tails.