Two problems, classifying an individual into one of several populations and estimating the regression in that population, are simultaneously treated as one problem. This can be viewed as a problem of a missing observation on a categorical variable. When all variables are jointly distributed multivariate normal, the maximum likelihood solution is the intuitively appealing one: classify the individual using the usual likelihood ratio procedure, then estimate the regression using the observations from the selected population.
"Classification and Estimation of Several Multiple Regressions." Ann. Statist. 2 (3) 558 - 561, May, 1974. https://doi.org/10.1214/aos/1176342716