A characterization of conjugate priors in exponential families with application to inverse regression

It is often convenient to assume that XX and X|YX|Y are in the same exponential family. By considering XX as the “parameter” and YY as the “data”, the problem becomes determining which exponential families X|YX|Y have conjugate priors. We develop a necessary condition for conjugacy. One-dimensional exponential families can be conjugate only if they have exactly two support points.