In multivariate statistics, the sample space is usually the Cartesian product of the ranges of the variables involved. Even the simplest structures used in statistics, like independence, rely heavily on this structure. The talk introduces basic concepts of statistical modeling, which can be applied when the structure of the sample space is different. First, motivating examples are presented, then coordinate-free exponential families of probability distributions are introduced, which postulate simple multiplicative structures. In these families, the sample space is not necessarily a Cartesian product and the effects are not necessarily associated with cylinder sets. Also, the existence of an overall effect is not implied. Some of the properties of these families are similar to that of log-linear models, but the maximum likelihood estimates under these models have a few very surprising characteristics depending on whether the distributions in the family do or do not have an overall effect.

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