The current paper derives restrictions on an agent's behavior from invariance principles, in particular, from the requirement that an agent's choices be independent of the units of measurement. It is shown that invariance principles are sufficient for approximating an agent's behavior in environments where little information is available. Strong restrictions on admissible shape of tails of the posterior distribution of an unknown variable are derived. For the case of independently and identically distributed draws, closed form solution for the family of tails of posterior distribution is obtained. It is shown that even if an agent assumes that the variable in question is drawn from a finite parametric family of distributions with exponential-like tails, the posterior distribution obtained by integrating out the unknown parameters has very fat tails. Many results obtained in this paper do not rely on expected utility axioms and thus could be combined with either expected or non-expected utility theories. Decision problems arising in the areas ranging from industrial R&D planning to risk management provide motivation and potential applications for the theory developed herein.