On the Born rule in a new quantum approach

In the context of a new approach towards quantum foundation, the Born rule is proved under weak conditions through several steps. Some of the steps turn out to have connections to statistical theory. One step gives a generalized likelihood principle, a result of independent interest. The sole additional assumption behind the Born rule is a consequence of the likelihood principle of statistics. This principle is discussed from several points of view. The whole approach towards quantum foundation advocated here, leads to a simpler and more intuitive pure state concept than the traditional one in quantum mechanics. It also provides links to statistical theory and to Andrei Khrennikov's macroscopical quantum-like models, in particular, models connected to cognition and decisions. Finally, the approach has connections to relativity theory and to quantum field theory. It is a hope that this approach now can be seen as a good alternative to the traditional formal foundation of quantum theory. The approach is here discussed in the case of finite-values variables, but continuous variables and variables taking a countable set of values can be included in the theory by approximating them with finite-valued ones.