We identify principles characterizing Solomonoff Induction by demands on an agents external behaviour. Key concepts are rationality, computability, indifference and time consistency. Furthermore, we discuss extensions to the full AI case to derive AIXI.
We provide a formal, simple and intuitive theory of rational decision making including sequential decisions that affect the environment. The theory has a geometric flavor, which makes the arguments easy to visualize and understand. Our theory is for
complete decision makers, which means that they have a complete set of preferences. Our main result shows that a complete rational decision maker implicitly has a probabilistic model of the environment. We have a countable version of this result that brings light on the issue of countable vs finite additivity by showing how it depends on the geometry of the space which we have preferences over. This is achieved through fruitfully connecting rationality with the Hahn-Banach Theorem. The theory presented here can be viewed as a formalization and extension of the betting odds approach to probability of Ramsey and De Finetti.
We provide a variable metric stochastic approximation theory. In doing so, we provide a convergence theory for a large class of online variable metric methods including the recently introduced onli
We define an entropy based on a chosen governing probability distribution. If a certain kind of measurements follow such a distribution it also gives us a suitable scale to study it with. This scale will appear as a link function that is applied to t
he measurements. A link function can also be used to define an alternative structure on a set. We will see that generalized entropies are equivalent to using a different scale for the phenomenon that is studied compared to the scale the measurements arrive on. An extensive measurement scale is here a scale for which measurements fulfill a memoryless property. We conclude that the alternative algebraic structure defined by the link function must be used if we continue to work on the original scale. We derive Tsallis entropy by using a generalized log-logistic governing distribution. Typical applications of Tsallis entropy are related to phenomena with power-law behaviour.
