We introduce a new class of real-valued monotones in preordered spaces, injective monotones. We show that the class of preorders for which they exist lies in between the class of preorders with strict monotones and preorders with countable multi-utilities, improving upon the known classification of preordered spaces through real-valued monotones. We extend several well-known results for strict monotones (Richter–Peleg functions) to injective monotones, we provide a construction of injective monotones from countable multi-utilities, and relate injective monotones to classic results concerning Debreu denseness and order separability. Along the way, we connect our results to Shannon entropy and the uncertainty preorder, obtaining new insights into how they are related. In particular, we show how injective monotones can be used to generalize some appealing properties of Jaynes’ maximum entropy principle, which is considered a basis for statistical inference and serves as a justification for many regularization techniques that appear throughout machine learning and decision theory.

The concept of free energy has its origins in 19th century thermodynamics, but has recently found its way into the behavioral and neural sciences, where it has been promoted for its wide applicability and has even been suggested as a fundamental principle of understanding intelligent behavior and brain function. We argue that there are essentially two different notions of free energy in current models of intelligent agency, that can both be considered as applications of Bayesian inference to the problem of action selection: one that appears when trading off accuracy and uncertainty based on a general maximum entropy principle, and one that formulates action selection in terms of minimizing an error measure that quantifies deviations of beliefs and policies from given reference models. The first approach provides a normative rule for action selection in the face of model uncertainty or when information processing capabilities are limited. The second approach directly aims to formulate the action selection problem as an inference problem in the context of Bayesian brain theories, also known as Active Inference in the literature. We elucidate the main ideas and discuss critical technical and conceptual issues revolving around these two notions of free energy that both claim to apply at all levels of decision-making, from the high-level deliberation of reasoning down to the low-level information processing of perception.

In its most basic form, decision-making can be viewed as a computational process that progressively eliminates alternatives, thereby reducing uncertainty. Such processes are generally costly, meaning that the amount of uncertainty that can be reduced is limited by the amount of available computational resources. Here, we introduce the notion of elementary computation based on a fundamental principle for probability transfers that reduce uncertainty. Elementary computations can be considered as the inverse of Pigou-Dalton transfers applied to probability distributions, closely related to the concepts of majorization, T-transforms, and generalized entropies that induce a preorder on the space of probability distributions. As a consequence we can define resource cost functions that are order-preserving and therefore monotonic with respect to the uncertainty reduction. This leads to a comprehensive notion of decision-making processes with limited resources. Along the way, we prove several new results on majorization theory, as well as on entropy and divergence measures.

Expected utility models are often used as a normative baseline for human performance in motor tasks. However, this baseline ignores computational costs that are incurred when searching for the optimal strategy. In contrast, bounded rational decision-theory provides a normative baseline that takes computational effort into account, as it describes optimal behavior of an agent with limited information-processing capacity to change a prior motor strategy (before information-processing) into a posterior strategy (after information-processing). Here, we devised a pointing task where subjects had restricted reaction and movement time. In particular, we manipulated the permissible reaction time as a proxy for the amount of computation allowed for planning the movements. Moreover, we tested three different distributions over the target locations to induce different prior strategies that would influence the amount of required information-processing. We found that movement endpoint precision generally decreases with limited planning time and that non-uniform prior probabilities allow for more precise movements toward high-probability targets. Considering these constraints in a bounded rational decision model, we found that subjects were generally close to bounded optimal. We conclude that bounded rational decision theory may be a promising normative framework to analyze human sensorimotor performance.

Although there is a somewhat standard formalization of computability on countable sets given by Turing machines, the same cannot be said about uncountable sets. Among the approaches to define computability in these sets, order-theoretic structures have proven to be useful. Here, we discuss the mathematical structure needed to define computability using order-theoretic concepts. In particular, we introduce a more general framework and discuss its limitations compared to the previous one in domain theory. We expose four features in which the stronger requirements in the domain-theoretic structure allow to improve upon the more general framework: computable elements, computable functions, model dependence of computability and complexity theory. Crucially, we show computability of elements in uncountable spaces can be defined in this new setup, and argue why this is not the case for computable functions. Moreover, we show the stronger setup diminishes the dependence of computability on the chosen order-theoretic structure and that, although a suitable complexity theory can be defined in the stronger framework and the more general one posesses a notion of computable elements, there appears to be no proper notion of element complexity in the latter.