This volume concerns Rational Agents - humans, players in a game, software or institutions - which must decide the proper next action in an atmosphere of partial information and uncertainty. The book collects formal accounts of Uncertainty, Rationality and Agency, and also of their interaction. It will benefit researchers in artificial systems which must gather information, reason about it and then make a rational decision on which action to take.

A sense of agency (SoA) is the experience of subjective awareness regarding the control of one’s actions. Humans have a natural tendency to generate prediction models of the environment and adapt their models according to changes in the environment. The SoA is associated with the degree of the adaptation of the prediction models, e.g., insufficient adaptation causes low predictability and lowers the SoA over the environment. Thus, identifying the mechanisms behind the adaptation process of a prediction model related to the (...) SoA is essential for understanding the generative process of the SoA. In the first half of the current study, we constructed a mathematical model in which the SoA represents a likelihood value for a given observation (sensory feedback) in a prediction model of the environment and in which the prediction model is updated according to the likelihood value. From our mathematical model, we theoretically derived a testable hypothesis that the prediction model is updated according to a Bayesian rule or a stochastic gradient. In the second half of our study, we focused on the experimental examination of this hypothesis. In our experiment, human subjects were repeatedly asked to observe a moving square on a computer screen and press a button after a beep sound. The button press resulted in an abrupt jump of the moving square on the screen. Experiencing the various stochastic time intervals between the action execution button-press) and the consequent event square jumping caused gradual changes in the subjects' degree of their SoA. By comparing the above theoretical hypothesis with the experimental results, we concluded that the update (adaptation) rule of the prediction model based on the SoA is better described by a Bayesian update than by a stochastic gradient descent. (shrink)

In the context of a general framework for belief dynamics which interprets revision as doxastic constraint satisfaction, we discuss a proposal for revising quasi-probabilistic belief measures with finite sets of graded conditionals. The belief states are ranking measures with divisible values (generalizing Spohn’s epistemology), and the conditionals are interpreted as ranking constraints. The approach is inspired by the minimal information paradigm and based on the principle-guided canonical construction of a ranking model of the input conditionals. This is achieved by extending (...) techniques known from conditional default reasoning. We give an overview of how it handles different principles for conditional and parallel revision and compare it with similar accounts. (shrink)

In this paper we discuss the process of building a joint probability distribution from an input set of low-dimensional probability distributions. Since the solution of the problem for a consistent input set of probability distributions is known we concentrate on a setup where the input probability distributions are inconsistent. In this case the iterative proportional fitting procedure, which converges in the consistent case, tends to come to cycles. We propose a new algorithm that converges even in inconsistent case. The important (...) property of the algorithm is that it can be efficiently implemented exploiting decomposability of considered distributions. (shrink)

One well-known objection to the principle of maximum entropy is the so-called Judy Benjamin problem, first introduced by van Fraassen. The problem turns on the apparently puzzling fact that, on the basis of information relating an event’s conditional probability, the maximum entropy distribution will almost always assign to the event conditionalized on a probability strictly less than that assigned to it by the uniform distribution. In this article, I present an analysis of the Judy Benjamin problem that can help to (...) make sense of this seemingly odd feature of maximum entropy inference. My analysis is based on the claim that, in applying the principle of maximum entropy, Judy Benjamin is not acting out of a concern to maximize uncertainty in the face of new evidence, but is rather exercising a certain brand of epistemic charity towards her informant. This epistemic charity takes the form of an assumption on the part of Judy Benjamin that her informant’s evidential report leaves out no relevant information. Such a reconceptualization of the motives underlying Judy Benjamin’s appeal to the principle of maximum entropy can help to further our understanding of the true epistemological grounds of this principle and correct a common misapprehension regarding its relationship to the principle of insufficient reason. 1Introduction2The Principle of Maximum Entropy3An Apologia for Judy Benjamin4Conclusion: Entropy and Insufficient Reason. (shrink)

The principle of maximum entropy is a general method to assign values to probability distributions on the basis of partial information. This principle, introduced by Jaynes in 1957, forms an extension of the classical principle of insufficient reason. It has been further generalized, both in mathematical formulation and in intended scope, into the principle of maximum relative entropy or of minimum information. It has been claimed that these principles are singled out as unique methods of statistical inference that agree with (...) certain compelling consistency requirements. This paper reviews these consistency arguments and the surrounding controversy. It is shown that the uniqueness proofs are flawed, or rest on unreasonably strong assumptions. A more general class of inference rules, maximizing the so-called Re[acute ]nyi entropies, is exhibited which also fulfill the reasonable part of the consistency assumptions. (shrink)

A paper on how to adapt your probabilisitc beliefs when learning a conditional.

The inference process in a probabilistic and conditional environment under minimum relative entropy, permits the acquisition of basic knowledge, the consideration of - even uncertain - ad hoc knowledge, and the response to queries. Even if these procedures are well known in the relevant literature their realisation for large-scale applications needs a sophisticated tool, allowing the communication with the user as well as all relevant logical transformations and numerical calculations. SPIRIT is an Expert-System-Shell for these purposes. Even for hundreds of (...) – consistent – facts about the involved variables’ dependencies the shell automatically generates the corresponding epistemic state, thus permitting the derivation of conclusions from the acquired knowledge. These conclusions’ reliability or precision can be checked, inviting the user to enrich the knowledge by further facts, if desired. Any inconsistencies among provided facts are detected, and their elimination will be supported by the shell. Knowledge acquisition can come from provided facts by a knowledge engineer as well as from real world data; inductive learning supports the use of such data. An important capability of the shell is the calculation of impacts upon ideas or concepts from a given stimulus. This paper is a brief survey of theoretical concepts and the corresponding features of the system, which are accompanied by illustrative examples. (shrink)

Probabilistic belief contraction has been a much neglected topic in the field of probabilistic reasoning. This is due to the difficulty in establishing a reasonable reversal of the effect of Bayesian conditionalization on a probabilistic distribution. We show that indifferent contraction, a solution proposed by Ramer to this problem through a judicious use of the principle of maximum entropy, is a probabilistic version of a full meet contraction. We then propose variations of indifferent contraction, using both the Shannon entropy measure (...) as well as the Hartley entropy measure, with an aim to avoid excessive loss of beliefs that full meet contraction entails. (shrink)

Sometimes we receive evidence in a form that standard conditioning (or Jeffrey conditioning) cannot accommodate. The principle of maximum entropy (MAXENT) provides a unique solution for the posterior probability distribution based on the intuition that the information gain consistent with assumptions and evidence should be minimal. Opponents of objective methods to determine these probabilities prominently cite van Fraassen’s Judy Benjamin case to undermine the generality of maxent. This article shows that an intuitive approach to Judy Benjamin’s case supports maxent. This (...) is surprising because based on independence assumptions the anticipated result is that it would support the opponents. It also demonstrates that opponents improperly apply independence assumptions to the problem. (shrink)

This article connects recent work in formal epistemology to work in economics and computer science. Analysing the Dutch Book Arguments, Epistemic Utility Theory and Objective Bayesian Epistemology we discover that formal epistemologists employ the same argument structure as economists and computer scientists. Since similar approaches often have similar problems and have shared solutions, opportunities for cross-fertilisation abound.

I present two measures of information for both consistentand inconsistent sets of sentences in a finite language ofpropositional logic. The measures of information are based onmeasures of inconsistency developed in Knight (2002).Relative information measures are then provided corresponding to thetwo information measures.

The principle of insufficient reason assigns equal probabilities to each alternative of a random experiment whenever there is no reason to prefer one over the other. The maximum entropy principle generalizes PIR to the case where statistical information like expectations are given. It is known that both principles result in paradoxical probability updates for joint distributions of cause and effect. This is because constraints on the conditional P P\left result in changes of P P\left that assign higher probability to those (...) values of the cause that offer more options for the effect, suggesting “intentional behavior.” Earlier work therefore suggested sequentially maximizing entropy according to the causal order, but without further justification apart from plausibility on toy examples. We justify causal modifications of PIR and MaxEnt by separating constraints into restrictions for the cause and restrictions for the mechanism that generates the effect from the cause. We further sketch why causal PIR also entails “Information Geometric Causal Inference.” We briefly discuss problems of generalizing the causal version of MaxEnt to arbitrary causal DAGs. (shrink)

Inductive probabilistic reasoning is understood as the application of inference patterns that use statistical background information to assign (subjective) probabilities to single events. The simplest such inference pattern is direct inference: from “70% of As are Bs” and “a is an A” infer that a is a B with probability 0.7. Direct inference is generalized by Jeffrey’s rule and the principle of cross-entropy minimization. To adequately formalize inductive probabilistic reasoning is an interesting topic for artificial intelligence, as an autonomous system (...) acting in a complex environment may have to base its actions on a probabilistic model of its environment, and the probabilities needed to form this model can often be obtained by combining statistical background information with particular observations made, i.e., by inductive probabilistic reasoning. In this paper a formal framework for inductive probabilistic reasoning is developed: syntactically it consists of an extension of the language of first-order predicate logic that allows to express statements about both statistical and subjective probabilities. Semantics for this representation language are developed that give rise to two distinct entailment relations: a relation ⊨ that models strict, probabilistically valid, inferences, and a relation that models inductive probabilistic inferences. The inductive entailment relation is obtained by implementing cross-entropy minimization in a preferred model semantics. A main objective of our approach is to ensure that for both entailment relations complete proof systems exist. This is achieved by allowing probability distributions in our semantic models that use non-standard probability values. A number of results are presented that show that in several important aspects the resulting logic behaves just like a logic based on real-valued probabilities alone. (shrink)

The lore is that standard information theory provides an analysis of information quantity, but not of information content. I argue this lore is incorrect, and there is an adequate informational semantics latent in standard theory. The roots of this notion of content can be traced to the secret parallel development of an information theory equivalent to Shannon’s by Turing at Bletchley Park, and it has been suggested independently in recent work by Skyrms and Bullinaria and Levy. This paper explicitly articulates (...) the semantics latent in information theory and defends it as an adequate theory of information content, or natural meaning. I argue that this theory suggests a new perspective on the classic misrepresentation worry for correlation-based semantics. (shrink)

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. (shrink)

Though we have access to a wealth of information, the main issue is always how to process the available information. How to make sense of all we observe and know. Just like the English alphabet: we know there are 26 letters but unless we put these letters together in a meaningful way, they convey no information. There are infinitely many ways of putting these letters together. Only a small number of those make sense. Only some of those convey exactly what (...) we wish to convey though the message may be interpreted differently by different individuals. That same issue comes up with information: how can we process the information we have? How can we infer and reason under conditions of incomplete observed information? In his seminal book on the philosophy of information, Floridi (2011a) raises a number of open questions. I discuss here one of these questions. That question is how to process information. To do so, I take the more realistic view that information is always limited, incomplete and possibly noisy. I define types of information, relate it to Floridi’s definitions and discuss a basic formulation for processing information under a unified framework. I relate it to some of the basic concepts discussed in the book. (shrink)

The technique of minimizing information (infomin) has been commonly employed as a general method for both choosing and updating a subjective probability function. We argue that, in a wide class of cases, the use of infomin methods fails to cohere with our standard conception of rational degrees of belief. We introduce the notion of a deceptive updating method and argue that non-deceptiveness is a necessary condition for rational coherence. Infomin has been criticized on the grounds that there are no higher (...) order probabilities that ‘support’ it, but the appeal to higher order probabilities is a substantial assumption that some might reject. Our elementary arguments from deceptiveness do not rely on this assumption. While deceptiveness implies lack of higher order support, the converse does not, in general, hold, which indicates that deceptiveness is a more objectionable property. We offer a new proof of the claim that infomin updating of any strictly-positive prior with respect to conditional-probability constraints is deceptive. In the case of expected-value constraints, infomin updating of the uniform prior is deceptive for some random variables but not for others. We establish both a necessary condition and a sufficient condition (which extends the scope of the phenomenon beyond cases previously considered) for deceptiveness in this setting. Along the way, we clarify the relation which obtains between the strong notion of higher order support, in which the higher order probability is defined over the full space of first order probabilities, and the apparently weaker notion, in which it is defined over some smaller parameter space. We show that under certain natural assumptions, the two are equivalent. Finally, we offer an interpretation of Jaynes, according to which his own appeal to infomin methods avoids the incoherencies discussed in this paper. (shrink)

A physical and mathematical framework for the analysis of probabilities in quantum theory is proposed and developed. One purpose is to surmount the problem, crucial to any reconciliation between quantum theory and space-time physics, of requiring instantaneous “wave-packet collapse” across the entire universe. The physical starting point is the idea of an observer as an entity, localized in space-time, for whom any physical system can be described at any moment, by a set of (not necessarily pure) quantum states compatible with (...) his observations of the system at that moment. The mathematical starting point is the theory of local algebras from axiomatic relativistic quantum field theory. A function defining thea priori probability of mistaking one local state for another is analysed. This function is shown to possess a broad range of appropriate properties and to be uniquely defined by a selection of them. Through a general model for observations, it is argued that the probabilities defined here are as compatible with experiment as the probabilities of conventional interpretations of quantum mechanics but are more likely to be compatible, not only with modern developments in mathematical physics, but also with a complete and consistent theory of measurement. (shrink)

I discuss the design of the method of entropic inference as a general framework for reasoning under conditions of uncertainty. The main contribution of this discussion is to emphasize the pragmatic elements in the derivation. More specifically: (1) Probability theory is designed as the uniquely natural tool for representing states of incomplete information. (2) An epistemic notion of information is defined in terms of its relation to the Bayesian beliefs of ideally rational agents. (3) The method of updating from a (...) prior to a posterior probability distribution is designed through an eliminative induction process that singles out the logarithmic relative entropy as the unique tool for inference. The resulting framework includes as special cases both MaxEnt and Bayes’ rule. It therefore unifies entropic and Bayesian methods into a single general inference scheme. I find that similar pragmatic elements are an integral part of Putnam’s internal realism, of Floridi’s informational structural realism, and also of van Fraasen’s empiricist structuralism. I conclude with the conjecture that their valuable insights can be incorporated into a single coherent doctrine—an informational pragmatic realism. (shrink)

We discuss the issue of quantum-classical transition in a system of a single particle with and without external potential. This is done by elaborating the notion of self-trapped wave function recently developed by the author. For a free particle, we show that there is a subset of self-trapped wave functions which is particle-like. Namely, the spatially localized wave packet is moving uniformly with undistorted shape as if the whole wave packet is indeed a classical free particle. The length of the (...) spatial support of the wave packet is given by the Compton wavelength so that the wave packet is more localized for particle with larger mass. Whereas for a particle of mass m in a macroscopic external potential, we show that the time needed by the corresponding self-trapped wave function to depart from a classical trajectory is of the order ∼m 2 c/ℏ. We argue that it is the Compton wavelength that matters and not the de Broglie wavelength as in conventional semiclassical approach. (shrink)

Handbook of the History of Logic, vol. 10, eds. Dov Gabbay, Stephan Hartmann, and John Woods, forthcoming.