Subjective probability plays an increasingly important role in many fields concerned with human cognition and behavior. Yet there have been significant criticisms of the idea that probabilities could actually be represented in the mind. This paper presents and elaborates a view of subjective probability as a kind of sampling propensity associated with internally represented generative models. The resulting view answers to some of the most well known criticisms of subjective probability, and is also supported by empirical work in neuroscience and (...) behavioral psychology. The repercussions of the view for how we conceive of many ordinary instances of subjective probability, and how it relates to more traditional conceptions of subjective probability, are discussed in some detail. (shrink)

When does it make sense to act randomly? A persuasive argument from Bayesian decision theory legitimizes randomization essentially only in tie-breaking situations. Rational behaviour in humans, non-human animals, and artificial agents, however, often seems indeterminate, even random. Moreover, rationales for randomized acts have been offered in a number of disciplines, including game theory, experimental design, and machine learning. A common way of accommodating some of these observations is by appeal to a decision-maker’s bounded computational resources. Making this suggestion both precise (...) and compelling is surprisingly difficult. Toward this end, I propose two fundamental rationales for randomization, drawing upon diverse ideas and results from the wider theory of computation. The first unifies common intuitions in favour of randomization from the aforementioned disciplines. The second introduces a deep connection between randomization and memory: access to a randomizing device is provably helpful for an agent burdened with a finite memory. Aside from fit with ordinary intuitions about rational action, the two rationales also make sense of empirical observations in the biological world. Indeed, random behaviour emerges more or less where it should, according to the proposal. (shrink)

Cause and effect relationships play a central role in how we perceive and make sense of the world around us, how we act upon it, and ultimately, how we understand ourselves. Almost two decades ago, computer scientist Judea Pearl made a breakthrough in understanding causality by discovering and systematically studying the “Ladder of Causation” [Pearl and Mackenzie 2018], a framework that highlights the distinct roles of seeing, doing, and imagining. In honor of this landmark discovery, we name this the Pearl (...) Causal Hierarchy (PCH). In this chapter, we develop a novel and comprehensive treatment of the PCH through two complementary lenses, one logical-probabilistic and another inferential-graphical. Following Pearl’s own presentation of the hierarchy, we begin by showing how the PCH organically emerges from a well-specified collection of causal mechanisms (a structural causal model, or SCM). We then turn to the logical lens. Our first result, the Causal Hierarchy Theorem (CHT), demonstrates that the three layers of the hierarchy almost always separate in a measure theoretic sense. Roughly speaking, the CHT says that data at one layer virtually always underdetermines information at higher layers. Since in most practical settings the scientist does not have access to the precise form of the underlying causal mechanisms – only to data generated by them with respect to some of PCH’s layers – this motivates us to study inferences within the PCH through the graphical lens. Specifically, we explore a set of methods known as causal inference that enable inferences bridging PCH’s layers given a partial specification of the SCM. For instance, one may want to infer what would happen had an intervention been performed in the environment (second-layer statement) when only passive observations (first-layer data) are available. We introduce a family of graphical models that allows the scientist to represent such a partial specification of the SCM in a cognitively meaningful and parsimonious way. Finally, we investigate an inferential system known as docalculus, showing how it can be sufficient, and in many cases necessary, to allow inferences across PCH’s layers. We believe that connecting with the essential dimensions of human experience as delineated by the PCH is a critical step towards creating the next generation of AI systems that will be safe, robust, human-compatible, and aligned with the social good. (shrink)

Many tasks in statistical and causal inference can be construed as problems of entailment in a suitable formal language. We ask whether those problems are more difficult, from a computational perspective, for causal probabilistic languages than for pure probabilistic (or “associational”) languages. Despite several senses in which causal reasoning is indeed more complex—both expressively and inferentially—we show that causal entailment (or satisfiability) problems can be systematically and robustly reduced to purely probabilistic problems. Thus there is no jump in computational complexity. (...) Along the way we answer several open problems concerning the complexity of well-known probability logics, in particular demonstrating the ${\exists \mathbb {R}}$ -completeness of a polynomial probability calculus, as well as a seemingly much simpler system, the logic of comparative conditional probability. (shrink)

A probabilistic Chomsky–Schützenberger hierarchy of grammars is introduced and studied, with the aim of understanding the expressive power of generative models. We offer characterizations of the distributions definable at each level of the hierarchy, including probabilistic regular, context-free, (linear) indexed, context-sensitive, and unrestricted grammars, each corresponding to familiar probabilistic machine classes. Special attention is given to distributions on (unary notations for) positive integers. Unlike in the classical case where the "semi-linear" languages all collapse into the regular languages, using analytic tools (...) adapted from the classical setting we show there is no collapse in the probabilistic hierarchy: more distributions become definable at each level. We also address related issues such as closure under probabilistic conditioning. (shrink)

Recent ideas about epistemic modals and indicative conditionals in formal semantics have significant overlap with ideas in modal logic and dynamic epistemic logic. The purpose of this paper is to show how greater interaction between formal semantics and dynamic epistemic logic in this area can be of mutual benefit. In one direction, we show how concepts and tools from modal logic and dynamic epistemic logic can be used to give a simple, complete axiomatization of Yalcin's [16] semantic consequence relation for (...) a language with epistemic modals and indicative conditionals. In the other direction, the formal semantics for indicative conditionals due to Kolodny and MacFarlane [9] gives rise to a new dynamic operator that is very natural from the point of view of dynamic epistemic logic, allowing succinct expression of dependence (as in dependence logic) or supervenience statements. We prove decidability for the logic with epistemic modals and Kolodny and MacFarlane's indicative conditional via a full and faithful computable translation from their logic to the modal logic K45. (shrink)

People often engage in “offline simulation”, considering what would happen if they performed certain actions in the future, or had performed different actions in the past. Prior research shows that these simulations are biased towards actions a person considers to be good—i.e., likely to pay off. We ask whether, and why, this bias might be adaptive. Through computational experiments we compare five agents who differ only in the way they engage in offline simulation, across a variety of different environment types. (...) Broadly speaking, our experiments reveal that simulating actions one already regards as good does in fact confer an advantage in downstream decision making, although this general pattern interacts with features of the environment in important ways. We contrast this bias with alternatives such as simulating actions whose outcomes are instead uncertain. (shrink)

While much of semantic theorizing is based on intuitions about logical phenomena associated with linguistic constructions—phenomena such as consistency and entailment—it is rare to see axiomatic treatments of linguistic fragments. Given a fragment interpreted in some class of formally specified models, it is often possible to ask for a characterization of the reasoning patterns validated by the class of models. Axiomatizations provide such a characterization, often in a perspicuous and efficient manner. In this paper, we highlight some of the benefits (...) of providing axiomatizations for the purpose of semantic theorizing. We illustrate some of these benefits using three examples from the study of modality. (shrink)

The problem of inferring probability comparisons between events from an initial set of comparisons arises in several contexts, ranging from decision theory to artificial intelligence to formal semantics. In this paper, we treat the problem as follows: beginning with a binary relation ≥ on events that does not preclude a probabilistic interpretation, in the sense that ≥ has extensions that are probabilistically representable, we characterize the extension ≥+ of ≥ that is exactly the intersection of all probabilistically representable extensions of (...) ≥. This extension ≥+ gives us all the additional comparisons that we are entitled to infer from ≥, based on the assumption that there is some probability measure of which ≥ gives us partial qualitative information. We pay special attention to the problem of extending an order on states to an order on events. In addition to the probabilistic interpretation, this problem has a more general interpretation involving measurement of any additive quantity: e.g., given comparisons between the weights of individual objects, what comparisons between the weights of groups of objects can we infer? (shrink)

This LNCS book is part of the FOLLI book series and constitutes the proceedings of the 8th International Workshop on Logic, Rationality, and Interaction, LORI 2021, held in Xi`an, China, in October 2021. The 15 full papers presented together with 7 short papers in this book were carefully reviewed and selected from 40 submissions. The workshop covers a wide range on the following topics such as doxastic and epistemic logics, deontic logic, intuitionistic and subsstructural logics, voting theory, and causal inference.