Paul D. Thorn

The article begins by describing two longstanding problems associated with direct inference. One problem concerns the role of uninformative frequency statements in inferring probabilities by direct inference. A second problem concerns the role of frequency statements with gerrymandered reference classes. I show that past approaches to the problem associated with uninformative frequency statements yield the wrong conclusions in some cases. I propose a modification of Kyburg’s approach to the problem that yields the right conclusions. Past theories of direct inference have postponed treatment of the problem associated with gerrymandered reference classes by appealing to an unexplicated notion of projectability . I address the lacuna in past theories by introducing criteria for being a relevant statistic . The prescription that only relevant statistics play a role in direct inference corresponds to the sort of projectability constraints envisioned by past theories.

In this article, I introduce the term “cognitivism” as a name for the thesis that degrees of belief are equivalent to full beliefs about truth-valued propositions. The thesis (of cognitivism) that degrees of belief are equivalent to full beliefs is equivocal, inasmuch as different sorts of equivalence may be postulated between degrees of belief and full beliefs. The simplest sort of equivalence (and the sort of equivalence that I discuss here) identifies having a given degree of belief with having a full belief with a specific content. This sort of view was proposed in [C. Howson and P. Urbach, Scientific reasoning: the Bayesian approach. Chicago: Open Court (1996)].In addition to embracing a form of cognitivism about degrees of belief, Howson and Urbach argued for a brand of probabilism. I call a view, such as Howson and Urbach’s, which combines probabilism with cognitivism about degrees of belief “cognitivist probabilism”. In order to address some problems with Howson and Urbach’s view, I propose a view that incorperates several of modifications of Howson and Urbach’s version of cognitivist probabilism. The view that I finally propose upholds cognitivism about degrees of belief, but deviates from the letter of probabilism, in allowing that a rational agent’s degrees of belief need not conform to the axioms of probability, in the case where the agent’s cognitive resources are limited.

Meta-induction, in its various forms, is an imitative prediction method, where the prediction methods and the predictions of other agents are imitated to the extent that those methods or agents have proven successful in the past. In past work, Schurz demonstrated the optimality of meta-induction as a method for predicting unknown events and quantities. However, much recent discussion, along with formal and empirical work, on the Wisdom of Crowds has extolled the virtue of diverse and independent judgment as essential to maintenance of 'wise crowds'. This suggests that meta-inductive prediction methods could undermine the wisdom of the crowd inasmuch these methods recommend that agents imitate the predictions of other agents. In this article, we evaluate meta-inductive methods with a focus on the impact on a group's performance that may result from including meta-inductivists among its members. In addition to considering cases of global accessibility (i.e., cases where the judgments of all members of the group are available to all of the group's members), we consider cases where agents only have access to the judgments of other agents within their own local neighborhoods.

The present article illustrates a conflict between the claim that rational belief sets are closed under deductive consequences, and a very inclusive claim about the factors that are sufficient to determine whether it is rational to believe respective propositions. Inasmuch as it is implausible to hold that the factors listed here are insufficient to determine whether it is rational to believe respective propositions, we have good reason to deny that rational belief sets are closed under deductive consequences.

It is well known that there are, at least, two sorts of cases where one should not prefer a direct inference based on a narrower reference class, in particular: cases where the narrower reference class is gerrymandered, and cases where one lacks an evidential basis for forming a precise-valued frequency judgment for the narrower reference class. I here propose (1) that the preceding exceptions exhaust the circumstances where one should not prefer direct inference based on a narrower reference class, and (2) that minimal frequency information for a narrower (non-gerrymandered) reference class is sufficient to yield the defeat of a direct inference for a broader reference class. By the application of a method for inferring relatively informative expected frequencies, I argue that the latter claim does not result in an overly incredulous approach to direct inference. The method introduced here permits one to infer a relatively informative expected frequency for a reference class R', given frequency information for a superset of R' and/or frequency information for a sample drawn from R'.

The applicability of Bayesian conditionalization in setting one’s posterior probability for a proposition, α, is limited to cases where the value of a corresponding prior probability, PPRI(α|∧E), is available, where ∧E represents one’s complete body of evidence. In order to extend probability updating to cases where the prior probabilities needed for Bayesian conditionalization are unavailable, I introduce an inference schema, defeasible conditionalization, which allows one to update one’s personal probability in a proposition by conditioning on a proposition that represents a proper subset of one’s complete body of evidence. While defeasible conditionalization has wider applicability than standard Bayesian conditionalization (since it may be used when the value of a relevant prior probability, PPRI(α|∧E), is unavailable), there are circumstances under which some instances of defeasible conditionalization are unreasonable. To address this difficulty, I outline the conditions under which instances of defeasible conditionalization are defeated. To conclude the article, I suggest that the prescriptions of direct inference and statistical induction can be encoded within the proposed system of probability updating, by the selection of intuitively reasonable prior probabilities

In previous work, we studied four well known systems of qualitative probabilistic inference, and presented data from computer simulations in an attempt to illustrate the performance of the systems. These simulations evaluated the four systems in terms of their tendency to license inference to accurate and informative conclusions, given incomplete information about a randomly selected probability distribution. In our earlier work, the procedure used in generating the unknown probability distribution (representing the true stochastic state of the world) tended to yield probability distributions with moderately high entropy levels. In the present article, we present data charting the performance of the four systems when reasoning in environments of various entropy levels. The results illustrate variations in the performance of the respective reasoning systems that derive from the entropy of the environment, and allow for a more inclusive assessment of the reliability and robustness of the four systems.

Formal and empirical work on the Wisdom of Crowds has extolled the virtue of diverse and independent judgment as essential to the maintenance of ‘wise crowds’. In other words, com-munication and imitation among members of a group may have the negative effect of decreasing the aggregate wisdom of the group. In contrast, it is demonstrable that certain meta-inductive methods provide optimal means for predicting unknown events. Such meta-inductive methods are essentially imitative, where the predictions of other agents are imitated to the extent that those agents have proven successful in the past. Despite the (self-serving) optimality of meta-inductive methods, their imitative nature may undermine the ‘wisdom of the crowd’, since these methods recommend that agents imitate the predictions of other agents. In this paper, I present a replication of selected results of Thorn and Schurz, illustrating the effect on a group’s performance that may result from having members of a group adopt meta-inductive methods. I then expand on the work of Thorn and Schurz by considering three simple measures by which meta-inductive prediction methods may improve their own performance, while simultaneously mitigating their negative impact on group performance. The effects of adopting these maneuvers are investigated using computer simulations.

In attempting to form rational personal probabilities by direct inference, it is usually assumed that one should prefer frequency information concerning more specific reference classes. While the preceding assumption is intuitively plausible, little energy has been expended in explaining why it should be accepted. In the present article, I address this omission by showing that, among the principled policies that may be used in setting one’s personal probabilities, the policy of making direct inferences with a preference for frequency information for more specific reference classes yields personal probabilities whose accuracy is optimal, according to all proper scoring rules, in situations where all of the relevant frequency information is point-valued. Assuming that frequency information for narrower reference classes is preferred, when the relevant frequency statements are point-valued, a dilemma arises when choosing whether to make a direct inference based upon relatively precise-valued frequency information for a broad reference class, R, or upon relatively imprecise-valued frequency information for a more specific reference class, R*. I address such cases, by showing that it is often possible to make a precise-valued frequency judgment regarding R* based on precise-valued frequency information for R, using standard principles of direct inference. Having made such a frequency judgment, the dilemma of choosing between and is removed, and one may proceed by using the precise-valued frequency estimate for the more specific reference class as a premise for direct inference.

Systems of logico-probabilistic (LP) reasoning characterize inference from conditional assertions interpreted as expressing high conditional probabilities. In the present article, we investigate four prominent LP systems (namely, systems O, P, Z, and QC) by means of computer simulations. The results reported here extend our previous work in this area, and evaluate the four systems in terms of the expected utility of the dispositions to act that derive from the conclusions that the systems license. In addition to conforming to the dominant paradigm for assessing the rationality of actions and decisions, our present evaluation complements our previous work, since our previous evaluation may have been too severe in its assessment of inferences to false and uninformative conclusions. In the end, our new results provide additional support for the conclusion that (of the four systems considered) inference by system Z offers the best balance of error avoidance and inferential power. Our new results also suggest that improved performance could be achieved by a modest strengthening of system Z

The article proceeds upon the assumption that the beliefs and degrees of belief of rational agents satisfy a number of constraints, including: consistency and deductive closure for belief sets, conformity to the axioms of probability for degrees of belief, and the Lockean Thesis concerning the relationship between belief and degree of belief. Assuming that the beliefs and degrees of belief of both individuals and collectives satisfy the preceding three constraints, I discuss what further constraints may be imposed on the aggregation of beliefs and degrees of belief. Some possibility and impossibility results are presented. The possibility results suggest that the three proposed rationality constraints are compatible with reasonable aggregation procedures for belief and degree of belief.

An agent’s belief in a proposition, E0, is justified by an infinite regress of deferred justification just in case the belief that E0 is justified, and the justification for believing E0 proceeds from an infinite sequence of propositions, E0, E1, E2, etc., where, for all n ≥ 0, En+1 serves as the justification for En. In a number of recent articles, Atkinson and Peijnenburg claim to give examples where a belief is justified by an infinite regress of deferred justification. I argue here that there is no reason to regard Atkinson and Peijnenburg’s examples as cases where a belief is so justified. My argument is supported by careful consideration of the grounds upon which relevant beliefs are held within Atkinson and Peijnenburg’s examples.

There are numerous formal systems that allow inference of new conditionals based on a conditional knowledge base. Many of these systems have been analysed theoretically and some have been tested against human reasoning in psychological studies, but experiments evaluating the performance of such systems are rare. In this article, we extend the experiments in [19] in order to evaluate the inferential properties of c-representations in comparison to the well-known Systems P and Z. Since it is known that System Z and c-representations mainly differ in the sorts of inheritance inferences they allow, we discuss subclass inheritance and present experimental data for this type of inference in particular

We describe a prediction method called "Attractivity Weighting" (AW). In the case of cue-based paired comparison tasks, AW's prediction is based on a weighted average of the cue values of the most successful cues. In many situations, AW's prediction is based on the cue value of the most successful cue, resulting in behavior similar to Take-the-Best (TTB). Unlike TTB, AW has a desirable characteristic called "access optimality": Its long-run success is guaranteed to be at least as great as the most successful cue. While access optimality is a desirable characteristic, concerns may be raised about the short-term performance of AW. To evaluate such concerns, we here present a study of AW's short-term performance. The results suggest that there is little reason to worry about the short-run performance of AW. Our study also shows that, in random sequences of paired comparison tasks, the behavior of AW and TTB is nearly indiscernible.

Systems of logico-probabilistic reasoning characterize inference from conditional assertions that express high conditional probabilities. In this paper we investigate four prominent LP systems, the systems _O, P_, _Z_, and _QC_. These systems differ in the number of inferences they licence _. LP systems that license more inferences enjoy the possible reward of deriving more true and informative conclusions, but with this possible reward comes the risk of drawing more false or uninformative conclusions. In the first part of the paper, we present the four systems and extend each of them by theorems that allow one to compute almost-tight lower-probability-bounds for the conclusion of an inference, given lower-probability-bounds for its premises. In the second part of the paper, we investigate by means of computer simulations which of the four systems provides the best balance of reward versus risk. Our results suggest that system _Z_ offers the best balance.

Individuals often revise their beliefs when confronted with contradicting evidence. Belief revision in the spatial domain can be regarded as variation of initially constructed spatial mental models. Construction and revision usually follow distinct cognitive principles. The present study examines whether principles of revisions which follow constructions under high task demands differ from principles applied after less demanding constructions. We manipulated the task demands for model constructions by means of the continuity with which a spatial model was constructed. We administered tasks with continuous, semi-continuous, and discontinuous conditions as between-subject factor (experiment 1) and as within-subject factor (experiment 2). Construction and revision followed distinct cognitive principles in the changeless conditions of experiment 1. With increased task demands due to switches between different continuity conditions (experiment 2), reasoners adapted the principles they used for model revisions to the principles which they had used during antecedent constructions.

According to the paradigm of adaptive rationality, successful inference and prediction methods tend to be local and frugal. As a complement to work within this paradigm, we investigate the problem of selecting an optimal combination of prediction methods from a given toolbox of such local methods, in the context of changing environments. These selection methods are called meta-inductive strategies, if they are based on the success-records of the toolbox-methods. No absolutely optimal MI strategy exists—a fact that we call the “revenge of ecological rationality”. Nevertheless one can show that a certain MI strategy exists, called “AW”, which is universally long-run optimal, with provably small short-run losses, in comparison to any set of prediction methods that it can use as input. We call this property universal access-optimality. Local and short-run improvements over AW are possible, but only at the cost of forfeiting universal access-optimality. The last part of the paper includes an empirical study of MI strategies in application to an 8-year-long data set from the Monash University Footy Tipping Competition

I here aim to show that a particular approach to the problem of induction, which I will call “induction by direct inference”, comfortably handles Goodman’s problem of induction. I begin the article by describing induction by direct inference. After introducing induction by direct inference, I briefly introduce the Goodman problem, and explain why it is, prima facie, an obstacle to the proposed approach. I then show how one may address the Goodman problem, assuming one adopts induction by direct inference as an approach to the problem of induction. In particular, I show that a relatively standard treatment of what some have called the “Reference Class problem” addresses the Goodman Problem. Indeed, plausible and relatively standard principles of direct inference yield the conclusion that the Goodman inference (involving the grue predicate) is defeated, so it is unnecessary to invoke considerations of ‘projectibility’ in order to address the Goodman problem. I conclude the article by discussing the generality of the proposed approach, in dealing with variants of Goodman’s example.

Agents typically revise their beliefs when confronted with evidence that contradicts those beliefs, selecting from a number of possible revisions sufficient to reestablish consistency. In cases where an individual’s beliefs concern spatial relations, belief revision has been fruitfully treated as a decision about which features of an initially constructed spatial mental model to modify. A normative claim about belief revision maintains that agents should prefer minimal belief revisions. Yet recent studies have rebutted the preceding claim, where minimality is understood to consist in modifying the position of the fewest objects, showing instead that reasoners prefer revisions that modify the position of an object x while retaining the position of an object y, when the agent’s new evidence is a relational statement of the form ‘xRy’. We here present cases where the preceding effect is reduced, and show an effect of minimality as measured by the number of initial premises preserved.

This article presents results from a simulation‐based study of inheritance inference, that is, inference from the typicality of a property among a “base” class to its typicality among a subclass of the class. The study aims to ascertain which kinds of inheritance inferences are reliable, with attention to the dependence of their reliability upon the type of environment in which inferences are made. For example, the study addresses whether inheritance inference is reliable in the case of “exceptional subclasses” (i.e., subclasses that are known to be atypical in some respect) and attends to variations in reliability that result from variations in the entropy level of the environment. A further goal of the study is to show that the reliability of inheritance inference depends crucially on which sorts of base classes are used in making inferences. One approach to inheritance inference treats the extension of any atomic predicate as a suitable base class. A second approach identifies suitable base classes with the cells of a partition (of a preselected size k) of the domain of objects that satisfies the condition of maximizing the similarity of objects that are assigned to the same class. In addition to permitting more inferences, our study shows that the second approach results in inheritance inferences that are far more reliable, particularly in the case of exceptional subclasses.