The Cognitive Basis of Joint Probability...

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ACTA UNIVERSITATIS UPSALIENSIS UPPSALA 2019 Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Social Sciences 166 The Cognitive Basis of Joint Probability Judgments Processes, Ecology, and Adaption JOAKIM SUNDH ISSN 1652-9030 ISBN 978-91-513-0608-7 urn:nbn:se:uu:diva-380099

Transcript of The Cognitive Basis of Joint Probability...

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ACTAUNIVERSITATIS

UPSALIENSISUPPSALA

2019

Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Social Sciences 166

The Cognitive Basis of JointProbability Judgments

Processes, Ecology, and Adaption

JOAKIM SUNDH

ISSN 1652-9030ISBN 978-91-513-0608-7urn:nbn:se:uu:diva-380099

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Dissertation presented at Uppsala University to be publicly examined in Humanistiska teatern,Engelska parken, Thunbergsv. 3H, Uppsala, Monday, 20 May 2019 at 10:15 for the degree ofDoctor of Philosophy. The examination will be conducted in English. Faculty examiner: JörgRieskamp (University of Basel, Faculty of Psychology).

AbstractSundh, J. 2019. The Cognitive Basis of Joint Probability Judgments. Processes, Ecology,and Adaption. Digital Comprehensive Summaries of Uppsala Dissertations fromthe Faculty of Social Sciences 166. 60 pp. Uppsala: Acta Universitatis Upsaliensis.ISBN 978-91-513-0608-7.

When navigating an uncertain world, it is often necessary to judge the probability of aconjunction of events, that is, their joint probability. The subject of this thesis is how people inferjoint probabilities from probabilities of individual events. Study I explored such joint probabilityjudgment tasks in conditions with independent events and conditions with systematic risk thatcould be inferred through feedback. Results indicated that participants tended to approach thetasks using additive combinations of the individual probabilities, but switch to multiplication(or, to a lesser extent, exemplar memory) when events were independent and additive strategiestherefore were less accurate. Consequently, participants were initially more accurate in thetask with high systematic risk, despite that task being more complex from the perspective ofprobability theory. Study II simulated the performance of models of joint probability judgmentin tasks based both on computer generated data and real-world data-sets, to evaluate whichcognitive processes are accurate in which ecological contexts. Models used in Study I and othermodels inspired by current research were explored. The results confirmed that, by virtue of theirrobustness, additive models are reasonable general purpose algorithms, although when one isfamiliar with the task it is preferable to switch to other strategies more specifically adaptedto the task. After Study I found that people adapt strategy choice according to dependencebetween events and Study II confirmed that these adaptions are justified in terms of accuracy,Study III investigated whether adapting to stochastic dependence implied thinking according tostochastic principles. Results indicated that this was not the case, but that participants insteadworked according to the weak assumption that events were independent, regardless of theactual state of the world. In conclusion, this thesis demonstrates that people generally do notcombine individual probabilities into joint probability judgments in ways consistent with thebasic principles of probability theory or think of the task in such terms, but neither does thereappear to be much reason to do so. Rather, simpler heuristics can often approximate equally ormore accurate judgments.

Keywords: Judgment and decision-making, Joint probability judgment, Probability theory,Ecological rationality

Joakim Sundh, Department of Psychology, Box 1225, Uppsala University, SE-75142 Uppsala,Sweden.

© Joakim Sundh 2019

ISSN 1652-9030ISBN 978-91-513-0608-7urn:nbn:se:uu:diva-380099 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-380099)

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“I failed math twice, never fullygrasping probability theory. I mean,

first off, who cares if you pick ablack ball or a white ball out of the

bag? And second, if you’re bent over about the color, don’t leave it to

chance. Look in the damn bag andpick the color you want.”

- Janet Evanovich

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List of Papers

This thesis is based on the following papers, which are referred to in the text by their Roman numerals.

I Sundh, J., & Juslin, P. (2018). Compound risk judgment in

tasks with both idiosyncratic and systematic risk: The “Robust Beauty” of additive probability integration. Cognition, 171, 25–41

II Sundh, J. (2019). The ecological scopes of cognitive models for joint probability judgment. Manuscript.

III Sundh, J., Juslin, P., & Millroth, P. (2019). Appreciation for in-dependence: Does adaptation to stochastic dependence imply thinking according to stochastic principles?. Manuscript.

Reprints were made with permission from the respective publishers. For all studies included in this thesis, Joakim Sundh planned and designed the experiments, analyzed the data, and wrote the manuscripts, along with contributions to all aforementioned areas from supervisor and, where appli-cable, co-authors.

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Contents

Introduction ..................................................................................................... 9 

Probability Theory and the Human Mind ..................................................... 11 The Rational Animal ................................................................................ 11 Heuristics and Biases ............................................................................... 12 Correspondence and Linear Models ......................................................... 13 Ecological Rationality and the Adaptive Toolbox ................................... 15 The Bayesian Brain .................................................................................. 17 Summary .................................................................................................. 17 

Probability Theory and the External World .................................................. 19 Small samples ........................................................................................... 19 Dependence .............................................................................................. 20 Generative Models ................................................................................... 21 Summary .................................................................................................. 22 

Joint Probability Judgment ........................................................................... 23 

Cognitive Resources: Reasoning, Intuitive Judgment, and Memory ............ 25 

Aims .............................................................................................................. 28 

Study I ........................................................................................................... 29 Background .............................................................................................. 29 Experiment I ............................................................................................. 30 

Methods ............................................................................................... 30 Results ................................................................................................. 31 

Experiment II ............................................................................................ 32 Methods ............................................................................................... 32 Results ................................................................................................. 33 

Experiment III .......................................................................................... 33 Methods ............................................................................................... 33 Results ................................................................................................. 33 

Conclusions .............................................................................................. 34 

Study II ......................................................................................................... 35 Background .............................................................................................. 35 Methods .................................................................................................... 35 

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Results ...................................................................................................... 36 Conclusions .............................................................................................. 41 

Study III ........................................................................................................ 42 Background .............................................................................................. 42 Experiment I ............................................................................................. 42 

Methods ............................................................................................... 42 Results ................................................................................................. 43 

Experiment II ............................................................................................ 44 Methods ............................................................................................... 44 Results ................................................................................................. 45 

Conclusions .............................................................................................. 46 

General Discussion ....................................................................................... 47 General Conclusions ................................................................................ 48 Practical Implications ............................................................................... 50 Limitations ............................................................................................... 51 Future Directions ...................................................................................... 52 In Conclusion ........................................................................................... 53 

Acknowledgments......................................................................................... 54 

References ..................................................................................................... 56 

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Introduction

In order to successfully navigate life it is necessary to accommodate the un-certain nature of the world. If the past is a foreign country, then the future is rather like terra incognita; a region for which no reliable maps exist. Never-theless, induction from past experience in concert with our understanding of how the universe works allows us to make certain assumptions about future events, just as it allows us to make educated guesses at what lies on the other side of a mountain (the best guess usually being “another mountain”).

One of the most basic questions we can ask regarding the future is wheth-er a specific event will occur or not (e.g., if the bus will be on time, if an investment will pay off, if it will rain tomorrow, etc.). In many cases the question will concern a conjunction of events, either because the simultane-ous occurrence of certain events is in itself relevant (e.g., if all their invest-ments fail an investor might go bankrupt) or because the probability of an event can only be assessed through the combination of constituent events (e.g., a manufacturing process that is contingent on several discrete steps). These conjunctions, their associated probabilities, and the way in which we infer them from probabilities of constituent events are the subjects of this thesis.

As a tool for helping us answer such questions, humanity has developed probability theory, a systematic framework for understanding and quantify-ing uncertainty regarding the occurrence or non-occurrence of events. In probability theory, the probability of a conjunction of events is commonly known as a joint probability, and the joint probability P(A&B) of two events A and B is defined as

& | , (1)

where P(A|B) is the conditional probability of A given that B has occurred. If the events are independent, that is, if the outcome of event B does not affect the probability of event A (and vice versa), this reduces to

& . (2)

Being a branch of mathematics, probability theory is based on using formal logic to derive solutions from pre-defined axioms; Equation 1 and 2 follows

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from the Kolmogorov Axioms (Kolmogorov, 1933), which constitutes the basis of modern probability calculus. This means that, as with any system based on deductive logic, the conclusions of probability theory are only “true” insofar as the premises being true. Thus, if we can ensure that our estimates of P(A|B) and P(B) are representative of the external world, then we can also calculate a representative estimate of P(A&B). The question is how often this type of precise and reliable information is possible to attain.

The roots of probability theory lie in lotteries and games of chance, in which we can often ascertain within a (theoretically) infinite level of preci-sion the expectations of random variables. But the probability judgments we make in our daily lives are seldom so neatly specified. Our past experiences are limited and, even if they were not, the dynamic nature of the world usu-ally precludes deterministic cause-effect relationships. As a consequence, most probability estimates we deal with on a daily basis are likely to be noisy (i.e., subject to error) to some degree.

Conditional probabilities create additional complications, because each event must be considered in parallel with other events; to estimate the condi-tional probability P(A|B) one must have an impression not only of the ex-pected frequency of B, but the expected outcome of A given B. This extra layer of information might not always be available and, because it entails combining two different sources of information (i.e., the expected frequency of A with the expected frequency of B) it is arguably even more susceptible to noise than non-conditional probability estimates. In a strictly frequentist paradigm where probabilities are based on proportions of outcomes in a giv-en sample this is necessarily the case; logically, the sample of outcomes of A given B cannot be larger than the sample of B.

Given these complications, probability theory might not always supply us with the best tools for judging the joint probability of two or more events; not because probability theory is in itself flawed, but because its application is based on assumptions that cannot always be satisfied in the external world. In a world of perfect information, probability theory would be the obvious yardstick for making any kind of probability judgment, but that is not a world in which we are currently living. This prompts the question, if not probability theory then what? And, given that there are other more effective ways of approaching joint probability judgment in the external world, do these coincide with what people actually do?

Thus, the purpose of this thesis is to explore how people produce joint probability judgments from constituent probabilities under different forms of noise and dependence. By extension, I also aim to evaluate the relative effi-cacy of the processes that people use in these tasks and to which extent peo-ple think about the task in terms of probability theory.

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Probability Theory and the Human Mind

The Rational Animal According to Aristotle, it is the very capacity for rational thought that sepa-rates humans from other animals. Since Aristotle, this perspective has been the basis for many theories on not only how the mind ideally should work but also how the mind generally does work. After the first formalization of probability theory during the 17th and 18th centuries, Pierre-Simon Laplace famously claimed that probability theory is nothing but common sense re-duced to calculus. The quote is worth reproducing in its entirety, and is as follows:

"One sees, from this Essay, that the theory of probabilities is basically just common sense reduced to calculus; it makes one appreciate with exactness that which accurate minds feel with a sort of instinct, often without being able to account for it." – Théorie Analytique des Probabilités (1820)

The notion of probability theory as “common sense” is reproduced in several economic frameworks on human behavior, such as Expected Utility Theory. Originally developed by Daniel Bernoulli (1738) as a solution to the so called “St. Petersburg Paradox,” this theory postulates that humans act in ways that will maximize the statistical expectation of the subjective value of an outcome. For example, this implies that a lottery ticket might have greater expected value for a poor person than for a very rich person; although the (low) probability of winning is the same for both individuals, the winnings would make a greater difference for the poor person than for the rich person.

The modern rendition of Expected Utility Theory is defined by the Neu-mann-Morgenstern Axioms (Neumann & Morgenstern, 1953), consisting of completeness (i.e., for any set of alternatives, any two alternatives can be compared to each other), transitivity (i.e., if A is preferred to B and B is pre-ferred to C then A must be preferred to C), independence of irrelevant alter-natives (i.e., if A is preferred to B, then the inclusion of C will not change the preference order of A and B), and continuity (i.e., if A is preferred to B and B is preferred to C then there should exist a possible combination of A and C with equivalent preference as B).

Expected Utility Theory, and other frameworks based on adherence to ax-ioms based in logic and probability theory, are generally referred to as the

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“classic” interpretation of rationality. This perspective is central to neo-classical economic models, which postulates that people have rational pref-erences, act to maximize expected utility, and act on the basis of complete and relevant information. Although, as we shall see, this position has been challenged by research on judgment and decision making, it has recently been experiencing somewhat of a revival.

Heuristics and Biases During the latter half of the 20th century it became clear that Expected Utility Theory is a poor approximation of human behavior. Research demonstrated quite clearly that people tend to systematically violate the Neumann-Morgenstern Axioms (e.g., Kahneman & Tversky, 1979; McFadden, 1974; Tversky, 1969; 1972; Tversky & Kahneman, 1974; 1983), indicating that human beings are (at least according to the classic view of rationality) not a very rational animal at all.

The most influential criticism of Expected Utility Theory and related the-ories arguably stems from the Heuristics and Biases research program (Tversky & Kahneman, 1974; Kahneman & Frederick, 2005), in which it is argued that people tend to defer judgments and decisions to simplifying heu-ristics. As a consequence, this creates biases and cognitive fallacies such as intransitivity (Tversky, 1969) and base-rate neglect (Bar-Hillel, 1980; Tversky & Kahneman, 1974). In the current formulation of Heuristics and Biases, attribute substitution is thought to be one of the most foundational of such cognitive short-cuts, and the cause of many biases and logical fallacies (Kahneman & Frederick, 2002). The basic idea is that, when an attribute that is the subject of a judgment is computationally complex or otherwise inac-cessible, a more easily available attribute is substituted instead.

One of the most studied cognitive fallacies, and one that is inexorably connected to the study of joint probability judgment, is the conjunction fal-lacy. It follows from Equation 1 and 2 that, regardless if events are depend-ent or independent, the joint probability P(A&B) can never be higher than the probability of either constituent event, that is P(A) or P(B). Yet, people often behave in a manner not consistent with this rule (e.g., Costello & Watts, 2014; Tentori, Crupi, & Russo, 2013; Tversky & Kahneman, 1983). The most well-known example of this effect is the so called “Linda prob-lem” (Tversky & Kahneman, 1983). In this experiment, a woman called Linda is described in a manner consistent with the popular conception of a feminist. Participants are subsequently asked if they believe that it is more likely that Linda is a bank teller or that Linda is a bank teller and a feminist. A large number of participants tend to respond that “feminist and bank teller” is more likely than “bank teller,” despite the logical fallacy in that the conjunction cannot be more probable than the conjunct. Many factors have

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been shown to reduce the propensity of the conjunction fallacy to some de-gree, such as changes in wording (Hertwig & Gigerenzer, 1999), frequency formats (Gigerenzer, 1991; Hertwig & Gigerenzer, 1999), and higher cogni-tive ability of the respondents (Oechssler, Roider, & Schmitz, 2009), but so far no single intervention has been found to eliminate the effect entirely. Tversky and Kahneman (1983) originally suggested that people rely on a representativeness heuristic (a type of attribute substitution) when making such judgments; because Linda appears more representative of (i.e., similar to) the common notion of a feminist than the common notion of a bank teller, the option that includes Linda being a feminist appears more likely.

For the most part, the Heuristics and Biases framework is firmly rooted in the classic perspective of rationality in that heuristics are commonly viewed as “irrational” and the associated biases and fallacies tend to be emphasized. This is not to say that proponents of the Heuristics and Biases perspective necessarily claim that humans are fundamentally irrational. Rather, the pro-pensity to use heuristics (and make errors) has traditionally been linked to an adherence to intuition as opposed to reasoning, a dualism made explicit in so called dual-systems frameworks. This term comprises a number of different theories that all presume that human cognitive processes are based in one of two systems: System 1, which is “intuitive” and System 2, which is “analyt-ic” (e.g., Evans, 2008; Kahneman, 2003; Sloman, 1996). The exact meanings and definitions of the two systems varies somewhat, but in general System 1 is thought to be unconscious, implicit, automatic, and non-verbal, while Sys-tem 2 is thought to be conscious, explicit, voluntary, and symbolic. Although historically quite popular, this way of thinking has been increasingly criti-cized in recent years, the main points of the criticisms being that the systems are too loosely defined to draw meaningful inferences from (Moors & De Houwer, 2006) and that the characteristics that are usually cited as discrimi-native between the two systems are often orthogonal (Melnikoff & Bargh, 2018; Zbrodoff & Logan, 1986).

Correspondence and Linear Models While research on Heuristics and Biases painted a grim vision of human rationality in the context of decision making, research on multiple-cue judg-ment reached similar conclusions based on different metrics. In the Heuris-tics and Biases framework, rationality is generally defined as coherence with the axioms of logic and probability theory, and experiments often take the form of choices between probabilistic prospects. A multiple-cue judgment task, on the other hand, is based on inferring the value of a criterion variable based on the values of two or more cue variables. Accordingly, accuracy in a multiple-cue task is typically assessed by correspondence to an objective value, rather than by coherence with formal axioms.

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Multiple-cue judgment relates to probability theory mainly through the concept of probabilistic feedback. Many multiple-cue judgment tasks are based on feedback learning in the form of an initial training period, during which the participants are given feedback on the correct criterion value after each produced judgment. Probabilistic feedback entails inclusion of a proba-bilistic error, so that the feedback rarely represents the exact correct value. Research on such judgments have shown that people often use few cues, use those cues inconsistently, and that cues are generally interpreted and inte-grated linearly (e.g., Brehmer, 1994; Cooksey, 1996; Kruschke & Johansen, 1999; Gluck, Shohamy, & Myers, 2002; but see also Lagnado, Newell, Ka-han, & Shanks, 2006 for an alternative perspective). These results all contra-dict Expected Utility Theory, because maximizing expected utility would require taking all available information into consideration, consistently use this information to act according to maximum expected utility, and (because expected utility is generally represented by non-linear functions) process information non-linearly. Notably, using only a subset of available cues is an important component in the Adaptive Toolbox (see below).

There is a strong association between research on multiple-cue judgment and the works of Egon Brunswik, most particularly in connection with Brunswik’s Lens Model (Brunswik, 1952). The rationale of the Lens Model is that whenever a person observes the world, they do so through a “lens” made up by cues that are probabilistically associated with the actual state of the world (i.e., the criterion, in the context of multiple-cue judgment) and with each other. Because, as previously observed, people apparently find probabilistic and non-linear associations difficult to learn, the cues are gen-erally interpreted and integrated in a linear manner regardless of the actual cue-criterion relationships (Karelia & Hogarth, 2008).

Although, from the standpoint of the classic perspective on rationality, this would have been seen as a deficiency, some researchers have empha-sized the capacity of linear models to successfully approximate also distinct-ly non-linear tasks (Dawes, 1979; Dawes & Corrigan, 1974; Einhorn & Ho-garth, 1975; Karelia & Hogarth, 2008). Dawes (1979) made a distinction between “proper” and “improper” linear models, based on whether predictor variables are optimally weighted or not. He further concluded that proper linear models work well, because people are good at picking predictive vari-ables and coding them so that they are monotonically related to the criterion, while improper linear models tend to also work well, because linear models are robust over deviations from optimal weighting. Based on this, later re-search has emphasized adaptive shifts between linear additive models and exemplar based memory (e.g., Juslin, Karlsson, & Olsson, 2008).

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Ecological Rationality and the Adaptive Toolbox Decades before the Heuristics and Biases approach popularized Linda the feminist bank teller as the face of human irrationality, Herbert A. Simon introduced the concept of bounded rationality as an alternative to the classic view (Simon, 1955). According to Simon, human decision making is always bounded by, on the one hand, cognitive limitations of the individual and, on the other hand, the structure of the environment in which the decision is made. As a consequence, Simon suggested, humans often base decisions on heuristics rather than optimization, a concept that was later adapted by the Heuristics and Biases framework.

Bounded rationality laid the foundations for the concept of ecological ra-tionality, but while the former aims towards a more realistic appreciation of human capabilities, the idea of the latter is to redefine the concept of ration-ality itself. According to the proponents of ecological rationality, the relative rationality of a behavior should be assessed based on cognitive assets, the structure of the environment, and the goals of the organism rather than ac-cording to abstract rules (Gigerenzer & Goldstein, 1996; Gigerenzer & Todd, 2012). In short, ecological rationality defines rationality according to success in the world, with the consequence that it is not measurable in the same sense as under the classic perspective. Like multiple-cue judgment tasks, ecological rationality emphasizes correspondence, but correspondence to the world at large rather than correspondence to particular objective crite-ria. A consequence of this standpoint is that rationality is not necessarily “computable” in the traditional sense; mainly because the information re-quired to make an informed choice of the most rational behavior is usually not available in the complex environments of our daily lives, but also be-cause the subjective nature of utility precludes objective assessment.

Based on the concept of ecological rationality, the Adaptive Toolbox re-search program stands in direct contrast to Heuristics and Biases in that heu-ristics are viewed as adaptions to certain circumstances in the interest of fulfilling certain goals, rather than symptoms of shortcomings in human cognitive capabilities (Gigerenzer, Hertwig, & Pachur, 2011; Gigerenzer & Selten, 2002; Gigerenzer & Todd, 1999). Adherents to this perspective ar-gues, firstly, that under many circumstances “fast-and-frugal heuristics” can be equally or more accurate than methods that adhere to logic and probabil-ity theory, and, secondly, that many biases and fallacies observed in research on Heuristics and Biases can be mitigated or even made to disappear entirely depending on the format in which the task is presented. It is argued that much research on Heuristics and Biases has little to do with actual cognitive capability, because it is based on tasks that humans are very unlikely to en-counter and thus success on those tasks would be of little benefit. From an evolutionary standpoint, this follows logically; if human beings were bad at

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tasks that are relevant to our survival, then it is unlikely that our species would exist at all today.

The logic of accuracy-effort trade-offs is generally accepted in most fields; sooner or later one will reach a point in any task where it is no longer worth putting in more work. This does not in itself contradict Expected Utili-ty Theory. However, the Adaptive Toolbox posits the existence of less-is-more effects, where additional resources are actually a hindrance rather than an advantage (Gigerenzer & Brighton, 2009). The rationale lies in the bias-variance dilemma. In short, a bias is an error in the average predictions of a function, while variance denotes variability in the predictions dependent on current input. Gigerenzer and Brighton argues that, because judgments and choices are usually based on noisy information, a biased algorithm can actu-ally produce more accurate judgments than an unbiased algorithm in the long run, by virtue of reducing variance.

Fast-and-frugal heuristics often entail filtering information and prioritiz-ing the most valid cues. This principle is perhaps most clearly exemplified by the “take-the-best” heuristic, in which decisions are made by comparing alternatives only according to the single most relevant characteristic and then choosing to the “best” based on that sole metric (Gigerenzer & Goldstein, 1996). This heuristic has been shown to produce successful inferences in a variety of contexts, such as medicine (Gigerenzer & Goldstein, 1996) and political forecasting (Graefe & Armstrong, 2012).1 For more involved deci-sion making, “fast-and-frugal trees” are often suggested (Green & Mehr, 1997; Luan, Schooler, & Gigerenzer, 2011). Like in the take-the-best heuris-tic, informational cues are ranked according to importance, but rather than the choice being made only on the basis of the first cue, each cue either leads to a final decision or a move downwards in the tree.

However, the main idea of the adaptive toolbox concept is that people adaptively shift between different heuristics according to circumstance, of which these are only examples. The basic formulation of Adaptive Toolbox posits that people adaptively shift between strategies according to the char-acteristics of the environment (Todd & Gigerenzer, 2012), but does not spec-ify how exactly these shifts are realized. Several models have been proposed as solutions of this “strategy selection problem” (e.g., Erev & Barron, 2005; Lieder & Griffiths, 2017; Rieskamp & Otto, 2006).

1 According to an anonymous researcher, this is also a useful way of choosing your meal at a restaurant.

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The Bayesian Brain Although research during the last few decades generally confirm that people tend not to adhere to probability theory, a more recent hypothesis that has been gaining traction is that, quite to the contrary, human behavior corre-sponds to optimal Bayesian solutions; in other words, that people are blessed with a probabilistic mind (Chater & Oaksford, 2008). Evidence to support this claim has been demonstrated in areas of perception (Ma & Jazayeri, 2014), motor control (Braun, Mehring, & Wolpert, 2010), and language ac-quisition (Chater & Manning, 2006; van den Bos, Christiansen, & Misyak, 2012).

Of course, the probabilistic mind hypothesis is difficult to reconcile with the bulk of research on judgment and decision making. To accommodate the biases that sometimes occur in human behavior, it has been suggested that people base judgments and decisions on sampling processes. These process-es are broadly consistent with the principles of probability theory, but biases and fallacies will sometimes occur as a consequence of shortcomings in the samples or sampling processes.

According to one such theory, the Probability Theory plus Noise model (Costello & Watts, 2014; 2016), probability judgments are created by sam-pling outcomes from memory (or, in the case of events not yet experienced, from prospective memory processes) and counting the proportions of posi-tive relative to negative outcomes. Biases and fallacies are thought to be a symptom of noise in the recall processes.

While the Probability Theory plus Noise postulates that people are natural frequentists, a growing body of work espouses that people are rather natural Bayesians (e.g., Chater & Oaksford, 2008; Chater, Oaksford, Hahn, & Heit, 2010; Sandborn & Chater, 2016). The basic idea is that, rather than sampling past events from memory, people sample from a posterior distribution, in-formed by a combination of prior beliefs and current data. Through mecha-nisms such as Markov Chain Monte Carlo algorithms (Dasgupta, Schulz, & Gershman, 2017) or importance sampling through likelihood-weighting (Shi, Griffiths, Feldman, & Sandborn, 2010), accurate inferences can be made even from very few samples. However, the exact nature of the sampling mechanisms, and how they relate to our psychological reality, still requires more research.

Summary We can observe three distinct themes in research on the role of probability theory in regards to the human mind. First, as expressed by Laplace and embraced by neo-classical models of economic behavior, the perspective that probability theory describes both how people should act and also how

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people (generally) do act. Second, following the research of the Heuristics and Biases research program, the perspective that while probability theory describes the optimal and “rational” way to behave, people do not in fact tend to behave in this way. Third, as suggested by adherents to ecological rationality, the perspective that although people do not tend to adhere to the principles of probability theory, there is usually no compelling reason why they should.

In this context, the more recent research on Bayesian models of human behavior can, arguably, be viewed as a resurgence of the Laplacian perspec-tive that people do act according to probability theory. However, this discus-sion necessitates a clearer definition of what “acting according to probability theory” actually means. According to the classic perspective of rationality, this involves behaving in a manner consistent with the basic principles of probability theory, as exemplified by the Neumann-Morgenstern Axioms. Bayesian models, on the other hand, often implies that although a behavior might not be consistent with such principles on a visible level, the actual processes are consistent with Bayesian rationality conditional on the availa-ble information and computational resources. Thus, although a person might not be able to verbalize their behavior in terms of probability theory, their behavior can still approximate the same solutions (just as Laplace himself observed). We must be clear then, whether we discuss if people’s behavior coheres with the logical axioms of probability theory (which it apparently does not) or if people’s behavior can be justified in terms of probability the-ory (which it apparently can). From this perspective, it might very well be that heuristics are successful precisely because they approximate an optimal Bayesian solution when information is unreliable or incomplete.

In the context of joint probability judgments, most researchers today would agree that people do not generally approach such tasks using probabil-ity theory as expressed by Equation 1 or 2, though opinions would differ regarding why this is and what people do instead. As I demonstrate in the following section, noisy probability estimates and unspecified dependence between events often precludes this manner of straightforward and explicit application of probability theory to joint probability judgments. Instead, alternative approaches such as heuristics or additive processes might be more effective in such cases and might even correspond closely to Bayesian rationality. Thus, apparent incoherence with the principles of probability theory is not necessarily a symptom of the limited capabilities of the human mind. Rather, I hypothesize that it is an adaption to the situations in which the behavior is generally executed.

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Probability Theory and the External World

Small samples To recapitulate, if a person has access to perfect estimates of the constituent probabilities P(A) and P(B), and if the events are guaranteed to be independ-ent, then the joint probability P(A&B) can easily be calculated using Equa-tion 2. However, if P(A) and P(B) are noisy it is no longer necessarily the case that Equation 2 represents the optimal judgment, even if A and B can be proven to be independent. The reasons for this might not be immediately apparent, but can easily be illustrated by an example.

Imagine two coins, A and B. Our goal is to estimate the joint probability of both coins coming up heads, that is, P(A = H & B = H). For whatever reason, we do not know the probability of either coin coming up heads (per-haps we do not trust the owner of the coins). The only information available to us is from having seen each coin tossed once, that is, two outcome sam-ples of n = 1 that will each be either 1 (if heads) or 0 (if tails)

Now let us imagine that our distrust is unjustified and that both coins are fair, that is, P(A = H) = P(B = H) = .5 and, accordingly, P(A = H & B = H) = .25. Because the possible combinations we can observe are heads-heads, heads-tails, tails-heads, and tails-tails (i.e., four possible combinations), and because these combinations are equally likely, multiplication of our n = 1 samples according to Equation 2 will result in an average error of

|1 .25| |0 .25| |0 .25| |0 .25| /4 .375. (3)

Interestingly, taking the average of both outcomes (a method emphatically

not in coherence with the axioms of probability theory) will also result in an average error of .375. By contrast, a weighted summation in which the esti-mates are added together and weighted by a factor of .25 will result in an average error of

|. 5 .25| |. 25 .25| |. 25 .25| |0 .25| /4 .125. (4)

This example is admittedly trivial, but it does illustrate two important

points: First, effective explicit application of probability theory requires a minimum of adequate information (in this case, a representative outcome

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sample of sufficient size) and if this information is not available then other methods can be equally or more accurate. Second, the fact that most of us instinctively recognize why the example is trivial (few people would trust a probability judgment based on one single outcome) suggests the presence of some manner of Laplacian “common sense.”

As outcome samples grow larger, and probability estimates of the constit-uent events grow more accurate, multiplication of the proportions will (trivi-ally) also be more accurate, and as sample sizes approach infinity the aver-age error approaches zero. Although, as demonstrated, the performance of additive algorithms can be enhanced by adapting weights calibrated to the task, this will never outperform multiplication when n → ∞. More generally, this indicates that alternative algorithms (additive or otherwise) will often outperform multiplication of probability estimates of independent events under certain levels of noise, but as precision increases multiplication will be more accurate.

Noisy estimates also put the conjunction fallacy in a somewhat different light. It is true that the probability of the conjunction can never be higher than the probability of either constituent event, but if the estimates of the constituent events do not represent the objective probabilities, then this ob-viously does not hold. To return to the example above, if the probabilities of the two coins are P(A = H) = P(B = H) = .5 and P(A = H & B = H) = .25, and if the observed outcomes are tails-heads, that is A = T and B = H, then the probability of both coins coming up heads is most certainly higher than the sampled estimate of B (i.e., zero). One can comfortably conclude that, if the probability estimates of the constituent events are unreliable, committing a conjunction fallacy might occasionally be in our best interest. Of course, this does not make fallacious conclusions such as the one observed in the Linda problem less wrong, but it does indicate that adhering to the axioms of prob-ability theory will not always carry an adaptive function.

Dependence The above examples presume that events are independent, but when events are dependent then application of Equation 1 and estimation of the condi-tional probability P(A|B) is required. If probability estimates are based on samples of previous outcomes, estimation of P(A|B) implies that we must have observed simultaneous outcomes of events A and B in order to estimate the proportions of A give that B has occurred. But if we have observed the simultaneous outcomes of A and B, then that will also constitute a sample of the conjunction A&B and thus approximate P(A&B). Therefore, if we have a reliable estimate of P(A|B) then we likely also have a reliable estimate of P(A&B), in which case estimating the joint probability reduces to sampling inference. The more interesting case, and the one this thesis is focused on, is

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when we have only observed A and B on each own and consequently lack a reliable estimate of both P(A&B) and P(A|B).

Given these constraints it appears that precise information regarding de-pendence between events in the external world is rare, though it can poten-tially be approximated. If, for example, there is a positive dependence be-tween A and B, that is P(A|B) > P(A), then application of Equation 2 will result in a negative bias. If we are conscious of this fact, we could potentially adjust for this by, for example, adjusting either P(A) or P(A) × P(B) positive-ly according to our expectation of the strength of the dependence. However, this would imply adding an additional estimation and also presupposes that we are able to explicitly quantify our estimate of the strength of the depend-ency. In comparison, a model that is calibrated to the task but does not rely on explicit assumptions derived from probability theory, for example an additive model, might not require additional information per se, only adap-tion to the context in a more general sense.

Generative Models As previously observed, meaningful application of probability theory re-quires certain information (e.g., reliable estimates of probabilities and de-pendencies). Generally, probability theory also requires that the variables in question can be conceptualized as stochastic processes. This is possible to do in many cases, but might not always be the best alternative.

In the play Rosencrantz and Guilderstern are Dead by Tom Stoppard (1968), the two titular characters toss 92 coins and have them all end up heads. Well aware of the extremely low probability of such a thing occur-ring,2 Guilderstern is understandably perplexed by this and suggests a few different possible explanations, including “[a] spectacular vindication of the principle that each individual coin spun individually […] is as likely to come down heads as tails and therefore should cause no surprise each individual time it does,” (Stoppard, 1968, p. 6). The coin-tosses are indicative of a theme of the play; predetermination. They symbolize that, from the moment the play starts, Rosencrantz and Guilderstern’s fates are not their own, and therefore their world can no longer be described by stochastic processes. Yet, they are not aware of this fact, which is why the results of the coin-tosses upset Guilderstern so.

This example serves as a reminder that stochasticity is not just a function of the task itself, but also our perspective of the task. Tossing coins is a very popular example of a stochastic process, but as Diaconis, Holmes, and Montgomery observed, “[…] naturally tossed coins obey the laws of me-chanics (we neglect air resistance) and their flight is determined by their

2 P = 2.02 × 10-28 to be exact.

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initial conditions. […] With careful adjustment, the coin started heads up always end up heads up – one hundred percent of the time. We conclude that coin tossing is ‘physics’ not ‘random,’” (Diaconis, Holmes, & Montgomery, 2007, p. 211). This is, of course, technically true. However, because we are usually not able to account for the physics of the coin toss, we have no rea-son to assume that the coin will end up one side or the other. Thus, the pro-cess remains stochastic from our perspective and can easily and accurately be described by probability theory. On the other hand, if we became aware that the coins were tossed using a coin-tossing machine that could be cali-brated so that the coins ended up the same side each time, then the situation would obviously change.

We can conclude that the applicability of probability theory to a judgment task is not only dependent on reliable information regarding noise and de-pendence, but also on whether the task can be reasonably approximated by a stochastic model. In the context of coin tosses the answer is usually obvious, but in other situations things might not be so clear cut. Consider the Linda problem. As long as the extent of Linda’s existence is her fictional descrip-tion she is somewhat of a “Schrödinger’s bank teller” whose feminism re-mains in a state of flux unless observed, which means that we might as well consider her a stochastic variable. However, the situation might be different if Linda was a real person, in which case she either is a feminist or she is not, and the optimal approach would be contingent on our relation to Linda.

Summary According to the classic interpretation of rationality, a behavior is rational if it coheres with the principles of logic and probability theory. As I have demonstrated in this section, applying those principles is not always straight-forward and other methods can sometimes result in equally or more accurate judgments. This is not to say that probability theory as a whole should be discarded as a basis for understanding human judgment; I have obviously used probability theory myself in order to reach these very conclusions. For example, all other things being equal, the Bayesian approximation of the probability of an event based on a single occurrence in a sample n = 1 is p̂ = .667 and not p̂ = 1, which might very well be closer to what a person would actually act upon in such a situation. However, it is not likely that people have explicit knowledge of the relatively advanced theorems that form the basis of such inference; rather it might be the case that these conclusions can be approximated through heuristics and other simpler models.

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Joint Probability Judgment

Much research on joint probability judgment has, directly or indirectly, con-cerned the conjunction fallacy. In such cases, the task is often based on the ranking of probabilities (i.e., choosing which alternative is more likely) ra-ther than direct estimation (i.e., estimating how likely a conjunction of events is). This thesis is primarily focused on the latter; therefore I will use the term joint probability judgment as indicating direct estimation rather than ranking. Although one could argue that ranking probabilities is simply an extension of making a probability judgment, decision research often argues that decisions are based on processes that are qualitatively different from ranking quantitative judgments (e.g., attribute substitution and fast-and-frugal heuristics).

The research that has been done on explicit joint probability judgments is broadly consistent with research on the conjunction fallacy in that joint probabilities tend to be overestimated (e.g., Bar-Hillel, 1973; Brockner, Pa-ruchuri, Idson, & Higgins, 2002; Cohen, Chesnick, & Haran, 1971; Doyle, 1997; Slovic, 1969).3 Similar results have also been found in the related field of cumulative probability; that is, judging the probability of an event during an increasing period of time (e.g., McCloy, Byrne, & Johnson-Laird, 2010; Shaklee & Fischhoff, 1990). This effect appears to be fairly robust; Nilsson, Rieskamp, and Jenny (2013) observed that although accuracy of joint proba-bility judgments were improved by increased experience with constituent events, joint probabilities were still overestimated. Reducing cognitive load via memory aids did reduce overestimation, however.

Self-reports made by participants in one study have indicated significant heterogeneity in people’s strategy choices (Doyle, 1997), examples being additive and truncated additive strategies as well as anchoring and adjust-ment. Interestingly, this particular study did not observe use of either the representativeness heuristic or fast-and-frugal algorithms, possibly because of the response format (estimation rather than ranking) and possibly because of the structure of the task (sequential estimates over time). Relatedly, an exploration of the conjunctive fallacy in the context of joint probability judgment (Gavanski, & Roskos-Ewoldsen, 1991) concluded that, while peo-ple did commit conjunction fallacies, this did not appear to be on account of

3 Conversely, the probability of disjunctions tends to be underestimated, but see Doyle (1997) for an exception.

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representativeness, but rather because probabilities were combined in ways not consistent with the extension rule.

The propensity for additive processes is a noteworthy similarity to behav-ior observed in multiple-cue judgment. A multiple-cue judgment task is de-fined as the estimation of an unknown criterion on the basis of two or more cues, where criterion and cues can be either categorical or associated with a discrete or continuous scale. Thus, joint probability judgment represents a special case of multiple-cue judgment, in which the joint probability is the criterion and the constituent probability estimates are the cues. Indeed, re-search confirms that people intuitively tend to integrate cues additively in joint probability judgment tasks, either by “signed summation” in which the constituent probabilities are coded as likely, unlikely, or indifferent and sub-sequently summed (Yates & Carlson, 1986), or by a weighted average (Juslin, Lindskog, & Mayerhofer, 2015; Nilsson, Rieskamp, & Jenny, 2013; Nilsson, Winman, Juslin, & Hansson, 2009). In multiple-cue judgment peo-ple have also been shown to switch to exemplar-based processes when they attain enough familiarity with the task (Juslin, Olsson, & Olsson, 2003; Karlsson, Juslin, & Olsson, 2007), but it is less clear if this also applies to joint probability judgments. When events are independent and when given some amount of feedback, people sometimes do appear to make use of Equa-tion 2 (Juslin, Lindskog, & Mayerhofer, 2015). Similar results were ob-served in a recent study proposing a model of probability judgment inspired by dual-systems theories, which posited that people use a combination of additive and multiplicative processes to produce joint probability judgments (Khemlani, Lotstein, & Johnson‐Laird, 2015).

So far, most research on joint probability judgments has concerned inde-pendent events. Although some researchers have observed that additive models and other heuristics are often particularly effective when probability estimates are noisy (Juslin et al. 2009; Leuker, Pachur, Hertwig, & Pleskac, 2018; Nilsson et al. 2009), it has so far not been explored whether the same is true when events are dependent. Considering that positive dependence defined by P(A|B) > P(A) implies that joint probabilities are greater than the product P(A)P(B) and that past research indicates that joint probability judgments made on independent events are often overestimated, people’s usual approaches to joint probability judgments might very well be more accurate when events are dependent than when they are independent. The studies included in this thesis all touched on this hypothesis.

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Cognitive Resources: Reasoning, Intuitive Judgment, and Memory

A straightforward way of conceptualizing cognitive mechanisms is in ac-cordance with the manner in which the information is processed. Juslin, Nilsson, Winman, and Lindskog (2011) have suggested a framework for judgment and decision making based on three different processes: analytic judgment, intuitive judgment, and exemplar memory (see Figure 1).

Figure 1. Theoretical framework for human judgment suggesting three generic cog-nitive processes by which one or several cues (e.g., probabilities) can be transformed into a judgment. Adapted from Juslin, P., Nilsson, H., Winman, A., & Lindskog, M. (2011). Reducing cognitive biases in probabilistic reasoning by the use of logarithm formats. Cognition, 120(2), 248-267.

Analytic judgment is based on conscious processing of declarative facts. In the context of joint probability judgment, this could take the form of apply-ing Equation 1 or 2. This requires consideration of several factors: First, the way in which constituent probabilities are combined in a joint probability judgment according to probability theory, second, the interpretation of the constituent probabilities as numerical values, and third, the calculation of a product (which in the case of probabilities, being on a decimal format, likely would require several steps in itself). Because this kind of processing re-

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quires previously acquired knowledge, this kind of process is generally cul-turally defined.

As noted above, this process can result in accurate judgments, but only contingent on the availability of adequate information. It is not only neces-sary to have precisely calibrated numerical estimates of the constituent prob-abilities and knowledge of Equation 1 and 2, it is also necessary to under-stand the generative model. For example, if one presumes that events A and B are independent, that is, P(A|B) = P(A), even when that is not the case, then one will obviously reach an inaccurate conclusion.

Intuitive judgment, on the other hand, involves sequential and non-symbolic processes. For example, consider the evaluation of a prospective apartment, where different aspects of the apartment is considered in turn and the total estimate is adjusted thereafter; the apartment might have a good location (adjust estimate upwards) and be relatively cheap (adjust estimate upwards) but on the other hand be very small (adjust estimate downwards). This principle is easily applicable to joint probability judgments, for example when making the assessment of whether everybody will arrive to a meeting on time; Albertina is known for always being on time (adjust estimate up-wards) and Bertrand will be in the building early (adjust estimate upwards) but Cassandra will be coming straight from another meeting (adjust estimate downwards).

Because this process will naturally result in additive combination of in-formation (Juslin et al., 2008) while the normative process of a joint proba-bility judgment is typically multiplicative, this type of process will often result in biases and fallacies such as the conjunction fallacy. Nevertheless, additive models have often proven to be very robust predictors even when the predicted function is non-linear (e.g., Dawes, 1979; Karelia & Hogarth, 2008), so although some bias is unavoidable in this type of process, it can potentially reach relatively high precision with respect to average error. This can, to a certain degree, be further mitigated by the use of configural algo-rithms that give greater weight to the lower probabilities (Nilsson et al., 2009). Because the joint probability P(A&B) can never be higher than either P(A) or P(B), the lowest probability effectively acts as a maximal bound of the joint probability and is consequently of particular importance in joint probability judgments.

Exemplar memory consists of the comparison with the current situation to past experience, and making a judgment based on the most similar past situa-tion or situations. For example, one might compare the prospective apart-ment to other apartments one has lived in. If one remembers really enjoying an apartment with a balcony despite the kitchen being very small, and the current apartment also has a balcony and a small kitchen, then that will have a positive effect on the judgment. On the other hand, if one resented living in an apartment with a walk-in-closet and the current apartment has a walk-in-closet, then that will have a negative effect on the judgment. Note that in this

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case it is not the characteristics of the apartment in itself that will affect the evaluation but only the relative similarity to the past exemplars. In this case, application to joint probability judgments is slightly less straightforward in that it must be determined which characteristics determine similarity to past judgments. The most direct way is to presume that exemplars are compared to the current judgment via the relative probability estimates of the constitu-ent events; if one must judge the joint probability of two unlikely events, then past experience with two other unlikely events might very well come to mind (potentially constrained by the context in which the judgment is made). The criterion values of past exemplars can either consist of past outcomes or, in the case of many of the learning tasks administered in the studies included in this thesis, objective joint probabilities.

A distinct advantage of exemplar based processes in the context of joint probability judgment (or indeed probability judgments in general) is that the weighted mean of the outcomes of similar past exemplars will in most cases approximate the objective probability, given that the similarity function is representative of the task environment. Mathematically, if the similarity function is proportional to the differences in objective probability and past exemplars are representative of the true probability distribution, then as the number of exemplars approach infinity, error will approach zero. However, this is an idealization that might not necessarily be applicable to situations in the external world.

As previously stated, research indicates that people often tend to approach joint probability judgments using weighted additive processes. This is con-sistent with the notion that analytic judgment (i.e., explicit application of probability theory) and exemplar memory requires additional information that might only be available with extensive experience with the task envi-ronment. In this case, the “structural neutrality” of the initial additive process might constitute an advantage, particularly when dealing with unknown gen-erative models.

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Aims

The purpose of this thesis is to explore how people produce joint probability judgments from constituent probabilities under different forms of noise and dependence. By extension, I also aim to evaluate the relative efficacy of the processes that people use in these tasks, and to which extent people think about the task in terms of probability theory. Research indicates that people generally approach joint probability judgment tasks using additive combina-tion or other heuristic processes, but also suggests significant heterogeneity in strategy choice. The question is whether people adapt their strategies ac-cording to their environment, particularly in regards to dependence between events, and, if so, then how effective these adaptions are.

In Study I, we aimed to determine how people approach joint probability judgment in a task with perfectly calibrated constituent probabilities versus the same task with an unknown systematic risk component that could only be inferred through feedback training. Computational cognitive modeling was used to determine both how people intuitively approach the tasks and to what extent people adapted their judgments according to the task. In Study II I evaluated the ecological scope of the cognitive processes ob-served in Study I as well as other candidate processes informed by current research; that is, in which ecological environments those models perform more or less accurately. To this end, models representative of the selected processes were applied to judgment tasks involving both computer generated and real-world data, both of which varied in regards to sampling noise, de-pendency, and number of events. This way, it is possible to evaluate when and why people are more likely to use which cognitive strategies. In Study III our purpose was to determine whether the successful adaptions observed in Study I imply an understanding of the generative model with respect to dependence. Thus, a learning task on the same format as in Study I was used, with an added survey component designed to screen for knowledge of the generative model. Additionally, we wished to complement the experimental paradigm of Study I by using conditional dependence in-stead of an unknown systematic risk factor and by comparing judgment from description with judgment from experience.

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Study I

Background The aim of Study I was to determine how people approach joint probability judgments in a numerical format, both when constituent probabilities are precise and events are independent and when an unknown systematic risk is added (i.e., a generative model implying dependence). By using computa-tional cognitive modeling we explored how people initially tended to ap-proach joint probability judgment tasks, if they adapted their strategies ac-cording to the task, and if so then how. Focus lay on the previously de-scribed general cognitive processes analytic judgment (i.e., multiplication), intuitive sequential judgment (i.e., weighted addition), and exemplar memory. Although the effect of noisy constituent probability estimates have previously been studied in the context of joint probability judgments and has been shown to encourage additive processes (Nilsson et al., 2009), the effect of different types of dependence have not been similarly explored. It was hypothesized that, if people are well adapted to perform joint probabilities in the external world, and if independent events are generally rare in the exter-nal world, then people might initially be better at producing joint probability judgments when events are dependent than when they are not.

The experimental paradigm was based on a task used by Juslin, Lindskog, and Mayerhofter (2015), in which participants judged the joint probability that all four steps in a factory production cycle were successful; the results indicated that although participants did not completely lack qualitative un-derstanding of the principles of probability theory, they tended to approach the task using additive combination of constituent probabilities. In the cur-rent study, the basic concept was the same but the context was changed so that the task concerned whether all four stocks in a stock portfolio would decrease in value simultaneously (see Figure 2).

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Figure 2. The experimental paradigm used in Study I.

In all experiments included in the study the task was administered in the manner of a feedback learning task, so that participants first completed a training phase in which they were given feedback on the correct joint proba-bility after each judgment. After this, participants were presented with a test phase in which no feedback was given. Based on the current economic litera-ture, the concept of systematic and non-systematic risk was used (Bodie, Kane, & Marcus, 2013). Non-systematic risk concerns risk inherent to the company as a single entity, such as production rate, the quality of manage-ment, and so on. Conversely, systematic risk denotes risk associated with a market as a whole and will, consequently, affect all the assets in the market approximately equally.

The systematic risk was conceptualized in such a way that the objective joint probability could be accurately approximated by the average of the constituent probabilities (i.e., an additive heuristic). Computational modeling confirmed that this was so, and that when no systematic risk was present multiplication was more accurate than weighted additive processes.

Experiment I Methods A factorial 2 × 3 design was used. Both a conjunctive (the probability that all stocks decrease in value) and a disjunctive (the probability that at least one stock does not decrease in value) frame was used for the joint probability judgment task, crossed with three different levels of systematic risk; one condition without systematic risk (i.e., independent events), one condition where a low systematic risk component was added, and one condition where a high systematic risk component was added. The systematic risk component was not known to the participants at the start of the task, but its presence could be inferred by using the feedback given during the initial training tri-als.

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Root Mean Squared Difference (RMSD) was used to determine perfor-mance, and computational cognitive modeling was used to determine which cognitive processes participants had used when producing their judgments. Modeling included a multiplicative model, an additive model, and an exem-plar based model, corresponding to the basic cognitive processes of analytic judgment, intuitive judgment, and exemplar memory (see previous section). Additionally, a normative model was included for comparison purposes; although we did not expect that any participant would be able to perfectly approximate the equations of the systematic risk, it was relevant to see if any participant responses correlated more with the objective criterion than any of the three candidate models. If this were the case, then this would indicate that those models had a relatively low explanatory value. The Bayesian In-formation Criterion (BIC) was used as a metric of comparison between mod-els.

Results The results indicated that people were generally able to make accurate joint probability judgments (RMSD ≤ .101 in all conditions). Quick learning was observed in all conditions except for the high systematic risk condition, in which performance was accurate from the start (see Figure 3). This suggests that participants initially approached the task using an averaging algorithm (in line with previous research) and that in the condition with high systemat-ic risk this default algorithm was adequate enough so that there was no need to learn further. Conversely, in the conditions without systematic risk and with low systematic risk participants were compelled to adapt their strategies in order to perform well.

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Figure 3. Boxplots of the RMSD for each of 12 presentation intervals with 10 items in each interval during training for the no systematic risk and high systematic risk conditions with conjunctive frame in Experiment 1, denoting the median, the quar-tiles, the non-outlier max/min, the outliers, and the extreme values. RMSD = 0 is perfect accuracy.

Based on BIC differences, individual modeling results indicated that most participants indeed used an additive strategy when systematic risk was high (19/29), while in the condition without systematic risk they often learned to successfully apply multiplication (23/30). Support was more evenly split between the additive and multiplicative models in the condition with low systematic risk, where there were also more participants for which a single model could not be reliably distinguished (14/30). Over all conditions only a very few participants appeared to have used an exemplar based model (7/89). Results were broadly similar between the conjunctive and disjunctive conditions.

Experiment II Methods The same joint probability judgment task was used as in Experiment I, with slight modifications. The low systematic risk condition was replaced by a “fixed” systematic risk condition, in which the systematic risk was similar in magnitude to the high systematic risk condition but calculated in a subtly different way. A pre-training phase without feedback was also included, to discern how participants initially approached the task, and thus confirm the results in Experiment I suggesting that participants initially tended to ap-proach the task using additive strategies.

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Results As in Experiment I, participants learned to make accurate judgments (RMSD ≤ .141 in all conditions). Modeling of the pre-training phase data confirmed that many participants indeed appeared to initially approach the task using a weighted additive process (16/45), although many participants could not be reliably categorized to any single model (14/45). Subsequent test data showed that support for the multiplicative model in the condition without systematic risk was lower than in Experiment 1 (4/15, the rest of participants evenly split between additive, exemplar based, and uncategorized) but the additive model was still strongly supported in the conditions with high and fixed systematic risk (10/15 and 11/15 respectively).

Experiment III Methods The same task was used in Experiment III as in the previous experiments, with a factorial 2 × 2 × 2 design with the dependent variables frame (con-junctive or disjunctive), systematic risk (none or high), and feedback format (description or outcome). The framing and systematic risk conditions were identical to Experiment I, and while the descriptive feedback was also iden-tical to the previous experiments, the outcome feedback condition implied that the objective joint probabilities were represented by a sample of out-comes rather than a numerical value. This latter manipulation was intended to explore whether a description-experience gap could be observed (e.g., Hertwig & Erev, 2009).

As an alternative method of discriminating multiplicative versus additive processes, training in this experiment consisted only of items with three con-stituent events rather than four, while the test phase consisted of items with either three or four probabilities. The items were chosen so that an additive process would result in higher judgments and a multiplicative process would results in lower judgments when the additional risk was added.

Results Again, participants learned to make accurate judgments, although perfor-mance was significantly lower in the outcome condition (RMSD descriptive = .151; RMSD outcome = .254; Mann-Whitney, U = 458, Z = -3.76, p < .001). Apart from this difference, there was little indication of a description-experience gap. Interestingly, the modal response to the items with four ra-ther than three constituent probabilities was neither to increase nor decrease estimates, but rather not to change the estimates at all (33/80), though more evidence of additive integration (32/80) was observed than multiplicative

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integration (15/80); no significant differences were observed between condi-tions. It appears that people had troubles adjusting to the addition of a fourth probability after having been trained on judgments with only three.

Conclusions In Study I, a joint probability judgment task with exact information on the constituent probabilities and independent events were compared to a similar task with a systematic risk component that was initially unknown but could be inferred through feedback. Despite the fact that the former task appears (from the perspective of probability theory) to be much more straightfor-ward, performance after training was comparable in both tasks, as demon-strated in three different experiments. Not only that, but people apparently performed well on the task with high systematic risk from the start, while they had to learn through feedback to perform well on the task without sys-tematic risk. This is particularly noteworthy considering the fact that, be-cause of how the task was constructed, the task with high systematic risk required participants to commit conjunction fallacies in order to perform well, something participants appeared quite ready to do.

The results of Study I indicate that people tend to initially approach joint probability judgment using additive processes similar to an averaging heuris-tic but that they adapt their approaches depending on the task; it appears that when events are independent people are capable of applying Equation 2 and when events are dependent then people tend to stick with additive processes. However, it also appears that this is not universal and that there is notable heterogeneity in ultimate strategy choice between participants. Furthermore, the lack of adjustment for the added constituent probabilities in Experiment III indicates that, when specifically trained to perform well under particular conditions, these adaptions do not necessarily encourage generalization. It might be that additive processes are generally used as an initial approach not only because they are well suited to dependence, but also because they are reasonably robust over many different task environments. Study II further explored this idea.

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Study II

Background Study I demonstrated that with some training people can, and do, adapt their approaches to joint probability judgments according to different task envi-ronments. This suggests that, if we know the ecological scope of different cognitive strategies (i.e., which strategies are effective in which environ-ments), we can predict which strategies a majority of people are likely to use based on the task. Thus, the purpose of Study II was to systematically ex-plore the ecological scope of various models suggested as explanatory mod-els of joint probability judgments. By extension, this would indicate if the additive processes that people seem to initially use for joint probability judgments are more likely to be adaptions to particular conditions or effec-tive all-purpose algorithms for judgments in a wide variety of environments. In order to accomplish this, the models under study were applied to joint probability judgment tasks based on computer generated data that was sys-tematically varied in order to simulate ecological contexts with varying number of events, dependence, and sampling noise. In order to confirm if the conclusions reached when using computer generated data hold in a real-world context, models were also applied to data-sets comprised of real-world data.

Many cognitive processes have been proposed as explanatory models for how joint probability judgments are produced, and they all generally fit ex-perimental data well. However, accounting for experimental data is only part of the puzzle; if a model is suggested as an account for judgments outside of the laboratory, then it must be demonstrated that the model is indeed capable of producing accurate inferences in representative contexts. Otherwise, the generalizability of the model must be put into question.

Methods The same basic task as in Study I was used, that is, producing a joint proba-bility judgment based on probability estimates of the constituent events. However, rather than participants producing the judgments and the responses subsequently being fitted by cognitive models, responses were produced by the cognitive models in judgment tasks based on either computer generated

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or real-world input data and the accuracy of the judgments was assessed according to correspondence with the objective joint probabilities, measured by Mean Average Error (MAE). Five models were applied to data: Naïve multiplication (i.e., Equation 2), an adaption of the representativeness heuris-tic, a “fast-and-frugal” heuristic deferring to the lowest constituent probabil-ity estimate, a weighted additive model, and an exemplar based model.

The computer generated data was systematically manipulated in order to vary the number of constituent probabilities, the degree of sampling noise in the constituent probability estimates, and the dependence between events. Only positive dependence was considered, and was defined by conditional dependence, that is, P(A|B) > P(A). The models were applied to the computer generated data both with optimal parameter estimates and global, or “naïve,” parameter estimates, the latter based on a mean over the possible task envi-ronments. This latter distinction is important in order to distinguish situa-tions when an algorithm is “trained” from when it is not; in our daily lives we cannot always be expected to be familiar with the specifics of every par-ticular joint probability judgment task. Relatedly, the exemplar based model was applied to computer generated data using both a large battery of exem-plars and a number of smaller batteries of exemplars.

Subsequently, the models were also applied to judgment tasks based on real-world data, using optimal parameter estimates. The number of events and sample sizes were varied according to the same principle as when using computer generated data. Because dependence could not be controlled in the same manner, three different datasets were selected according to expected degree of dependence; results from premier league soccer matches for inde-pendent events, variations in stock prices from Standard and Poor’s 500 for medium levels of dependence, and precipitation data from the twenty biggest cities in Sweden for high levels of dependence. Although computer simula-tions of the performance of cognitive models are often performed as prelude to behavioral experiments, these are more rarely performed using real-world decision environments.

Results The results indicated that the models under consideration could be sorted into two general categories: First, processes that were effective only in spe-cific contexts and significantly less so in others. Unsurprisingly, naïve mul-tiplication belonged to this category, only being useful when events were independent and precision in estimates were high, and only barely outper-forming other processes even in such cases. The fast-and-frugal heuristic deferring to the lowest constituent probability also belonged to this category, by virtue of performing to a satisfactory standard only when dependence was very high. The representativeness heuristic also belonged to this category;

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although performance was acceptable under a larger number of different levels of dependence than the two former models, this heuristic never per-formed particularly well in cases when precision was low. The second cate-gory denotes models that performed well in most contexts, but which was dependent on calibration to the task environment to perform optimally. The weighted additive model belongs to this category, performing very well in all conditions when parameters were optimal and fairly adequate even when they were not. (See Figure 4 and 5 for illustrations of model performance with optimal and naïve parameters respectively).

The exemplar based model was also a part of the latter category, as it per-formed very well in any context as long as it had a large supply of exem-plars. As expected this model performed significantly worse when exemplars were very few but, interestingly, it turned out that the exemplars did not necessarily have to represent the same specific task environment as the one in which the judgment was made for the model to perform well; even when each exemplar was produced from a randomized task environment the num-ber of exemplars were generally far more important for performance than the representativeness of the exemplars (see Figure 6).

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Figure 4. Performance (MAE) of the naïve probability theory (NPT), representa-tiveness heuristic (REP), take-the-lowest (TTL), weighted addition (WAD), and exemplar memory (EXM) models applied to generated data with independence and high dependence; two and four events; and binomial samples of one through 20 outcomes. Optimal parameters were estimated for each condition.

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Figure 5. Performance (MAE) of the naïve probability theory (NPT), representa-tiveness heuristic (REP), take-the-lowest (TTL), weighted addition (WAD), and exemplar memory (EXM) models applied to generated data with independence and high dependence; two and four events; and binomial samples of one through 20 outcomes. Naïve parameters were estimated from a combination of possible envi-ronments.

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Figure 6. Performance (MAE) of the exemplar memory (EXM) model with exem-plars drawn from the same environment (EXMlocal) or with each exemplar drawn from a randomized environment (EXMgeneral) applied to generated data with inde-pendence and high dependence; two and four events; and one, two, three, five, ten, 20, 50, and 100 past exemplars. Optimal parameters were estimated for each condi-tion.

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Application of the models to real-world data confirmed that the conclusions of this study hold for probabilistic relationships in the external world as well as for computer generated data.

Conclusions The purpose of Study II was to evaluate the ecological scope of five models of joint probability judgment and, by extension, assess whether the additive models people apparently tend to gravitate towards are robust and effective, which was confirmed. The ecological scope of the models under study var-ied with respect to both size and locality but in general a weighted additive approach proved to be a useful general-purpose algorithm, even when the task is unfamiliar, validating the ecological rationality of such an approach. Exemplar based processes was equally robust over task environments and could potentially produce more accurate judgments, but only contingent on a large supply of exemplars (i.e., extensive experience with the task). Other models only performed adequately in relatively specific situations, but in those situations they could potentially perform very well indeed. Thus, one can conclude that, while an additive approach appears to be reasonably accu-rate in most cases, an agent who wishes to maximize precision would do well to change their approach according to circumstances.

The potential inherent in the exemplar based model, particularly the fact that the number of exemplars were apparently more important than their specificity, implies that this might indeed be a promising mechanisms for models of Bayesian rationality, as has been previously suggested (Shi et al., 2010). While additive processes might be less cognitively demanding than exemplar based models, it is possible that alternative sampling algorithms can mitigate the need for large numbers of exemplars (Vul, Goodman, Grif-fiths, & Tenenbaum, 2014). The results from this study also suggest that there might exist a process hierarchy of sorts, in which people initially ap-proach these joint probability judgment tasks using additive strategies and, either if this proves inadequate or if familiarity with the task invites even more accurate processes, switch to other strategies such as analytic applica-tion of probability theory or exemplar memory.

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Study III

Background While Study I confirmed that people successfully adapt to the presence or absence of dependence between events when making joint probability judg-ments, it did not explore to what extent this implies an understanding of those same dependencies. In other words, when people change their ap-proaches, either by adjusting their additive processes to better suit the task or by choosing another strategy altogether, is this done based on thinking of the environment in probabilistic terms such as dependence, or on content-independent and error-driven shifts between more generic cognitive re-sources?

The same learning task was used as in Study I, but rather than using the systematic risk concept from Study I the conditional dependence from Study II was used. The difference is subtle but important; the generative model in Study I included an unknown additional risk component while the generative model in Study II was defined by co-occurrence between events greater than what would have been expected by chance. The effect on the joint probabil-ity is similar but not identical; while in both cases positive dependence re-sults in a higher joint probability, this effect is larger in the systematic risk condition than with conditional dependence.

Experiment I Methods Three conditions were used: independence, low dependence, and high de-pendence. Participant performance was assessed according to Mean Average Error (MAE). Four models were applied to data by computational cognitive modeling: A normative model defined by the objective joint probability, a weighted additive model, a model based on the weighting of the lowest con-stituent probability, and an exemplar based model. The Bayesian Infor-mation Criterion (BIC) was used to quantify model fit. When events were independent, the normative model will coincide with the naïve multiplication used in Study II, but the naïve multiplication model was not further explored because results in Study I and II did not indicate that participants are likely

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to apply this strategy outside of when events are indeed independent and, as a consequence, multiplication is an accurate strategy.

In addition to the joint probability judgment task, a survey was adminis-tered subsequent to the task, with questions designed to assess participants’ understanding of the structure of the task with respect to dependence be-tween events. The bulk of the survey was made up by questions where the participants were instructed that a particular stock had a certain probability estimate, and then told either that all other stocks had decreased in value (question 1-8) or that all other stocks had not decreased in value (question 9-16) and asked whether this would affect their assessment of the first stock. If so, then the participant was asked what their new assessment would be. The idea was that, if participants presumed that events were independent they would not adjust the estimate of one stock given that the others change in value, but if they presumed that the stocks were dependent then they would. Lastly, the concept of correlation was explained to participants and they were asked whether they believed that the stocks had been correlated or not.

Results The results from Study I was broadly replicated, in that participants learned to make accurate judgments in all conditions (mean MAE = .088). Unlike in Study I, learning was observed in all conditions, presumably because the conditional dependency used in this study did not affect the joint probability to the same degree as the high systematic risk component in Study I. Thus, while participants in the high systematic risk condition in Study I had little reason to adapt, it was necessary in Study III even when events were highly dependent.

Also consistent with Study I, modeling indicated that the greatest number of participants were best described by the normative model (i.e., consistent with multiplication) in the condition with independence (8/30). In the condi-tions with dependence between events, model support was generally split between weighted addition (9/30) and weighted minimum probability (14/30).

Results from the survey indicated that people generally did not adjust their estimates of a single stock dependent on the outcomes of other stocks, regardless if they had experienced dependencies or not (see Figure 3). Nei-ther could any difference in adjustments depending on which model best explained participant data be observed. It appears that participants generally approached the task with no conception of dependencies between events and that this assumption did not change even after they were obliged to adjust their approaches.

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Figure 7. Boxplots of participant mean adjustments in each condition for survey questions 1-8 and 9-16. Reference lines indicate optimal mean adjustments for ques-tions 1-8 (independence = 0, low dependence = .136, high dependence = .313) and questions 9-16 (independence = 0, low dependence = -.039, high dependence = -.150).

Experiment II Methods The same joint probability task was used as in Experiment I, with the excep-tion that three constituent events were used rather than four and that the low dependency condition was replaced by a “single cause” condition in which two of the events were dependent on one single other event rather than each event being dependent on each other. Additionally, an outcome format con-dition was created in which participants were given outcome proportions in lieu of numeric probabilities, but feedback and responses were still adminis-tered in a symbolic format. (This differs from the outcome feedback condi-tion used in Experiment III in Study I in that the constituent probabilities were in an outcome format, rather than the feedback.)

The same computational modeling procedure was used as in Experiment I, with the addition of an outcome proportion sampling model. Because par-ticipants in the outcome format condition could potentially defer to the pro-portions of conjunctions in the outcome samples, the model was directly based on this number; the model was differentiated from the normative mod-el because it explicitly included sampling noise.

Like in Experiment I, a survey was created to assess participants’ under-standing of potential dependencies between events, based on questions in a similar format than in Experiment I. In this variation, the questions stipulated that one specific stock in the portfolio decreased in value a particular day

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and participants were asked whether this made all other stocks more, less, or equally likely to decrease in value. In the survey from Experiment I, partici-pants had generally elected not to adjust the estimates of the single stocks, but it is unclear if this was because of participants assumed that events were independent or because they lacked strong convictions either way. There-fore, participants were asked to supply their confidence that each response they gave was correct, on a scale from chance level and up to certainty. The assumption was that, if participants had explicit assumptions then confidence would be high and, if not, then confidence would be low. Using these confi-dence ratings in concert with participants responses, an index of understand-ing was calculated for each participant, defined on the scale -100 (high con-fidence and erroneous assumptions) to 100 (high confidence and correct assumptions).

Results There were no discernable difference between the “single cause” dependence and the dependence used in Experiment I, thus the two conditions were col-lapsed. Results in the symbolic condition mostly replicated Experiment I (mean MAE = .075), except that participants in in the independent condition surprisingly were not supported by the normative model to any great extent (i.e., they appeared not to multiply the constituent probabilities), instead support in the independent condition with symbolic format was relatively evenly split between all models. This was presumably because including only three events rather than four made a weighted additive process some-what better suited, and thus participants were not obliged to switch to multi-plication to the same extent as they were with four events. In the conditions with dependence and numerical format participants were generally best de-scribed by the weighted minimum probability (13/28). In all conditions with outcome format, people appeared to rely very heavily on counting conjunc-tions and this model accounted for a majority of participants (21/32).

Understanding of the probabilistic structure matched that in the previous experiment, in that participants generally did not adjust their estimates. In-dex of understanding did not differ between conditions (BF10 < 1.11 over both independent variables) and was not significantly different from zero (i.e., chance level; BF10 = .447) Whether participants did or did not make adjustments, confidence in their estimates was generally low (> 40 on a scale of 0-100 for all conditions), indicating that their choices were based on vague assumptions rather than explicit knowledge.

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Conclusions Study I indicated that people are able to approximate the conclusions of probability theory quite well, even in complex environments, and adapt their approaches to suit the task at hand. Study II confirmed that such approxima-tion can be successfully implemented by additive algorithms as well as ex-emplar memory and (in specific cases) also other heuristics. Building on these results, Study III further confirmed that people appear to adapt accord-ing to conditional dependence between events, but also indicated that these adaptions were not informed by an explicit understanding of dependence informed by probability theory. It appeared that people worked on the sim-plifying assumption that events were independent, not in terms of an explicit belief but rather defined by a lack of any strong assumptions either way. This held even though participants had to actively adjust their judgments and the effect was not reduced by observing the actual outcomes.

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General Discussion

What is the role of probability theory in the human mind? The mind has to operate in a fundamentally uncertain world, and it seems safe to assume that evolution has supplied us with tools to handle this uncertainty reasonably well, at the very least well enough to survive and procreate. More recently, humanity has developed probability theory, as a tool for systematically or-ganizing and quantifying uncertainty in the external world.

The aim of this thesis was to bring clarity to how people produce joint probability judgments from constituent probabilities under different forms of noise and dependence and, by extension, to evaluate the relative efficacy of the processes that people use in these tasks and to which extent they think about the task in terms of probability theory. Noise in probability estimates and dependence between events were given particular consideration by vir-tue of being two complications that often arise when dealing with probabili-ties in the external world but that, particularly in the case of dependence, seldom have been studied in this context. The three studies included in this thesis all took on these questions from different angles.

The main purpose of Study I was to determine how people approach joint probability judgments in different contexts and to what extent they adapt to different task environments. On the basis of previous research, and pre-experiment simulations, we expected participants to initially approach such judgments by combining probabilities additively, switching to multiplication if this proved to be unsuccessful. Results confirmed these predictions, and also suggested that the initial reliance on weighted addition might be due to a general adaption to an external world in which dependence between events is the rule rather than the exception; notably, participants appeared to initially be well adapted to situations where an unknown systematic risk component was present, but they were required to adjust their estimates according to feedback when events were independent.

Study II complemented Study I by exploring the ecological scope of the processes observed in Study I and other models informed by current re-search. This study further strengthened the case for weighted additive pro-cesses as a general adaption to the external world, by confirming that, rela-tive to other models, weighted additive processes are accurate under various different degrees of noise and dependency, even when not perfectly calibrat-ed to the task. However, other types of models can be very accurate in par-

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ticular situations, suggesting that successful adaption to the environment can involve adaptively switching between different cognitive processes.

While Study I and II indicated that people adaptively shift between cogni-tive processes when making joint probability judgments from constituent probabilities, it remained unclear to which extent these shifts engaged under-standing of the task in terms of probability theory. Study III indicated that people generally do not explicitly think of joint probability in terms of de-pendence between events but rather work under the implicit assuming that events are independent. This suggests that people are generally unsuited to explicitly apply probability theory to dependent events, while it creates a foundation for potentially applying probability theory when events are inde-pendent.

General Conclusions Results from the three studies included in this thesis indicate that people do not tend to intuitively approach joint probability judgments in ways con-sistent with explicit application of probability theory. Rather, people appear to make use of a range of different cognitive processes that approximate the solutions of probability theory, among which intuitive judgment based on weighted additive processes seems to be most generally applied. Results also indicate that this is generally an effective way of producing joint probability judgments from constituent probabilities, and particularly so when probabil-ity estimates of the constituent events are noisy and/or events are dependent. If one accepts the supposition that these characteristics are shared by many joint probability judgment tasks encountered in the external world, and if one defines the goal of joint probability judgments as correspondence to the objective joint probability as represented by frequencies in the external world, then one must conclude that weighted additive processes indeed con-stitute a suitable approach.

Results also suggest that people are very apt at adjusting their methods as circumstances dictate. Both Study I and Study III indicated that, if events are independent and probability estimates of those events are reliable, people are perfectly capable of simple explicit applications of probability theory (e.g., applying Equation 2). It should also be noted that, although exemplar memory did not appear to be a widely used process, all experiments in both studies included a small minority that was best described by such a model. In this, joint probability judgments apparently differ from multiple-cue judg-ment, in which exemplar memory is more often observed after a certain amount of experience with the task (Juslin et al., 2003; Karlsson et. al., 2007). The reason for this discrepancy is most likely that probability esti-mates, particularly when displayed in a symbolic format, is less discrimina-

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tory compared to other cues, a characteristic that has been demonstrated to reduce reliance on exemplar memory (Rouder & Ratcliff, 2004).

Study III also indicated that a configural weighting operation (e.g., a weighted minimum) can constitute a powerful method for producing joint probability judgments. This principle has been used previously in the context of joint probability judgments, as the main component of the Configural Weighted Average (CWA) model (Juslin et. al., 2009; Nilsson et al., 2009). However, because the CWA model takes both the maximum and minimum of constituent probabilities into consideration then, under the usually applied parameter constraints, this implies an inherent conjunction fallacy. A config-ural weighting that only considers the minimum of the constituent probabili-ties however, does not. On the contrary, as long as the weight is < 1 then it will produce a judgment coherent with the extension rule; because the joint probability P(A&B) can never be higher than either P(A) or P(B), the lowest probability effectively acts as a maximal bound of the joint probability, re-gardless of dependencies between events. Configural weighting is also an intuitively accessible heuristic; feminist bank-tellers aside, it is easy to real-ize that if you are hoping that all of your friends will show up to a party, then the friend who is least likely to show up is the one who is most important to convince.

Even though configural weighting appears to be an intuitively reasonable approach that will also prevent conjunction errors, this does not appear to be the manner in which people initially approach joint probability judgments. On the contrary, Study I indicated that people will initially defer to weighted additive processes that routinely produce conjunction errors. Why is this? The answer might lie in conservatism, the tendency for people to anchor their probability judgments on .5 and insufficiently adjust based on new evi-dence (Edwards, 1968; Tversky & Kahneman, 1992; Oechssler, Roider, & Schmitz, 2009). Although traditionally considered a bias, this has been shown to also have information-theoretic justification in that noisy conver-sion of objective information into subjective estimates can create this effect (Hilbert, 2012); by extension, one must suppose that noisy objective infor-mation will create the same effect regardless of errors in the processing of the information. One can see the logic in such conservatism if one consider small samples; as previously observed, if one tosses a coin once and it comes up heads, then it is patently absurd to presume that the chance of the coin coming up heads is P = 1. Even if one tosses the coin ten times and it comes up heads every single time, it is still sensible to assume a probability of P < 1. With noisy estimates, such as with small samples, conservatism some-times makes sense.

But even if this is the cause, should the effect not disappear if people are given noise-free probability estimates on a numerical format, as was the case in Study I? Not if participants do not explicitly apply probability theory, in which case the distinction between noisy and precise probability estimates

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will not necessarily be immediately apparent. Put quite bluntly, participants had no given reason to assume that the probability estimates we showed them were trustworthy just by virtue of being presented in a numerical for-mat.

Relatedly, Study III provided no evidence that people intuitively grasp the concept of dependence between events and the effects thereof. One should note that explicit understanding of probability theory is only beneficial if one has enough information available to profitably apply its theorems. As previ-ously observed, this is not always the case. Thus, understanding of probabil-ity theory need not be crucial to making joint probability judgments in the external world.

Practical Implications What does this mean for applications in the external world? First and fore-most, it demonstrates, mostly through Study II, that naïve application of probability theory when one lacks proper information on the world is rarely effective. Accordingly, one should only encourage this classic conception of “rationality” in situations where it is profitable, or even possible, to apply it. It is reasonable to invoke the concepts of ecological rationality and ask: What purpose does coherence with the axioms of probability serve? If one has a profession that concerns abstract and symbolic interpretations of prob-ability, then this understanding is obviously quite necessary and ignorance can have serious consequences indeed (e.g., base-rate fallacies in medical communication, see Hoffrage & Gigerenzer, 1998, and misuse of the p-value in scientific inference, see Badenes-Ribera, Frías-Navarro, Monterde-i-Bort, & Pascual-Soler, 2015). In such cases, biases and fallacies can and should be sought out and eliminated.

On the other hand, hunting for logical fallacies in contexts where formal logic is rarely relevant or necessary appears marginally useful. If I ponder whether the local store is likely to have all the ingredients I need for a pud-ding, the relevant question is not whether or not I make a conjunction fallacy but rather how often I go to the store in vain. This is not to belittle the im-portance of the Heuristics and Biases framework or related research, the existence of the biases must obviously be demonstrated before their im-portance can be assessed, but with our current knowledge, research might be better spent on delineating which processes are rational in which contexts rather than whether human beings as a whole adheres to some abstract defi-nition of “rationality” or not. Relatedly, the question whether or not people adhere to probability theory depends on how one chooses to define probabil-ity theory; in addition to the classic and Bayesian interpretations there exists recent research arguing for a quantum interpretation of probability as op-

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posed to one based on the Kolmogorov axioms (e.g., Busemeyer & Bruza, 2012; Busemeyer, Pothos, Franco, & Trueblood, 2011).

Limitations As is always the case with studies based on computational modeling, it is impossible to conclusively ascertain that there is not some other, untested, model that would fit data better. Additionally, the relative differences in fit between the models tested in the experiments included in this thesis tended to be low, which complicated model discrimination. In order to mitigate this challenge, I have attempted to define models to be, on the one hand, repre-sentative of general but qualitatively distinguishable processes (e.g., the lin-earity of an additive model versus the non-linearity of a multiplicative mod-el) and, on the other hand, as parsimonious as possible in order to avoid over-fitting. As such, I sometimes found it prudent to allow models to evolve following subsequent results; for example, because the multiplicative model defined in Study I tended to describe the unaltered product of the constituent events for the majority of participants, subsequent definitions of the model in Study II and III were consistent with this special case of the more flexible model included in Study I. In the same vein, the weighted minimum used in Study III is a more flexible variant of the unweighted “take-the-lowest” model explored in Study II. Thus, I believe that the computational modeling has, as far as possible, satisfactorily simulated processes representative of qualitatively different cognitive processes.

Regarding the ecological validity of the experimental paradigm, some re-searchers have argued that people are not inherently suited to process sym-bolically presented probabilities, but rather frequency formats (e.g., Gigerenzer, 1991; Gigerenzer, 1996; Hasher & Zacks, 1984; Sloman, Over, Slovak, & Stibel, 2003). In light of this, I have attempted to complement the symbolic format used in two of the studies in this thesis with an outcome format, both with respect to feedback (Study I) and constituent probability estimates (Study III). However, the bulk of the behavioral experiments were carried out using exclusively numerical format. Although I admit that nu-merical probabilities are not how people necessarily interact with probability in their daily life, I am confident that most people in a modern society are familiar enough with the concept that their internal representations propor-tionally map to the symbolic representations.

Additionally, it is not necessarily a given that what is commonly referred to as frequency formats is more representative of the external world. First, because people cannot be exposed to the actual physical events conceptual-ized in the experiment, frequency formats usually involve a symbolic repre-sentation of frequencies rather than the frequencies themselves. Second, many of our notions of probabilities are not necessarily based on frequencies

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at all. If I judge the probability that I will earn a Nobel Prize one day (pre-sumably low), then I have no sample of past frequencies to base my judg-ment on, rather I must base my judgment on completely different kinds of data (personal aptitude, career trajectory, etc.). The resultant estimate is pre-sumably quantitative, or at least quantifiable in the sense that it can be ranked in relation to other possibilities (higher than being killed in a meteor impact, lower than earning the title of professor), but will not necessarily be represented by frequencies. In the end, the question is how subjective proba-bility estimates are represented in the mind (if indeed they are represented at all), and until we know that we cannot be sure if numerical formats are more or less representative than, for example, frequency formats.

Future Directions This thesis demonstrates that joint probability judgments can, similar to mul-tiple-cue judgment, both descriptively and prescriptively be represented by weighted addition and other heuristic processes. However, the question still remains how these processes are implemented in the mind and the brain. Any model must, of course, always remain an abstraction; nevertheless the current models under study have determined only that joint probability judgments often correspond to weighted additive processes, but not precisely what is weighted and how. Delving deeper into these processes is presuma-bly a profitable continuation of this research.

Relatedly, Study II indicates that exemplar based processes can be very powerful tools for joint probability judgments, but little support for exemplar based processes could be discerned in the behavioral experiments. It is pos-sible that the symbolic format made participants disinclined to use exemplar memory, in which case it is relevant to discern which formats would make people more inclined to use these processes. This could be investigated fur-ther, possibly by including comparisons based on surface level characteris-tics as well as relative probability values.

By extension, all of the suggested modeling would profit from a deeper understanding of how probabilities are represented in the mind and the brain. For all the research on the subject during the last half century, the cognitive and neurological representations of probability is still up for debate, although there is currently increasing support for Bayesian sampling processes, which as previously noted can potentially be represented by the same type of mechanisms as exemplar based models. If this is indeed the way probabili-ties are typically conceptualized in the mind, then one must ask how this can be reconciled with the weighted additive processes observed in this study. It is clear that, when it comes to complex probability judgments, the work is not yet done.

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In Conclusion Close to two centuries ago, Laplace asserted that probability theory is basi-cally common sense reduced to calculus. Since then, a great amount of re-search has gone into demonstrating that people’s judgments and decisions often do not follow the basic principles of probability theory, indicating that common sense is not so common after all. Yet, humanity has apparently managed to muddle through anyway, suggesting that we must be doing something right. As this thesis demonstrates, at least in the context of joint probability judgment, weighted additive combination of probabilities and other simple processes that do not cohere with the principles of probability theory on a surface level can still approximate joint probabilities accurately. Although these approximations can potentially be justified in terms of prob-ability theory on a deeper level, it appears that people adapt to environments, not to probability theory.

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Acknowledgments

First, I would like to thank my supervisor Peter Juslin, for guiding me through the challenging process of crafting this thesis, whether this meant inspiring me when I’m stuck or restraining me when I get ahead of myself. It has been a pleasure!

I also extend my warmest gratitude to colleagues and friends at the De-partment of Psychology at Uppsala University. A special thank you goes out to Pär, for introducing me to Matlab and kindling a love for the software that still burns strong (despite some sputters along the way), and to Karin and Annika, for helping me make sense of study-plans, course credit, and other arcane matters. I would also like to thank everyone in the Cognition Lab, for always contributing stimulating discussions and ensuring that a presentation is never over as quickly as you expect it to be. I am particularly grateful to Ebba, Håkan, Marcus, Henry, and Anders, for your insightful comments along the way, and to Ronald, for your invaluable knowledge of Bayesian statistics and cognitive modeling. You have all improved my re-search immensely! I also want to thank Jerker Denrell, at the University of Warwick, for allowing me to spend a couple of months at the Warwick Business School and for helping me lay the foundations on which I built much of my latter two studies.

To my comrades in arms (past and present): Thank you Philip, for your cleverness and friendship, I could not have asked for a better person with whom to share office space! Thank you Mona, for all your kindness and support. Thank you Elina, for being a great travel companion on journeys to both London and Night Vale. Thank you Britt, for enabling my addiction to theatre and other forms of culture (may I never fully recover). Thank you Matilda, for angsting along with me these last heady days before our disser-tations. Thank you August, for always being ready to show things from a new and different angle. Thank you Kahl and Josh, fellow residents of the (now sadly defunct) cloning lab, it has been an honor! Thank you Laura, for always stopping for a chat. Thank you Tommie, Viktor, Kirsti, Emilia, Kristin, Moa, Sven, Eva, Sara, Gonçalo, Janna, Olof, Elisabeth, Johan, Kim, Anton, Mathias, Nils, Lisa, Linn, Pernilla, Johanna, Jörgen, Elin, Umay, and Linda, for your companionship down in the trenches; writing, teaching, and studying with you has made my time as a PhD student signifi-cantly brighter (p < .001).

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Of course, I also owe big thanks to all the people that have been a part of my life outside of work. A heartfelt thank you goes out to middags-psykologerna: Axel, Olof, Sandra, Anna, and Ulrika, for your companion-ship through the years of our undergraduate studies and beyond. I also want to thank the guys from Atlantis: Boman, Mats, Håkan, Dennis, Laffe and Peter, for all the years of rolling dice and gorging delicatobollar. And natu-rally, I raise a glass in thanks to Västgöta nations teatergrupp, for all of your passion and zest, on the stage and off.

Thank you Sebastian and Sonja, for always being ready with a glass of wine and a suitably scary movie. Thank you Emil, for staying grounded when other things wobble. Thank you Johan, for all the countless projects and schemes you have let me be a part of. Thank you Ida and Britta, for inspiring and encouraging my writing (the other writing). Thank you Åsa and Karin, for all the stories and all the countless cups of tea. Thank you Annika, för vanlig otvungen samvaro. Thank you Pierre, for a movie to discuss, a whiskey to drink, and a couch to crash on. Thank you Ernils, for talking, gaming, traveling, laughing, arguing, partying, grumbling, and rois-tering along with me through more years than I care to count. Thank you Jamie, for being my ally and accomplice on so many adventures, big and small.

Last but not least, I want to thank my parents Ingvar and Ulla, and my brother Mattias, for always supporting me in everything I do, even when it appears a bit unclear what that thing actually is. I also want to thank my grandparents Nippe and Olle, for always encouraging my scientific curiosity and contributing to putting me on the track that led to this thesis.

To everyone I have mentioned and everyone I might have forgotten: Thank you!

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