Natural environments and Bayesian reasoning: How inverse is the world?

Transcription

1 Natural environments and Bayesian reasoning: How inverse is the world? Stefan Krauss & Klaus Fiedler [Please Note: this is an unfinished draft from a larger project. Please excuse the lack of references and references to other chapters] In the previous chapters, we argued that the natural frequency approach allows us to cope efficiciently with many Bayesian problem situations. Natural frequencies were viewed as a psychological tool that can help people to understand Bayesian inferences. In this chapter the main focus is on the environments in which "Bayesian reasoning" is embedded. We argue that replacing probabilities by natural frequencies in certain environments means not only improving statistical problem solving but rather deleting artifacts that are caused by the introduction of probability theory, when in fact the environment does not call for probabilities. We show that formulating Bayesian tasks in such environments means artificially producing base-rate neglect and, furthermore, even artificia lly producing Bayesian reasoning. After specifying the term Bayesian reasoning, we introduce a distinction between environments where Bayesian reasoning is generally dispensable and environments where Bayesian reasoning is generally required. Having explained this distinction between two ecological ideal types, or extremes, we will finally consider possible degradations of environments. By analyzing real-life settings that correspond to these environments, we will conclude that Bayesian reasoning is required when the problem solver only can rely on provided information that is inverse to the question at hand (e.g., as is typically the case with probability tasks at school; see Chapter of Wassner, Martignon & Sedlmeier). In contrast, it is dispensable when the problem solver has access to directly observed data or to data stored in memory. Our claim is that in the latter cases the natural frequency concept is even more than a psychological concept for adaptive cognition; it is a repairing tool for psychological artifacts. Motivation The consideration of environments in which Bayesian reasoning tasks are usually formulated is motivated by an obvious discrepancy between the smoothness and easiness of many reasoning processes in every-day life and the following claims of the heuristics and biases program: Tversky and Kahneman argue, correctly, I think, that our minds are not built (for whatever reason) to work by the rules of probability (Gould, 1992) The genuineness, the robustness and the generality of the base-rate fallacy are matters of established fact (Bar-Hillel, 1980, p. 215) Mental illusions should be considered the rule rather than the exception (Thaler, 1991) We would like to pose the following questions: When did you last observe someone committing base-rate neglect? When did you yourself last commit base-rate neglect? In this chapter we want to explain, why many people s answer to both questions is probably never. We do this by analyzing the structure of information in real-life settings that correspond to the "Bayesian reasoning tasks" introduced in the preceding chapters by Wasssner, Martignon & Sedlmeier and by Kurzenhäuser & Hoffrage. What is Bayesian reasoning? Answering this question first requires a definition of what is meant by terms like Bayesian reasoning, Bayesian algorithm or Bayesian task. Unfortunately, in the psychological literature, the borders of the Bayesian world are not clearly marked and there is

2 no clear definition of the psychological concept of Bayesian reasoning yet. The pioneers who made base-rate neglect well-known (Kahneman, Slovic & Tversky, 1982) referred to this term as a written reasoning task with the following features: Three and only three pieces of information are given: p(b), p(a B), p(a B ) The required inference is: p(b A) =? Table 1: Definition of a Bayesian reasoning task, where A and B are arbitrary events The label Bayesian is justified by the name of the originally discussed algorithm, namely Bayes Formula p(b A) = p(a B) p(b) p(a B) p(b) + p(a B ) p(b ) (1) Consequently, we call a task according to Table 1 a Bayesian task. All algorithms that compute the required answer by integrating the three pieces of information specified above shall be called Bayesian algorithms. It does not matter, whether this given information is provided in terms of probabilities, percentages, frequencies (natural or others) or even as pictorial presentations. The inversion of conditional probabilities p(a B) p(b A) is a characteristic of Bayes formula. Assume for the following that A and B are features of individuals. In Bayesian tasks, according to Table 1, one has to consider the probability that an individual has A, given it has B in order to come up with the probability that an individual has B, given it has A. How is this inversion reflected when the same information is presented in terms of natural frequencies? The conditional probability p(a B) can be translated into a frequentistic expression of the kind n out of the m individuals with B, have A. The respective inverse frequentistic expression then would be n out of the m individuals with A, have B. Thus, in the natural frequency mode, the inversion refers to a semantic inversion of the main sentence and the relative sentence. Our definition of Bayesian reasoning makes no difference between format of information, computational demands or the explicit reasoning process and therefore is relatively broad compared to the restrictive Bayes formula, which is formulated exclusively for probabilities. Our definition is insofar better adjusted to psychological concerns as it also captures a variety of nonprobabilistic reasoning strategies. Because Bayesian tasks are among the most widely investigated in order to assess people s ability to deal with uncertainty, we are interested in the number and distributions of reallife settings that correspond to the information structure in Table 1. How Bayesian is the world? How widespread are environments where information is structured according to Table 1? In this section we investigate the degree to which written Bayesian reasoning tasks represent the information structure of corresponding real-life settings. We will see that this degree depends on the access that the problem environment affords to additional information. Let us first consider the ideal type of environments where all objects and their crucial features can be observed directly. Such environments we call observable environments. Observable environments

3 Imagine you are sitting in a restaurant and talking with your friends about vision problems and spectacles when the following question arises: What is the probability that a random person in this restaurant is male (M) given this person is wearing glasses (G), that is p(m G) =? In order to assess this probability, one can simply divide the number of males wearing glasses by the total number of people wearing glasses in the restaurant without struggling with the concept of conditional probability. Yet, in a Bayesian task, conditional probabilities have to be inverted. Let us thus consider the task of assessing in this environment the probability that is the inverse to p(m G), namely the probability that a person in this restaurant is wearing glasses (G), given he is male (M), that is p(g M). Again, this task will be no problem because in an observable environment the same algorithm can be applied for assessing the inverse conditional probability: In both cases one can count the number of elements in the total reference set and then assess the proportion of cases within that total set that corresponds to the conditional probability of the attribute in question. In the case of assessing p(g M), this means dividing the number of males wearing glasses by the number of all males in the restaurant. In observable environments we do not invert any probability using some abstract calculus, but we mentally structure or re-organize an immediately visible or imaginable sample of events using concrete mental operations. Thus, the notion of an observable environment is analogous to the notion of perceptual as opposed to inferential cognitive processes. As all objects (e.g., people in a restaurant) and their features (e.g., male, wearing glasses) are visible, there is no need to transform, or make inferences from, information that is inverse to the question at hand. Of course it is theoretically possible to formulate Bayesian tasks including presenting inverse information concerning such environments. However, these tasks would not be psychologically meaningful, just as it would not be meaningful to use a mirror for reading text that must then be deciphered from mirror image. Consider, for instance, the Bayesian task according to the real-life setting described above: Imagine you are sitting in a restaurant and talking about shortsightedness with your friends. You can see that the probability that someone in the restaurant is wearing glasses, P(G), is 25%. The probability that a person is male, given the person is wearing glasses P(M G), is 40% and the probability that a person is male, given the person is not wearing glasses, P(M G), is 60%. What is the probability that a random person in this restaurant is wearing glasses (G), given the person is male (M), that is p(g M) =? Given this probabilistic task one might struggle with the inversion of conditional probabilities 1. Yet, these problems are created artificially by omitting crucial information that is actually visible in the corresponding real-life setting. In order to answer the question, one would assess the number of males, say 22, and out of them the ones wearing glasses, say 4. So the frequentistic answer 4 out of 22 to the question p(g M) =? could easily be obtained. Our point is that in observable environments the frequentistic framing should not be viewed as derived from genuine probabilistic information patterns, it is rather the genuine description in such environments. Formulating information describing observable environments in terms of probabilities artificially introduces Bayesian reasoning and contains the danger of entailing artifacts like base-rate neglect. However, objects and their features cannot always be observed directly. In the following section, let us first examine environments where a set of enumerable, directly relevant cases is still available, but the information is not presently observable but has to be retrieved from past observations stored in the problem solver s memory. Retrievable environments 1 According to Bayes formula the correct answer is p(g M) = 40% 25% 40% 25%+ 60% 75% 18%

4 With respect to a judgment task under uncertainty, we call an environment retrievable if the relevant objects and their crucial features are not directly available but rather are stored in the problem solver s memory. Inferences in these environments are judgments based on experience. Gigerenzer and Hoffrage (1995) introduced their explanation of frequency formats with the following example: Imagine an old, experienced physician in an illiterate society. She has no books or statistical surveys and therefore mus t rely solely on her experience. Her people have been afflicted by a previously unknown and severe disease. Fortunately, the physician has discovered a symptom that signals the disease, although not with certainty. In her lifetime she has seen 1,000 people, 10 of whom had the disease. Of those 10, eight showed the symptom; of the 990 not afflicted, 95 did. Now a new patient appears. He has the symptom. What is the probability that he actually has the disease? The physician in the illiterate society does not need a pocket calculator to estimate the Bayesian posterior. All she needs is the number of cases that had both the symptom and the disease (here: 8) and the number of symptom cases (here: ). [...] The physician does not need to keep track of the base rate of the disease. Her modern counterpart, the medical student who struggles with single-event probabilities presented in medical textbooks, may on the other hand have to rely on a calculator and end up with little understanding of the result. Thus, in observable as well as in retrievable environments, all inferences on conditional probabilities can generally be assessed... - without taking into account the base-rate - without inverting information, not even at a semantic level - without the danger of struggling with base rate neglect Since in psychological literature Bayesian reasoning and medical diagnosis tasks are often intertwined, the question is: How Bayesian is the cognitive process concerning a medical diagnosis at all? Answering this question one has to sharply distinguish between a written Bayesian reasoning task and the corresponding real-life setting. In the next section ( technical environments") written Bayesian tasks will be handled. In this section, we concentrate on the corresponding real-life settings where physicians are allowed to use their expertise: Gigerenzer and Hoffrage s physician in the illiterate society has no need to retrieve information in the way it is presented in a Bayesian task (i.e., corresponding to Table 1). Gige renzer and Hoffrage (1995) correctly write: All she needs is the number of cases that had both the symptom and the disease (here: 8) and the number of symptom cases (here: ). Neither the reported total number of people (1000) nor the base-rate (990 ill people vs. 10 not afflicted people) is relevant for the diagnosis. Whether a physician indeed just recalls the two crucial subsets or whether she derives the answer by mentally subsetting all 1000 patients first, has not yet been distinguished by previous research. Let us therefore shed light on both possibilities: If she just recalled the two crucial subsets she would not have performed Bayesian reasoning. It would not even be a probabilistic inference, but rather just recalling evidence to estimate a proportion. Let us compare the cognitive demands of such a mental process with the Bayesian paradigm of medical questionnaire-diagnoses by Eddy (1982) in detail (Table 2). Physician in the illiterate society provided with a patient Question: Given a symptom, what is the probability that he actually has the disease? Base-rate is irrele vant. She only has to recall the people with symptom Physician in a paper and pencil environment provided with Eddy s task Question: Given a positive test result, what is the probability she actually has the disease? Base-rate has to be taken into account. Otherwise: base-rate neglect

5 Statistics on the symptom is irrelevant, she can just focus on all instances in her memory who suffered from the symptom No information has to be inverted Physician can rely on her experience and retrieve information that is relevant for the question from memory Statistics on the test (sensitivity and specificity) has to be taken into account Conditional probability has to be inverted Experience has to be turned off. Relying on memory might cause interference as the experienced and the provided input information may differ It can be seen, that apart from the intended diagnosis the two ways of reasoning are different. Let us consider now the second case that is based on recalling natural frequencies: Imagine, a physician derives the correct answer by subsetting the whole sample. This can happen by following two different directions of subsetting (Figure 1) breast no breast positive (M+) negative (M-) positive (M+) negative (M-) positive (M+) negative (M-) Figure 1 breast no breast breast no breast The right tree provides the required answer P(B+ M+) directly, whereas the left tree provides P(M+ B+), the sensitivity of the. Recalling information according to the left tree is not of relevance for a physician, because she does not have to examine patients in order to come up with the sensitivity of the applied test. In order to assess a statement on the state of illness of the patient based on a test result the tree on the right hand side would be more appropriate. The kind of subsetting information which is represented by the tree on the left hand side is relevant for developers of medical tests. In order to judge the sensitivity of a test, it has to be applied to healthy patients and in order to judge the specificity, it has to be applied to people suffering from the disease. The correspondence of the left tree to test developers interest and of the right tree to medical diagnoses is also reflected in the temporal order of considering information pieces: For the test developer the state of illness has to be specified before the test and then sensitivity and specificity can be assessed contingent on the outcome. In contrast, a medical doctor is first confronted with a test result before inferring from this the state of illness. From this viewpoint, "recalling natural frequencies" is a question-related reconstruction process of information.

6 Since the natural frequency concept makes no difference regarding the subsetting direction 2 (Krauss, Martignon, Hoffrage & Gigerenzer, 2001) it can provide frequentistic answers to all questions concerning breast and mammograms regardless of whether these answers are relevant for physicians or for test developers. In the absence of direct empirical evidence for the real recalling processes of physicians, we mention a third possibility: Recalling patients might also happen without any subsetting, but just by recalling the numbers of all conjunctions (B+&M+), (B+&M-), (B-&M+) and (B-&M-). These numbers that correspond to the numbers depicted in the lowest level of both trees also can be expressed by a 2x2 table (see Table 3): Positive Mammogramm (M+) Negative Mammogramm (M-) Breast (B+) 8 2 Not Breast (B-) However, the correct answer to any question formulated in terms of conditional probabilities can still be composed when numbers are recalled proportional to Table 3. Let us summarize our claims on observable and retrievable environments: All relevant conditional probabilities can be assessed by simply counting (in observable environments) or recalling (in retrievable environments) natural frequencies. No wonder that our answer to the question how often have we observed base-rate neglect in every-day life? was probably never : In naive people s real-life situations, Bayesian reasoning appears to be much less relevant than the flurry of research suggests. Note that this does not mean that the natural frequency concept is irrelevant: On the contrary, natural frequencies are nothing less than a genuine and natural format used by the human mind to observe, count, encode and recall (or reconstruct) information. Reasoning first becomes cumbersome when for instance, given a written problem sheet the inference is expressed in terms of probabilities: From that moment on, "Bayesian reasoning" is introduced and previously easy inferences entail distressing inversions. Well-known side effects of probability formats are the base-rate neglect and the confusion of different conditional probabilities. Let us try to devise a tutorial on how to construct the base-rate neglegt: First, one has to take out all relevant real-life features of provided information, for instance, the sample size. Probabilities do this, but be careful and avoid giving participants advantageous probabilities: Just provide participants with information that is inverse to the relevant question. All this is, unfortunately, still not sufficient for producing base-rate neglect. In addition, we have to provide an extreme base-rate in our written text! Then, indeed, it can be shown that the genuineness, the robustness and the generality of the base-rate fallacy are matters of established fact (Bar-Hillel, 1980, p. 215). Yet, the following question remains: So what? In the following section, we will now turn to situations where real Bayesian reasoning is required. What are the features of these environments and how can natural frequencies help in these environments? Technical environments In the preceding section, we substantiated the notion that in naive people s every-day life Bayesian reasoning is only rarely required. Yet, there are of course important situations, where the inversion of information is required. Human culture has developed techniques that were not available decades ago, as, for instance, AIDS-tests or DNA-fingerprinting etc. In contrast to the features of objects in observable environments, the features of such techniques cannot be 2 Not to be confused with the sampling direction (Fiedler et al., 2000). Whereas the sampling direction expresses how information is gathered, the subsetting direction illustrates the mental structuring of already sampled information.

7 observed directly. We will call environments that require people to understand and use technical devices like these technical environments. First we have to elucidate, why we put the breast example in the category retrievable environment, although a technical tool as a is involved in the decision process. If a diagnosis is a routine one it can be made exclusively by relying on data stored in memory. After having applied a lot of mammographies the technical features of the are irrelevant for inferring P(B+ M+) because it can be done by just recalling instances of the M+ category regardless how good the is. Of course, these data can only be recalled, if the physician also routinely gets feedback on her diagnosis. Cases where the physician sends away the patient after testing, without ever finding out about the true state of the patient, will be discussed shortly. There are basically two situations where the inversion of information and therefore Bayesian reasoning becomes relevant: First, if the decision maker has no previous experience with the technical tool; and second, if a diagnosis is counterintuitive and has to be made transparent. Let us consider the first situation: Assume a new medical test was developed and a physician wants to apply this test, although she has no expertise with it. If she relies on the instruction sheet, she can learn the sensitivity, that is P(Test+ Disease) and the specificity, that is P(Test no Disease), of the test. In order to come up with a diagnosis, she now indeed needs Bayesian reasoning. Yet, if this physician has applied the new test many times, one could speculate that she will represent the "quality" of the test no longer as P(Test+ Disease), but according to the tree on the right hand side of Figure 1 as the relation between P(Disease Test+) and P(Disease Test-). Expressing the sensitivity and the specificity of a test in terms of conditional probabilities is a mathematically clever way to communicate the quality of such a test independently of the baserate of the disease. Unfortunately, it means providing physicians information which is inverse to a medical diagnosis and, by all available evidence, inverting conditional probabilities is not a natural procedure as far as observable and retrievable environments are considered. Note that also the genuine information format of sensitivity and specificity is natural frequencies. The sensitivity of a, P(M+ B), as well as the specificity, P(M- B) originally stem from a counting algorithm: The developer of a has tested her tool with a group of women with breast and with a group of women without breast. From this procedure she obtained in both groups a hit-rate in terms of a proportion. Only normalisation of both values then leads to the sensitivity and the specificity reported in the instruction sheet. The cognitive problem concerning technical environments is that training and test set can often differ. If one wants to estimate the general probability of shortsightedness, given a person is a man, one has various possibilities to come up with an approximation. For instance, one might infer this probability by actual observation of people presently available (observable environment) or one might recall some friends and colleagues (retrievable environment). In any case: This inference is based on information that comes from one s own experience. As we have seen, the question at hand influences the way this information will be structured. Because in such environments all features of objects are either visible or retrievable, this organization can happen in favor of the requires cognitive inferences. As the features of a technique are not directly observable, it can happen that only information inverse to the question at hand is available. In the medical example, a test developer has to communicate the sensitivity and the specificity of the medical test in the instruction sheet with mathematically sound numbers. Unfortunately, the decision maker the physician cannot resort to the information stored in the memory of the test developer, but only to the provided sensitivity and specificity that are inverse to her queries. We speculate that in order to solve Eddy s (1982) written diagnosis task, we do not need an expert on diseases, but a statistician or expert on test characteristics such as sensitivity and specificity. A

8 physician who applies the test for the first time as well as the physician without feedback has no expertise of this kind. A frequently encountered question is, why in questionnaire tasks usually only 50% of participants solve Bayesian tasks correctly, even when provided with natural frequencies. The answer is: Eddy s task gives physicians information according to the tree on the left hand side of Figure 1. This means, even if the information is provided in terms of natural frequencies (Gigerenzer & Hoffrage, 1995), participants still have to invert this information if only in a semantic manner (Fiedler, Brinkmann, Betsch & Wild, 2000). In our view the opposite is remarkable: Participants 50% performance concerning written tasks that describe technical environments makes a strong claim for the natural frequency approach: Even if the provided information is inverse, natural frequencies can foster insight into the underlying situation. The second kind of situations where Bayesian reasoning is required is probably the most powerful application of natural frequencies: These kind of situations, which are discussed in the previous chapters in detail, go beyond making accurate inferences, but involve bringing across these inferences. Even if it is possible for a physician fastly to come up with a frequentistic diagnosis by relying on her expertise, this inference for the patient may be counterintuitive: The patient has no access to data stored in the physicians memory but maybe just knows the tests sensitivity of 90%. In order to understand the discrepancy between the seemingly powerful test and the low probability of actually having, the left tree of Figure 1, namely the Bayesian one, is an important tool to change despair into hope. The power of natural frequencies regarding the issue of risk communication in the medical domain is discussed in the chapter of Kurzenhäuser and Hoffrage in detail. So far, we analyzed the structure of environments with respect to the question whether the information available indeed calls for a "Bayesian" (in terms of Table 1; e.g., technical environments) or whether additional information can be observed or retrieved. In observable and retrievable environments, natural frequencie s afford a genuine format of information displayed by nature. Because natural frequencies in contrast to normalized probabilities always refer to one grand total, in the last section we will point to the impact that various degradations of the sample might have for the validity of inferences. Degraded samples At this point, many readers may not perfectly agree with the results of the analysis outlined so far. Our analysis only holds if the (observed or retrieved) sample is drawn randomly and not affected by interferences. In fact, many everyday examples may not be unproblematic and free of error. This is because the input of observed or retrieved information itself may not always be valid and reliable, just because the stimulus ecology to which cognitively functioning individuals are exposed does not provide equal access to all aspects of information, but always provides a relative, perspective -dependent picture of a world that may look different from another environmental perspective. Here is a brief list of examples to illustrate this problem: (a) Our physician who keeps natural frequency counts of the joint frequencies of all combinations of a positive or negative test, T+ or T-, with a present or absent disease, D+ or D-, and who, in order to estimate p(d+ T+), only has to compare the number of (D+ & T+) cases to the total number of T+ cases, might arrive at quite discrepant estimates depending on how valid or selective her data source is. For instance, the classification of cases as D+ or D- (i.e., having or not) may be based on patients self report, or on through histological examination, patients may be motivated to simulate or dissimulate a disease, the frequency count may refer to different subgroups with highly unequal baserates of D+, these subgroups may not be apparent at fist sight, etc. So just uncritically counting each and any data point that the

9 stimulus ecology provides can clearly mislead observers into inaccurate, strongly biased judgments. (b) One particularly error-prone example of a degraded environment that feeds observers with misleading frequency counts refers to those cases in which access to some cells of a contingency table is severely hindered or totally blocked. For instance, Einhorn and Hogarth (1978) referred to an illusion of validity when a personnel manager greatly overestimates his selection accuracy because the success rate in the sample of selected and hired employees is very high. In fact, the success rate may be even higher in the rejected applicants who however are not available for comparison, because they are simply gone. By analogy, our physician may have access to hospital records that only retain folders for patients with the disease and cast away after a while all folders of patients who turned out to be healthy and never returned for treatment or examination. (c) A related though less extreme case originates in the basic asymmetry of present versus absent information a phenomenon well-known as the feature-positive effect (Fiedler, xxx, 198x; Jenkins & Sainsbury, 19xx; Newman, Wolff & Hearst, 19xx). Observing the coincidence of a present effect (e.g., a disease) and a present signal (e.g., symptom) is much more likely to be remembered on purely informational grounds (Garner, 1978; Hovland & Weiss, 19xx ) than the coindicence of two absent events, or one present and one absent event. Imagine the chaos if new traffic laws would oblige car drivers to react to all absent traffic signs, as opposed to present traffic signs. (d) A similar asymmetry arises from the fact that the two poles of many antonyms (e.g., honest vs. dishonest; sane vs. insane; normal vs. abnormal) are not equally salient and diagnostic. Rather, one antonym is more diagnostic (i.e., justifies stronger inferences to other attributes) than the other. Typically, in the domain of morality, negative attributes (e.g., dishonest) have higher diagnosticity than positive attributes (e.g., honest). Therefore, a single dishonest act is often enough to infer a negative trait, whereas an extended sample of honest behavior is required before the positive trait, honesty, can be inferred. (e) For similar reasons, memory may not make all cells of a contingency scheme equally accessible and retrievable. (f) Environments often behave like a zoom objective. They enlarge certain events and diminish others. And they focus on one aspect, group, or phenomenon, and distract from others. For instance, American TV viewers may get a very different impression based on natural frequency counts of aggressive encounters between Islamists and Western societies than TV viewers in the middle East of Eurasia. Due to these various sorts of degraded samples, there can be no doubt that the solution of Bayesian conditional probability problems does not come with a guarantee of accuracy neither in observable nor in retrievable environments. Judges and decision makers can seriously fail when problems are in a genuinely ecological sense difficult, that is, when the environment does not support valid, non-selective stimulus presentation, but inhibits valid assessment from the beginning. However, it is important to note that these many reasons for erroneous judgments actually do not invalidate the present analysis. In fact, these undoubtedly true, ecological reasons for biased world views not only lead to distortions in natural frequency counts, but in the same fashion distort base-rates, hit rates, or false-alarm rates needed for Bayesian calculation. In other words, these ultimate reasons for erroneous judgment and assessment are independent of, and logically prior to the issue of cognitive reasoning that was the topic of the present chapter. These ultimate sources of error and failure are generated in the information environment well before cognitive processes come into play. Understanding these ultimate problem therefore is not a matter of Bayesian reasoning but a matter of developing new cognitive-ecological models that

10 consider environmental structures an integral part of cognitive psychology (see chapter of Gigerenzer and Fiedler).