# How to Improve Bayesian Reasoning Without Instruction: Frequency Formats

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3 686 GERD GIGERENZER AND ULRICH HOFFRAGE (the positive test), and the task is to arrive at a single-point probability estimate. The information is represented here in terms of single-event probabilities: All information (base rate, hit rate, and false alarm rate) is in the form of probabilities attached to a single person, and the task is to estimate a single-event probability. The probabilities are expressed as percentages; alternatively, they can be presented as numbers between zero and one. We refer to this representation (base rate, hit rate, and false alarm rate expressed as single-event probabilities) as the standard probability format. What is the algorithm needed to calculate the Bayesian posterior probability p(cancer positive) from the standard probability format? Here and in what follows, we use the symbols H and -H for the two hypotheses or possible outcomes (breast cancer and no breast cancer) and D for the data obtained (positive mammography). A Bayesian algorithm for computing the posterior probability p(h\ D) with the values given in the standard probability format amounts to solving the following equation: p(h)p(d\h) P(H\D) = p(h)p(d\h)+p(-h)p(d\-h) (.01X.80) 0) + (.99)(.096)' (O The result is.078. We know from several studies that physicians, college students (Eddy, 1982), and staff at Harvard Medical School (Casscells, Schoenberger & Grayboys, 1978) all have equally great difficulties with this and similar medical disease problems. For instance, Eddy (1982) reported that 95 out of 100 physicians estimated the posterior probability p(cancer positive) to be between 70% and 80%, rather than 7.8%. The experimenters who have amassed the apparently damning body of evidence that humans fail to meet the norms of Bayesian inference have usually given their research participants information in the standard probability format (or its variant, in which one or more of the three percentages are relative frequencies; see below). Studies on the cab problem (Bar- Hillel, 1980;Tversky&Kahneman, 1982), the light-bulb problem (Lyon & Slovic, 1976), and various disease problems (Casscells etal., 1978; Eddy, 1982; Hammerton, 1973) are examples. Results from these and other studies have generally been taken as evidence that the human mind does not reason with Bayesian algorithms. Yet this conclusion is not warranted, as explained before. One would be unable to detect a Bayesian algorithm within a system by feeding it information in a representation that does not match the representation with which the algorithm works. In the last few decades, the standard probability format has become a common way to communicate information ranging from medical and statistical textbooks to psychological experiments. But we should keep in mind that it is only one of many mathematically equivalent ways of representing information; it is, moreover, a recently invented notation. Neither the standard probability format nor Equation 1 was used in Bayes' (1763) original essay. Indeed, the notion of "probability" did not gain prominence in probability theory until one century after the mathematical theory of probability was invented (Gigerenzer, Swijtink, Porter, Daston, Beatty, & Kriiger, 1989). Percentages became common notations only during the 19th century (mainly for interest and taxes), after the metric system was introduced during the French Revolution. Thus, probabilities and percentages took millennia of literacy and numeracy to evolve; organisms did not acquire information in terms of probabilities and percentages until very recently. How did organisms acquire information before that time? We now investigate the links between information representation and information acquisition. Natural Sampling of Frequencies Evolutionary theory asserts that the design of the mind and its environment evolve in tandem. Assume pace Gould that humans have evolved cognitive algorithms that can perform statistical inferences. These algorithms, however, would not be tuned to probabilities or percentages as input format, as explained before. For what information format were these algorithms designed? We assume that as humans evolved, the "natural" format was frequencies as actually experienced in a series of events, rather than probabilities or percentages (Cosmides & Tooby, in press; Gigerenzer, 1991b, 1993a). From animals to neural networks, systems seem to learn about contingencies through sequential encoding and updating of event frequencies (Brunswik, 1939; Gallistel, 1990; Hume, 1739/1951; Shanks, 1991). For instance, research on foraging behavior indicates that bumblebees, ducks, rats, and ants behave as if they were good intuitive statisticians, highly sensitive to changes in frequency distributions in their environments (Gallistel, 1990; Real, 1991; Real & Caraco, 1986). Similarly, research on frequency processing in humans indicates that humans, too, are sensitive to frequencies of various kinds, including frequencies of words, single letters, and letter pairs (e.g., Barsalou & Ross, 1986; Hasher &Zacks, 1979;Hintzman, 1976; Sedlmeier, Hertwig, & Gigerenzer, 1995). The sequential acquisition of information by updating event frequencies without artificially fixing the marginal frequencies (e.g., of disease and no-disease cases) is what we refer to as natural sampling (Kleiter, 1994). Brunswik's (1955) "representative sampling" is a special case of natural sampling. In contrast, in experimental research the marginal frequencies are typically fixed a priori. For instance, an experimenter may want to investigate 100 people with disease and a control group of 100 people without disease. This kind of sampling with fixed marginal frequencies is not what we refer to as natural sampling. The evolutionary argument that cognitive algorithms were designed for frequency information, acquired through natural sampling, has implications for the computations an organism needs to perform when making Bayesian inferences. Here is the question to be answered: Assume an organism acquires information about the structure of its environment by the natural sampling of frequencies. What computations would the organism need to perform to draw inferences the Bayesian way? Imagine an old, experienced physician in an illiterate society. She has no books or statistical surveys and therefore must 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

7 690 GERD GIGERENZER AND ULRICH HOFFRAGE X /A Step 3: False alarm cut.01 % of no If prenatal, damage 99.5% no prenatal damage no prenatal damage Step 1: Base rate cut - 5% / prenatal damage Step 2: Hit rate cut 40% of prenatal damage Rare event shortcut If the outcome is rare, then just cut off p(di-h). p(hid) = Step 4: Comparison Step Prenatal damage & German measles All cases of German measles Comparison shortcut If the left piece (p(d&-h)) is very small compared to the right piece (p(d&h)), p(hid) = 1 - divide the smaller by the larger and take the complement. Big hit rate shortcut If p(dih) is very large, then skip Step 2. Figure 2. A Bayesian algorithm invented by one of our research participants. The "beam cut" is illustrated for the German measles problem in the standard probability format. H stands for "severe prenatal damage in the child," and D stands for "mother had German measles in early pregnancy." The information isp(h) = 0.5%,p(D\H) = 40%, andp(d\-h) = 0.01%. The task is to infer p(h\d). inference with little reduction in accuracy. If an event is rare, that is, ifp(h) is very small, and p(-h) is therefore close to 1.0, thenp(d \ -H)p(-H) can be approximated byp(d\ -H). That is, instead of cutting the proportion p(d\-h) of the left part of the beam (Step 3), it is sufficient to cut a piece of absolute size p(d] -H). The rare-event shortcut (see Figure 2) is as follows: IF the event is rare, THEN simplify Step 3: Cut a piece of absolute size p(d\-h). This shortcut corresponds to the approximation p(h\d)~p(h)p(d\h)/[p(h)p(d\h)+p(d\-h)]. The shortcut works well for the German measles problem, where the base rate of severe prenatal damage is very small, p(h) =.005. The shortcut estimates p(//1z>) as.9524, whereas Bayes' theorem gives It also works with the mammography problem, where it generates an estimate of.077, compared with.078 from Bayes' theorem. Big Hit-Rate Shortcut Large values ofp(d \H)(such as high diagnosticities in medical tests; that is, excellent hit rates) allow one to skip Step 2 with little loss of accuracy. If p(d\h) is very large, then the p(h) piece is practically the same size as the piece one obtains from cutting all but a tiny sliver from the p(h) piece. The big hit-rate shortcut is then as follows: IF p(d \ H) is very large, THEN skip Step 2. This shortcut corresponds to the approximation The big hit-rate shortcut would not work as well as the rareevent shortcut in the German measles problem because p(d H ) is only.40. Nevertheless, the shortcut estimate is only a few percentage points removed from that obtained with Bayes' theorem (.980 instead of.953 ). The big hit-rate shortcut works well, to offer one instance, in medical diagnosis tasks where the hit rate of a test is high (e.g., around.99 as in HIV tests). Comparison Shortcut If one of the two pieces obtained in Steps 2 and 3 is small relative to the other, then the comparison in Step 4 can be simplified with little loss of accuracy. For example, German measles in early pregnancy and severe prenatal damage in the child occur more frequently than do German measles and no severe damage. More generally, ifd&h cases are much more frequent

8 HOW TO IMPROVE BAYESIAN REASONING 691 than D&- H cases (as in the German measles problem), or vice versa (as in the mammography problem), then only two pieces (rather than three) need to be related in Step 4. The comparison shortcuts for these two cases are as follows: IFD& H occurs much more often than D&H, THEN simplify Step 4: Take the ratio of D&H( right piece) to D&-H (left piece) as the posterior probability. This shortcut corresponds to the approximation p(h\d)*,p(h)p(d\h)/p(-h)p(d\-h). Note that the right side of this approximation is equivalent to the posterior odds ratio p(h\ D)/p(-H\ D). Thus, the comparison shortcut estimates the posterior probability by the posterior odds ratio. IF D&H occurs much more often than D&-H, THEN simplify Step 4: Take the ratio ofd&-h (left piece) to D&H (right piece) as the complement of the posterior probability. This shortcut corresponds to the approximation p(ff D)«1 -p(-h)p(d\-h)/p(h)p(d\h). The comparison shortcut estimatesp(h\ D) as.950 in the German measles problem, whereas Bayes' theorem gives.953. The comparison shortcut is simpler when the D&-H cases are the more frequent ones, which is typical for medical diagnosis, where the number of false alarms is much larger than the number of hits, as in mammography and HIV tests. Multiple Shortcuts Two or three shortcuts can be combined, which results in a large computational simplification. What we call the quick-andclean shortcut combines all three. Its conditions include a rare event, a large hit rate, and many D& H cases compared with D&H cases (or vice versa). The quick-and-clean shortcut is as follows: IF an event H is rare, p(d/h) high, andd&-h cases much more frequent than D&H cases, THEN simply divide the base rate by the false alarm rate. This shortcut corresponds to the approximation p(h\d)~p(h)/p(d\-h). The conditions of the quick-and-clean shortcut seem to be not infrequently satisfied. Consider routine HIV testing: According to present law, the U.S. immigration office makes an HIV test a condition sine qua non for obtaining a green card. Mr. Quick has applied for a green card and wonders what a positive test result indicates. The information available is a base rate of.002, a hit rate of.99, and a false alarm rate of.02; all three conditions for the quick-and-clean shortcut are thus satisfied. Mr. Quick computes.002 /.02 =. 10 as an estimate of the posterior probability of actually being infected with the HIV virus if he tests positive. Bayes' theorem results in.09. The shortcut is therefore an excellent approximation. Alternately, if D&H cases are more frequent, then the quick-and-clean shortcut is to divide the false alarm rate by the base rate and to use this as an estimate for 1 - p(h\d). In the mammography and German measles problems, where the conditions are only partially satisfied, the quickand-clean shortcut still leads to surprisingly good approximations. The posterior probability of breast cancer is estimated at.01/.096, which is about.10 (compared with.078), and the posterior probability of severe prenatal damage is estimated as.98 (compared with.953). Shortcuts: Frequency Format Does the standard frequency format invite the same shortcuts? Consider the inference about breast cancer from a positive mammography, as illustrated in Figure 1. Would the rare-event shortcut facilitate the Bayesian computations? In the probability format, the rare-event shortcut uses p(d \ -H) to approximate p( H)p( D H); in the frequency format, the latter corresponds to the absolute frequency 95 (d& h) and no approximation is needed. Thus, a rare-event shortcut is of no use and would not simplify the Bayesian computation in frequency formats. The same can be shown for the big hit-rate shortcut for the same reason. The comparison shortcut, however, can be applied in the frequency format: IF d& h occurs much more often than d&h, THEN compute d&h/d&~h. The condition and the rationale are the same as in the probability format. To summarize, we proposed three classes of cognitive algorithms underlying Bayesian inference: (a) algorithms that satisfy Equations 1 through 3; (b) pictorial analogs that work with operations such as "cutting" instead of multiplying (Figure 2); and (c) three shortcuts that approximate Bayesian inference well when certain conditions hold. Predictions We now derive several predictions from the theoretical results obtained. The predictions specify conditions that do and do not make people reason the Bayesian way. The predictions should hold independently of whether the cognitive algorithms follow Equations 1 through 3, whether they are pictorial analogs of Bayes' theorem, or whether they include shortcuts. Prediction 1: Frequency formats elicit a substantially higher proportion of Bayesian algorithms than probability formats. This prediction is derived from Result 1, which states that the Bayesian algorithm is computationally simpler in frequency formats. 2 2 At the point when we introduced Result 1, we had dealt solely with the standard probability format and the short frequency format. However, Prediction 1 also holds when we compare formats across both menus. This is the case because (a) the short menu is computationally simpler in the frequency than in the probability format, because the frequency format involves calculations with natural numbers and the probability format with fractions, and (b) with a frequency format, the Bayesian computations are the same for the two menus (Result 6).

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