Apoyo espiritual para el alivio del sufrimiento

Gigerenzer and Hoffrage (1995) offered two related arguments to explain the beneficial effect of natural frequencies. The main argument for the facili-tative effect of natural frequencies is computational: Bayesian computations are simpler when the information is represented in natural frequencies rather than in any of the other formats in Table 3.1 (see also Kleiter, 1994).

With natural frequencies, the calculations necessary to arrive at the correct solution are equivalent to Equation 1; participants can derive the number of correct positive and false positive cases (8 and 95) directly from the problem text, without having to make any further calculations. With probabilities and relative or normalized frequencies, the calculation is more demanding (Equation 2) because all three formats contain normalized information that has to be multiplied with the respective base rates to arrive at the correct Bayesian solution.

The second explanation brings in an evolutionary perspective. Gigerenzer and Hoffrage (1995; see also Gigerenzer, 1998) argue that the human mind appears to be “tuned” to make inferences from natural frequencies rather than from probabilities and percentages, because for most of their existence, humans have made inferences from information encoded sequentially through direct experience. Natural frequencies are seen as the final tally of such a sequential sampling process (hence the term “natural” frequencies;

see Cosmides & Tooby, 1996; Kleiter, 1994). In contrast, mathematical probability did not emerge until the mid-17th century; in other words, probabilities and percentages are much more “recent” in evolutionary terms. Therefore, Gigerenzer and Hoffrage (1995) assume that minds have evolved to deal with natural frequencies rather than with probabilities.

Both explanations have been heavily disputed (see the discussions in Gigerenzer & Hoffrage, 1999; Hoffrage et al., 2002), and the evolutionary argument in particular has been met with scepticism (Fiedler et al., 2000;

Girotto & Gonzalez, 2001; Sloman, Over, Slovak, & Stibel, 2003). It should be noted that, strictly speaking, the evolutionary argument has yet to be tested, because it is still not clear how the effects of the two explanations (i.e., computational and evolutionary) can be disentangled (Hoffrage et al.,

2002; for a first step in this direction, see Brase, 2002). Furthermore, many researchers have argued that it is not the use of frequency formats per se, but rather some third factor that could be the explanation for the results obtained.

Most alternative accounts refer to the structure of the information entailed by the use of natural frequency formats. For instance, Girotto and Gonzalez (2001) argue that reasoning about conditional probability is mainly affected by two factors, the structure of the problem information and the form of the question asked. They adopt the “mental model theory of probabilistic reasoning” (Johnson-Laird, Legrenzi, Girotto, Sonino-Legrenzi, & Caverni, 1999) and assume that naïve reasoners infer con-ditional probabilities from so-called subset relations in mental models, rather than from Bayes’ rule. People make correct probability evaluations if they are encouraged to apply the subset principle, that is, to determine the proportion of the elements of the subset D&H (we have used the term

“subsample” so far) in the set of the elements of D. As we saw above, natural frequencies automatically contain these sets, but Girotto and Gonzalez (2001) argue that subset representations can also be elicited with-out natural frequencies (see also Sloman et al., 2003). They reported that when the form of the question and the structure of the problem were framed so as to encourage the application of the subset principle, naïve individuals solved problems equally well irrespective of whether they were stated in terms of probabilities or frequencies. Other studies showed that subset rep-resentations could also be activated by another factor, namely by using so-called partitive formulations in the text problems (Macchi, 2000; Macchi &

Mosconi, 1998). Partitive formulations clarify the relationship between the subsets to which the probabilities refer. Macchi (2000) reports that not only natural frequencies that are automatically partitive, but also relative fre-quencies with a partitive formulation, lead to high performance rates.

To date, there is no consensus on the explanation of statistical format effects in Bayesian reasoning (e.g., see Hoffrage et al., 2002; also Girotto &

Gonzalez, 2002, for a discussion on whether the subset principle is a mere redescription of a property of natural frequencies). Unfortunately, theor-etical advancement in this debate is slowed down by differing empirical methods and recurrent misunderstandings. First, the use of different per-formance criteria (see above) complicates the direct comparison of perform-ance across studies. Second, the wordings of the text problems often differ considerably between studies. This can be problematic since even small dif-ferences can affect performance (Cosmides & Tooby, 1996; Hoffrage et al., 2002). Third, some authors have misinterpreted the results on natural frequencies as a claim that any kind of frequency information would be more helpful than probabilities and percentages (e.g., Lewis & Keren, 1999;

Macchi, 2000; for an overview see Hoffrage et al., 2002).

However, although there is an ongoing debate on why the facilitating effect of natural frequencies in Bayesian inferences occurs and what its

boundary conditions are (e.g., Mellers & McGraw, 1999), there seems to be a consensus about its existence. As mentioned earlier, the effect has been shown several times for lay people (e.g., Hoffrage & Gigerenzer, 2004;

Macchi, 2000) as well as for medical (Hoffrage & Gigerenzer, 1998) and legal experts (Koehler, 1996a; Lindsey et al., 2003). The effect could also be observed in more complex diagnostic problems that invoke data from more than one cue for evaluating the hypothesis, for instance two medical tests in a row (Krauss, Martignon, & Hoffrage, 1999). Moreover, there is evidence that frequency representations can reduce or eliminate other well-known

“cognitive illusions” such as the conjunction fallacy (Hertwig & Gigerenzer, 1999; see also Chapter 1) or the overconfidence bias (Gigerenzer, Hoffrage,

& Kleinbölting, 1991; see also Chapter 13).

Despite the ongoing theoretical debate, the existing evidence has already inspired researchers to test whether the facilitating effect of natural frequen-cies could also be used to improve statistical thinking in applied settings. For instance, a 2-hour computerized tutorial was developed with the goal of helping people to deal with the probabilities and percentages that they encounter in textbooks and the media by teaching them how to translate probabilities into natural frequencies (Sedlmeier & Gigerenzer, 2001). This representation-learning approach has led to significantly higher perform-ance, especially in the long run, compared to a traditional approach that teaches how to use Bayes’ rule (see also Kurzenhäuser & Hoffrage, 2002, for an adaptation of the tutorial to the traditional classroom setting).

A domain for which the research on statistical formats in Bayesian inference is directly applicable has already been mentioned throughout this chapter: medical risk communication. Given the facilitating effect of natural frequencies in the diagnostic inference problems, it has been proposed that the meaning of medical test results should be communicated to patients in terms of natural frequencies in order to foster understanding (e.g., Gigerenzer, Hoffrage, & Ebert, 1998; Hamm & Smith, 1998).

To give a final example, the representation of risk information is also relevant in the legal domain. In criminal and paternity cases, the general practice in court is to present information in terms of probabilities or ratios of conditional probabilities, with the consequence that jurors, judges, and sometimes the experts themselves are confused and misinterpret the evi-dence (Koehler, 1996a; Lindsey et al., 2003). It will be most relevant for the development of such legal practices to follow the scientific debate about statistical formats and other representation features in Bayesian inference problems.

SUMMARY

• In Bayesian inferences, a prior probability estimate for a hypothesis is updated in light of new evidence.

• The statistical information that is used in Bayesian inferences can be represented in different statistical formats. The external representation of statistical information is a powerful factor for performance in Bayesian inference tasks.

• The natural frequency format is a statistical format that facilitates Bayesian computations, compared to single-event probabilities or rela-tive frequencies. Natural frequencies are not any kind of frequencies, but a specific type that results from the sequential partitioning of one sample into subsamples.

• While there seems to be a consensus that natural frequencies facilitate Bayesian inferences, there is an ongoing debate on the question of why this effect occurs.

• Nevertheless, natural frequencies can already be used as a tool to facili-tate statistical thinking in applied settings such as medical or legal risk communication.

FURTHER READING

Gigerenzer and Hoffrage (1995) wrote the classic paper that introduced natural frequencies as an alternative way of presenting statistical informa-tion in Bayesian inference tasks. An overview of the research on natural frequencies since then and a useful clarification of misunderstandings con-cerning natural frequencies can be found in the paper by Hoffrage et al.

(2002). One of the alternative explanations of the effect of natural frequen-cies in Bayesian inference tasks was proposed by Girotto and Gonzalez (2001). Interesting applications of natural frequencies, for instance in the legal or medical context, can be found in Gigerenzer (2002).

ACKNOWLEDGEMENTS

We are grateful to Barbara Fasolo, Ulrich Hoffrage, Julie Holzhausen, Stefan Krauss, and Gaëlle Villejoubert for helpful comments on an earlier version of this chapter. We also thank Anita Todd and Rona Unrau for editing the manuscript.

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APPENDIX

The four problems are taken from Hoffrage and Gigerenzer (2004). We present the full text for the two versions of one diagnostic problem. For the

other three problems, we present only the natural frequency version, from which the numerical information for the probability versions can easily be derived.