Everyday predictions 1 Running head: EVERYDAY PREDICTIONS Optimal predictions in everyday cognition

Human perception and memory are often explained as optimal statistical inferences, informed by accurate prior probabilities. In contrast, cognitive judgments are usually viewed as following error-prone heuristics, insensitive to priors. We examined the optimality of human cognition in a more realistic context than typical laboratory studies, asking people to make predictions about the duration or extent of everyday phenomena such as human life spans and the box-office take of movies. Our results suggest that everyday cognitive judgments follow the same optimal statistical principles as perception and memory, and reveal a close correspondence between people’s implicit probabilistic models and the statistics of the world. Everyday predictions 3 Optimal predictions in everyday cognition If you were assessing the prospects of a 60-year-old man, how much longer would you expect him to live? If you were an executive evaluating the performance of a movie that had made 40 million dollars at the box office so far, what would you estimate for its total gross? Everyday life routinely poses such challenges of prediction, where the true answer cannot be determined based on the limited data available, yet common sense suggests at least a reasonable guess. Analogous inductive problems arise in many domains of human psychology, such as identifying the three-dimensional structure underlying a two-dimensional image (Freeman, 1994; Knill & Richards, 1996), or judging when a particular fact is likely to be needed in the future (Anderson, 1990; Anderson & Milson, 1989). Accounts of human perception and memory suggest that these systems effectively approximate optimal statistical inference, correctly combining new data with an accurate probabilistic model of the environment (Anderson, 1990; Anderson & Milson, 1989; Anderson & Schooler, 1991; Freeman, 1994; Geisler, Perry, Super, & Gallogly, 2001; Huber, Shiffrin, Lyle, & Ruys, 2001; Knill & Richards, 1996; Körding & Wolpert, 2004; Shiffrin & Steyvers, 1997; Simoncelli & Olshausen, 2001; Weiss, Simoncelli, & Adelson, 2002). In contrast – perhaps as a result of the great attention garnered by the work of Kahneman, Tversky, and their colleagues (Kahneman, Slovic, & Tversky, 1982; Tversky & Kahneman, 1974) – cognitive judgments under uncertainty are often characterized as the result of error-prone heuristics, insensitive to prior probabilities. This view of cognition, based on laboratory studies, appears starkly at odds with the near-optimality of other human capacities, and with people’s ability to make smart predictions from sparse data in the real world. To evaluate how cognitive judgments compare with optimal statistical inferences in real-world settings, we asked people to predict the duration or extent of everyday Everyday predictions 4 phenomena such as human life spans and the gross of movies. We varied the phenomena that were described and the amount of data available, and we compared the predictions of human participants with those of an optimal Bayesian model, described in detail in the Appendix. To illustrate the principles behind this Bayesian analysis, imagine that we want to predict the total life span of a man we have just met, based upon the man’s current age. If ttotal indicates the total amount of time the man will live and t indicates his current age, the task is to estimate ttotal from t. The Bayesian predictor computes a probability distribution over ttotal given t, by applying Bayes’ rule: p(ttotal|t) ∝ p(t|ttotal)p(ttotal). (1) The probability assigned to a particular value of ttotal is proportional to the product of two factors: the likelihood p(t|ttotal) and the prior probability p(ttotal). The likelihood is the probability of first encountering a person at age t given that their total life span is ttotal. Assuming for simplicity that we are equally likely to meet a person at any point in his life, this probability is uniform, p(t|ttotal) = 1/ttotal, for all possible values of t between 0 and ttotal (and 0 for values outside that range). This assumption of uniform random sampling is analogous to the “Copernican anthropic principle” in Bayesian cosmology (Buch, 1994; Caves, 2000; Garrett & Coles, 1993; Gott, 1993, 1994; Ledford, Marriott, & Crowder, 2001) and the “generic view principle” in Bayesian models of visual perception (Freeman, 1994; Knill & Richards, 1996). The prior probability p(ttotal) reflects our general expectations about the relevant class of events – in this case, about how likely it is that a person’s life span will be ttotal. Analysis of actuarial data shows that the distribution of life spans in our society is (ignoring infant mortality) approximately Gaussian – normally distributed – with a mean, μ, of about 75 years and a standard deviation, σ, of about 16 years. Combining the prior with the likelihood according to Equation 1 yields a probability Everyday predictions 5 distribution p(ttotal|t) over all possible total life spans ttotal for a man encountered at age t. A good guess for ttotal is the median of this distribution – that is, the point at which it is equally likely that the true life span is longer or shorter. Taking the median of p(ttotal|t) defines a Bayesian prediction function, specifying a predicted value of ttotal for each observed value of t. Prediction functions for events with Gaussian priors are nonlinear: for values of t much less than the mean of the prior, the predicted value of ttotal is approximately the mean; once t approaches the mean, the predicted value of ttotal increases slowly, converging to t as t increases but always remaining slightly higher, as shown in Figure 1. Although its mathematical form is complex, this prediction function makes intuitive sense for human life spans: a predicted life span of about 75 years would be reasonable for a man encountered at age 18, 39, or 51; if we met a man at age 75 we might be inclined to give him several more years at least; but if we met someone at age 96 we probably would not expect him to live much longer. This approach to prediction is quite general, applicable to any problem that requires estimating the upper limit of a duration, extent, or other numerical quantity given a sample drawn from that interval (Buch, 1994; Caves, 2000; Garrett & Coles, 1993; Gott, 1993, 1994; Jaynes, 2003; Jeffreys, 1961; Ledford et al., 2001; Leslie, 1996; Maddox, 1994; Shepard, 1987; Tenenbaum & Griffiths, 2001). However, different priors will be appropriate for different kinds of phenomena, and the prediction function will vary substantially as a result. For example, imagine trying to predict the total box office gross of a movie given its take so far. The total gross of movies follows a power-law distribution, with p(ttotal) ∝ t −γ total for some γ > 0. 1 This distribution has a highly non-Gaussian shape (see Figure 1), with most movies taking in only modest amounts but occasional blockbusters making huge amounts of money. In the Appendix, we show that for power-law priors, the Bayesian prediction function picks a value for ttotal that is a multiple of the observed sample t. The exact multiple depends on the parameter γ. For the Everyday predictions 6 particular power law that best fits the actual distribution of movie grosses, an optimal Bayesian observer would estimate the total gross to be approximately 50% greater than the current gross: if we observe a movie has made $40 million to date, we should guess a total gross of around $60 million; if we had observed a current gross of only $6 million, we should guess about $9 million for the total. While such “constant multiple” prediction rules are optimal for event classes that follow power-law priors, they are clearly inappropriate for predicting life spans or other kinds of events with Gaussian priors. For instance, upon meeting a 10-year-old child and her 75-year-old grandfather, we would never predict that she will live a total of 15 years (1.5 × 10) and he will live to be 112 (1.5 × 75). Other classes of priors, such as the exponential-tailed Erlang distribution, p(ttotal) ∝ ttotal exp{−ttotal/β} for β > 0, 2 are also associated with distinctive optimal prediction functions. For the Erlang distribution, the best guess of ttotal is simply t plus a constant determined by the parameter β, as shown in the Appendix and illustrated in Figure 1. Our experiment compared these ideal Bayesian analyses with the judgments of a large sample of human participants, examining whether people’s predictions were sensitive to the distributions of different quantities that arise in everyday contexts. We used publicly available data to identify the true prior distributions for several classes of events (the sources of these data are given in Table A1 in the Appendix). For example, as shown in Figure 2, human life spans and the runtime of movies are approximately Gaussian, the gross of movies and the length of poems are approximately power-law distributed, and the number of years spent in office for members of the U.S. House of Representatives and the reigns of Pharaohs are approximately Erlang. The experiment examined how well people’s predictions corresponded to optimal statistical inference in these different settings. Everyday predictions 7