Predicting the future 1 Running head: PREDICTING THE FUTURE Predicting the future as Bayesian inference: People combine prior knowledge with observations when estimating duration and extent

Predicting the future is a basic problem that people have to solve every day and a component of planning, decision-making, memory, and causal reasoning. This paper presents five experiments testing a Bayesian model of predicting the duration or extent of phenomena from their current state. This Bayesian model indicates how people should combine prior knowledge with observed data. Comparing this model with human judgments provides constraints on possible algorithms that people might use to predict the future. The experiments examine the effects of multiple observations, the effects of prior knowledge, and the difference between independent and dependent observations, using both descriptions and direct experience of prediction problems. The results indicate that people integrate prior knowledge and observed data in a way that is consistent with our Bayesian model, ruling out some simple heuristics for predicting the future. We suggest some mechanisms that might lead to more complete algorithmic-level accounts. Predicting the future 3 Predicting the future as Bayesian inference: People combine prior knowledge with observations when estimating duration and extent Making predictions is hard. Especially about the future. Neils Bohr (or Yogi Berra) Despite the difficulty of predicting the future, we do it effortlessly every day. We are confident about being able to predict the durations of events, how much time we will need to get home after work, and how long it will take to finish the shopping. In many cases we have a great deal of information guiding our judgments. However, sometimes we have to make predictions based upon much less evidence. When faced with new situations our decisions about how much longer we can expect events to last are based on whatever evidence is available. When the only information we possess concerns how long a particular event has lasted until now, predicting the future becomes a challenging inductive problem. Being able to predict future events is an important component of many cognitive tasks. Our expectations about the future are certainly fundamental to planning and decision making, but inferences about time also play a role in other aspects of cognition. Anderson’s (1990; Anderson & Schooler, 1991) rational analysis of human memory takes the fundamental problem of memory to be predicting whether an item will be needed in the future, with retention favouring those items most likely to be in demand. Assessing the need for a particular item explicitly involves predicting the future, a problem that Anderson formulates as a Bayesian inference. Prediction is also intimately related to the discovery of causal relationships. The regularities induced by causal relationships lead us Predicting the future 4 to make predictions about future events that can alter our perceptions (Eagleman & Holcombe, 2002) and result in surprises when our predictions are incorrect (Huettel, Mack, & McCarthy, 2002). In this paper, we test the predictions of a model of predicting the future, formulating the problem as one of Bayesian inference. Our Bayesian model indicates how people should combine their prior knowledge about a phenomenon with the information provided by observed data. Prior knowledge is expressed in terms of a probability distribution over the extent or duration of a phenomenon, while the effect of observations is incorporated via a statistical argument that is used in cosmology, known as the “anthropic principle” (Gott, 1993). We explore the predictions of this model in depth and consider their implications for the psychological mechanisms that might guide human predictions. In previous work, we showed that people are sensitive to the statistical properties of everyday quantities in the way that is consistent with our model (Griffiths & Tenenbaum, 2006). These results indicate that people use prior knowledge, but they do not imply that people are making predictions by performing the calculations indicated by Bayes’ rule. Rather, they suggest that whatever algorithm people use for predicting the future, it is consistent with Bayesian inference in its sensitivity to prior knowledge. Many simple heuristics might have this property. For example, Mozer, Pashler, and Homaei (2008) have argued that our results could be explained by people following a simple heuristic, in which they use only a small number of previous experiences to inform their judgments rather than having access to the whole probability distribution. Our goal in this paper is to gain a deeper understanding of the extent to which people’s predictions are consistent with our Bayesian model, and thus to obtain stronger constraints on the algorithms that they might be using in making these predictions. Our Bayesian model makes strong predictions about the effects of providing further observations, how prior knowledge should be combined with these observations, and the Predicting the future 5 difference between independent and dependent observations. Each of these predictions provides an opportunity to identify a constraint that an algorithm would need to satisfy: If people behave in a way that is consistent with our model in each of these cases, then whatever algorithm they are using must also approximate Bayesian inference. We test these predictions empirically, using both cognitive and perceptual judgments about time, and consider their implications for simple heuristics that people might use in predicting the future. Rational models of predicting the future We begin our analysis by describing the “anthropic principle” used in predicting the future. We then present the Bayesian generalization of this argument and consider the effects of manipulating the prior and the likelihood within this model. This leads us to the model predictions that we test in the remainder of the paper. The Copernican anthropic principle The cosmologist J. Richard Gott III (1993) proposed a simple heuristic for predicting the future which was intended to provide insight into weighty matters like the prolonged existence of the human race, but to be just as valid when applied to matters arising in everyday life. Gott’s heuristic is based on the “Copernican anthropic principle,” which holds that . . . the location of your birth in space and time in the Universe is priveleged (or special) only to the extent implied by the fact that you are an intelligent observer, that your location among intelligent observers is not special but rather picked at random (1993, p. 316) Gott extends this principle to reasoning about our position in time – given no evidence to the contrary, we should not assume that we are in a “special” place in time. If we adopt Predicting the future 6 this principle, the time at which an observer encounters a phenomenon should be randomly located in the total duration of that phenomenon.1 This principle leads to a simple method for predicting the future: discover how long a phenomenon has endured until the moment it was observed and predict that it should last that long into the future. This follows from the anthropic principle because if you assume you encounter a phenomenon at a random point, it is equally likely you observed it in the first or second half of its total duration. Denoting the time between the start of a phenomenon and its observation tpast and its total duration ttotal, a good guess is ttotal = 2 tpast. The argument works just as well if you know how far a phenomenon will extend into the future (for example if you see a sign that is counting down to some event), but do not know how far it extends into the past. Again, you should guess ttotal = 2 tfuture. For simplicity, we will focus on predicting ttotal from tpast for the remainder of this section. More formally, Gott’s rule can be justified by a probabilistic analysis that he calls the “delta t argument”. If we define the ratio r = tpast ttotal , (1) the Copernican anthropic principle tells us that r should be uniformly distributed between 0 and 1. This lets us make probabilistic predictions about the value of r. In particular, the probability that r < 0.5 is 0.5, so there is a 50% chance that ttotal > 2 tpast. Likewise, there is a 50% chance that ttotal < 2 tpast, making ttotal = 2 tpast a good guess. We can also use this argument to define confidence intervals over the durations of events by evaluating confidence intervals on r. For example, r will be between 0.025 and 0.975 with a probability of 0.95, meaning that with 95% confidence 1 39 tpast < tfuture < 39tpast, Predicting the future 7 where tfuture = ttotal − tpast. This method of reasoning has been used to predict a wide range of phenomena. Gott (1993) gives the example of the Berlin Wall, which he first encountered in 1969. At this point the Berlin Wall had been in existence for 8 years, so tpast is 8 years. The 95% confidence interval on the future duration of the Berlin Wall, tfuture, based upon the assumption that Gott’s visit was randomly located in the period of its existence is 2.46 months to 312 years, firmly containing the actual tfuture of 20 years. Gott made similar calculations of tfuture for Stonehenge, the journal Nature, the U.S.S.R., and even the human race (the good news is that a 95% confidence interval gives us at least 5, 100 years, the bad news is that it also predicts less than 7.8 million). The principle has subsequently been applied to a surprisingly broad range of targets, including predicting the run of Broadway musicals (Landsberg, Dewynne, & Please, 1993). A Bayesian approach to predicting the future Gott’s (1993) Copernican anthropic principle suggests how we might formulate a rational statistical account of our ability to predict the future, but the context in which we make our daily predictions differs from that assumed by Gott in two important ways: prior knowledge and multiple observations. In many cases in the real world where it might be desirable to predict the future, we know more than simply how long a process has been underway. In particular, our interaction with the world often gives us some prior expectations about the duration of an event. For example, meeting a 78 year-old man on the street, we are unlikely to think that there is a 50% chance that he will be alive at the age of 156 (for a similar example, see Jaynes, 2003). Likewise, our predictions are often facilitated by the availability of multiple observations of a phenomenon. For example, if we were attempting to determine the period that passes between subway trains arriving at a station, we would probably have several trips upon which to base our judgment. If on Predicting the future 8 our first trip we discovered that a train had left the station 103 seconds ago, we might assume that trains run every few minutes. But, after three trips yield trains that have left 103, 34, and 72 seconds ago, this estimate might get closer to 103 seconds. And after ten trains, all leaving less than 103 seconds before we arrive, we might be inclined to accept a value very close to 103 seconds. Gott’s (1993) delta-t argument does not incorporate the prior knowledge about durations that people bring to the problem of predicting the future, or the possibility of multiple observations. However, it can be shown that the delta-t argument is equivalent to a simple Bayesian analysis of the problem of predicting the future (Gott, 1994). Bayesian inference naturally combines prior knowledge with information from one or many observations, making it possible to extend Gott’s argument to provide a more general account of predicting the future. Bayes’ rule states that P (h|d) = P (d|h)P (h) P (d) , (2) where h is some hypothesis under consideration and d is the observed data. By convention, P (h|d) is referred to as the posterior probability of the hypothesis, P (h) the prior probability, and P (d|h) the likeliihood, giving the probability of the data under the hypothesis. The denominator P (d) can be obtained by summing across P (d|h)P (h) for all hypotheses, giving P (hi|d) = P (d|hi)P (hi) ∫ H P (d|h)P (h)dh , (3) where H is the set of all hypotheses. In responding to a criticism offered by Buch (1994), Gott (1994) noted that his method for predicting the future could be expressed in Bayesian terms. In this setting, the data are the observation of the current duration of a phenomenon, tpast, and the hypotheses concern its total duration, ttotal. Using the prior P (ttotal) ∝ 1 ttotal and the Predicting the future 9 likelihood P (tpast|ttotal) = 1 ttotal if ttotal ≥ tpast (and 0 otherwise) yields the same results as his original formulation of the delta t argument.2 This can be seen if we substitute Gott’s choices of prior and likelihood into Equation 3, P (ttotal|tpast) = 1 t total ∫∞ tpast 1 t total dttotal , (4)