Abstract
According to the paradigm of adaptive rationality, successful inference and prediction methods tend to be local and frugal. As a complement to work within this paradigm, we investigate the problem of selecting an optimal combination of prediction methods from a given toolbox of such local methods, in the context of changing environments. These selection methods are called meta-inductive (MI) strategies, if they are based on the success-records of the toolbox-methods. No absolutely optimal MI strategy exists—a fact that we call the “revenge of ecological rationality”. Nevertheless one can show that a certain MI strategy exists, called “AW”, which is universally long-run optimal, with provably small short-run losses, in comparison to any set of prediction methods that it can use as input. We call this property universal access-optimality. Local and short-run improvements over AW are possible, but only at the cost of forfeiting universal access-optimality. The last part of the paper includes an empirical study of MI strategies in application to an 8-year-long data set from the Monash University Footy Tipping Competition.
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Notes
The lack of connection between a high cue correlation and TTB's success is also reported in Czerlinski et al. (1999, p. 116f). The implication between high cue-validity dispersion and TTB's optimality holds only in "naive Bayes" environments (cf. fn. 13 and Katsikopoulos and Martignon 2006, Corollary 1). Gigerenzer and Brighton (2009, p. 143) and Brighton and Gigerenzer (2012, p. 55) describe an environment with zero cue validity dispersion (a so-called Guttman environment) in which TTB works particularly well.
Exceptions to (a) are Hoffrage et al. (2000), Hogarth and Karelaia (2005) and Katsikopoulos et al. (2010), who study prediction tasks based on continuous-valued cues. Exceptions to (b) are Dieckmann and Todd (2012), and Rieskamp and Otto (2006), who study prediction tasks in the course of online learning.
Other frequently studied methods are Dawes' rule (equal weights), regression (optimal weights), and "naive Bayes" (Gigerenzer et al. 1999, part III).
A related idea is anticipated in Katsikopoulos and Martignon (2006, p. 491), who interpret a cue as a juror voting for one of two options in a social choice task.
A clairvoyant P ‘sees’ the future, and can be identified with a function fP: |N × Ω∞ → Ω.
For fixed n we approximate \( {\text{p}}(|{\text{suc}}_{\text{n}} - {\text{p}}| \ge\updelta) \approx {\text{c}}/({\text{n}}^{0.5} \cdot {\text{e}}^{{0.5 \cdot {\text{n}} \cdot\updelta^{2} }} ) \) (see de Finetti 1974, sect. VII.5.4). pδ is upper bounded by the infinite \( {\text{sum}}\,{\text{c}} \cdot \Sigma_{\text {n} \le \text {i} \le \infty } (1/({\text{i}}^{0.5} \cdot {\text{e}}^{{0.5 \cdot {\text{i}} \cdot\updelta^{2} }} )) \). This sum is lower-equal (c/n0.5)·Σn≤i≤∞xi), which is (by the sum-formula for a convergent geometric series) equal to (c/n0.5)·xn/(1−x).
This requirement guarantees that under the conditions of Theorem 3 the intermittent version of AW approximates TTB in the long run.
Cf. Cesa-Bianchi and Lugosi (2006, ch. 4.2). The randomization method presupposes that the event sequence does not react adversarially to AW's predictions. For adversarial event sequences, Theorem 4 can be transferred by assuming a collective of binary meta-inductivists who approximate real-valued predictions by the mean value of their binary predictions (cf. Schurz 2008, Theorem 5).
This dominance-claim can be strengthened (cf. Schurz 2008, Sect. 9.2). But AW is not universally access-dominant, since there are variations of AW with a different short-run performance.
(a) follows from the fact that there are uncountably many sequences but only countably many computable ones. (b) holds since the uniform prior distribution over {0,1}∞ implies p(ei|ej) = p(ei) = 1/2, i.e., the distribution is IID, which entails (b) by the strong law of large numbers.
See next section. Katsikopoulos and Martignon (2006) proved that under the condition of known validities and "naive Bayes environments" (conditionally independent cue validities and uniform prior), the logarithmic version of iSW that takes log(val(Pi)/(1 − val(Pi))) as the weight of cue Pi is probabilistically optimal among all possible methods.
References
Arnold, E. (2010). Can the best-alternative-justification solve Hume’s problem? Philosophy of Science, 77, 584–593.
Brighton, H. & Gigerenzer, G. (2012). How heuristics handle uncertainty. In Todd & Gigerenzer (eds.), (pp. 33–60). (Note: This article is an abbreviated version of Gigerenzer and Brighton 2009.)
Carnap, R. (1950). Logical foundations of probability. Chicago: Univ. of Chicago Press.
Cesa-Bianchi, N., & Lugosi, G. (2006). Prediction, learning, and games. Cambridge: Cambridge Univ. Press.
Czerlinski, J., Gigerenzer, G., & Goldstein, D. G. (1999). How good are simple heuristics? In: Gigerenzer et al. (Eds.), (pp. 97–118).
De Finetti, B. (1974). Theory of probability. New York: John Wiley.
Dieckmann, A., & Todd, P. M. (2012). Simple rules for ordering cues in one-reason decision making. In: Todd & Gigerenzer (Eds.), (pp. 274–306).
Feldbacher, C. (2012). Meta-induction and the wisdom of crowds. Analyse & Kritik, 34(2), 367–382.
Gigerenzer, G., & Brighton, H. (2009). Homo heuristicus: Why biased minds make better inferences. Topics in Cognitive Science, 1, 107–143.
Gigerenzer, G., Todd, P. M., & The ABC Research Group (Eds.). (1999). Simple heuristics that make us smart. Oxford: Oxford Univ. Press.
Hertwig, R., Hoffrage, U., & The ABC Research Group. (2013). Simple heuristics in a social world. New York: Oxford University Press.
Hoffrage, U., Hertwig, R., & Gigerenzer, G. (2000). Hindsight bias: A by-product of knowledge updating? Journal of Experimental Psychology. Learning, Memory, and Cognition, 26, 566–581.
Hogarth, R. M., & Karelaia, N. (2005). Ignoring information in binary choice with continuous variables: When is Less “More”? Journal of Mathematical Psychology, 49, 115–124.
Howson, C., & Urbach, P. (1996). Scientific reasoning: The Bayesian approach (2nd ed.). Chicago: Open Court.
Jekel, M., Glöckner, A., Fielder, S., & Bröder, A. (2012). The rationality of different kinds of intuitive decision processes. Synthese, 89(1), 147–160.
Katsikopoulos, K. V., & Martignon, L. (2006). Naive heuristics for paired comparisons. Journal of Mathematical Psychology, 50, 488–494.
Katsikopoulos, K. V., Schooler, L. J., & Hertwig, R. (2010). The robust beauty of ordinary information. Psychological Review, 117(4), 1259–1266.
Kelly, K. (1996). The logic of reliable inquiry. New York: Oxford Univ. Press.
Martignon, L., & Hoffrage, U. (1999). Why does one-reason decision making work? In: Gigerenzer et al. (Eds.), (pp. 119–140).
Popper, K. (1935/2002). Logic of discovery. London: Routledge.
Rieskamp, J., & Dieckmann, A. (2012). Redundancy: Environment structure that simple heuristics can exploit. In Todd & Gigerenzer (eds.), (pp. 187–215).
Rieskamp, J., & Otto, P. (2006). SSL: A theory of how people learn to select strategies. Journal of Experimental Psychology: General, 135(2), 207–236.
Schmitt, M., & Martignon, L. (2006). On the complexity of learning lexicographic strategies. Journal of Machine Learning Research, 7, 55–83.
Schurz, G. (2008). The meta-inductivist’s winning strategy in the prediction game: A new approach to Hume’s problem. Philosophy of Science, 75, 278–305.
Schurz, G. (2009). Meta-induction and social epistemology. Episteme, 6, 200–220.
Schurz, G. (2012). Meta-induction in epistemic networks and social spread of knowledge. Episteme, 9(2), 151–170.
Schurz, G. (2013). Philosophy of science: A unified approach. New York: Routledge.
Schurz, G., & Thorn, P. (2014). TTB vs. Franklin’s rule in environments of different redundancy. Appendix to G. Schurz, Instrumental justifications of normative systems of reasoning. Frontiers in Psychology, 5, article 625, July 2014. doi:10.3389/fpsyg.2014.00625.
Shanks, D. R., Douglas, L. M., & Holyoak, K. J. (Eds.). (1996). The psychology of learning and motivation. San Diego, CA: Academic Press.
Simon, H. A. (1982). Models of bounded rationality (Vol. 1). Cambridge, MA: MIT Press.
Solomonoff, R.J. (1964). A formal theory of inductive inference. Information and Control, 7, 1–22 (part I), 224–254 (part II).
Thorn, P., & Schurz, G. (2012). Meta-induction and the wisdom of crowds. Analyse & Kritik, 34(2), 339–366.
Todd, P. M., & Gigerenzer, G. (Eds.). (2012). Ecological rationality: intelligence in the world. New York: Oxford University Press.
Vickers, J. (2010). The problem of induction. In E. N. Zalta (Ed.) The Stanford Encyclopedia of Philosophy (Spring 2010 Edition). http://plato.stanford.edu/archives/spr2010/entries/induction-problem.
Weibull, J. (1995). Evolutionary game theory. Cambridge, MA: MIT Press.
Wells, A. J. (2005). Rethinking cognitive computation: Turing and the science of the mind. Basingstoke: Palgrave.
Wolpert, D. H. (1996). The lack of a priori distinctions between learning algorithms. Neural Computation, 8(7), 1341–1390.
Acknowledgments
Work on this paper was supported by the DFG Grant SCHU1566/9-1 as part of the priority program “New Frameworks of Rationality” (SPP 1516). For valuable help we are indebted to K.V. Katsikopolous, Ö. Simsek, A.P. Pedersen, R. Hertwig, M. Jekel, P. Grunwald, J.-W. Romeijn, and L. Martignon.
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Appendix
Appendix
Proof of Theorem 2
For 2.1(a): Since xMI’s imitates for each time n > 1 some deceiver Pi, xMI’s score for all times > 1 is 0, and so limsuc(xMI) = 0. For 3.1(b): On average, xMI imitates each player equally often, with a limiting frequency of 1/m. So the frequency of times for which each player earns a maximal score of 1 is (m − 1)/m. For (2.2): Let “p(fav-Pi)” be the limiting frequency of times for which player Pi was xMI’s favorite, and limsuc(Pi|fav) be player Pi’s limit success conditional on these times. Then by probability theory, limsuc(xMI) = Σ1≤i≤mp(fav-Pi)·limsuc(Pi|fav), which implies that limsuc(xMI) ≤ max({limsuc(Pi|fav):1 ≤ i ≤ m}). By the negative correlation assumption, limsuc(Pi|fav)) < limsuc(Pi) holds for all i ∈ {1, …, m}; so limsuc(xMI) < max({limsuc(Pi):1 ≤ i ≤ m}) =def maxlimsuc. Hence xMI is not long-run optimal.□
Proof of Theorem 3
For 3.1(a): Before the convergence time s, TTB may be, in the worst case, permanently deceived by the non-MI-players (or cues), by negatively correlated success-oscillations. So TTB’s worst-case success until time s is zero, whence his worst-case loss at times n ≥ s is s/n. After time point s, TTB’s earns for each kth-best player (or cue) \( {\text{P}}_{{{\text{s}}_{\text{k}} }} \) the sum-of-scores earned by \( {\text{P}}_{{{\text{s}}_{\text{k}} }} \), for all time points at which \( {\text{P}}_{{{\text{s}}_{\text{k}} }} \) but no player better than \( {\text{P}}_{{{\text{s}}_{\text{k}} }} \) delivered a prediction. The sum-expression in 3.1(a) is identical with the sum of these scores divided by time n. For 3.1(b): Additionally we assume that after time s each player’s validity is better than the success of a random guess, which is 1/2 in a binary prediction game. This implies that sucn(TTB) > maxsucn − (s/n), where maxsucn ≥ sucn(ITB) since ITB approximates maxsucn from below (by Theorem 1). (3.2) follows from (3.1) in the explained way, since limn→∞(s/n) = 0. □
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Schurz, G., Thorn, P.D. The Revenge of Ecological Rationality: Strategy-Selection by Meta-Induction Within Changing Environments. Minds & Machines 26, 31–59 (2016). https://doi.org/10.1007/s11023-015-9369-7
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DOI: https://doi.org/10.1007/s11023-015-9369-7