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Minds and Machines

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01.03.2016

The Revenge of Ecological Rationality: Strategy-Selection by Meta-Induction Within Changing Environments

verfasst von: Gerhard Schurz, Paul D. Thorn

Erschienen in: Minds and Machines | Ausgabe 1-2/2016

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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.

Vorheriger Artikel Building the Theory of Ecological Rationality

Nächster Artikel Rational Foundations of Fast and Frugal Heuristics: The Ecological Rationality of Strategy Selection via Improper Linear Models

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Cf. Gigerenzer et al. (1999), Todd and Gigerenzer (2012), and Hertwig et al. (2013).

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.

See (among others) Gigerenzer et al. (1999), ch. III, Brighton and Gigerenzer (2012), Rieskamp and Dieckmann (2012), and Katsikopoulos et al. (2010).

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 f_P: |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). p_δ 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/n^0.5)·Σ_n≤i≤∞xⁱ), which is (by the sum-formula for a convergent geometric series) equal to (c/n^0.5)·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(e_i|e_j) = p(e_i) = 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(P_i)/(1 − val(P_i))) as the weight of cue P_i is probabilistically optimal among all possible methods.

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Titel: The Revenge of Ecological Rationality: Strategy-Selection by Meta-Induction Within Changing Environments
verfasst von: Gerhard Schurz
Paul D. Thorn
Publikationsdatum: 01.03.2016
Verlag: Springer Netherlands
Erschienen in: Minds and Machines / Ausgabe 1-2/2016
Print ISSN: 0924-6495
Elektronische ISSN: 1572-8641
DOI: https://doi.org/10.1007/s11023-015-9369-7

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