Abstract
Richard Pettigrew [13, 14] defends the following theses: (1) epistemic disutility can be measured with strictly proper scoring rules (like the Brier score) and (2) at the beginning of their credal lives, rational agents ought to minimize their worst-case epistemic disutility (Minimax). This leads to a Principle of Indifference for ignorant agents. However, Pettigrew offers no argument in favour of Minimax, suggesting that the epistemic conservatism underlying it is a “normative bedrock.” Is there a way to test Minimax? In this paper, we argue that, since Pettigrew’s Minimax is impermissive, an argument against credence permissiveness constitutes an argument in favour of Minimax, and that arguments for credence permissiveness are arguments against Minimax.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
We borrowed this example from Schoenfield [18, p. 640].
- 2.
In such a case Pria is facing a situation akin to the one an agent faces in the famous Ellsberg paradox. In our framework, however, there is no need to make a distinction between risk and uncertainty, since in all cases, Pria is maximally ignorant. Indeed, since she has no prior information, she is always dealing with uncertainty. Minimax does not in fact solve the Ellsberg paradox: it simply avoids it when it is used by a maximally ignorant agent.
- 3.
- 4.
See [3].
- 5.
The expected result is sometimes called the weighted mean result. For example, suppose that, in a fair lottery, 5 participants each have 1 chance in 5 to win a single prize of $50. In that lottery, 4 participants won’t win a prize, and 1 participant will win $50. Since (40 + 150)/5 = 10, the weighted mean value of this lottery is $10. This means that $10 is the expected prize to each participant.
- 6.
- 7.
Specifically, while every agent could be bound to a specific impermissive scoring rule and a specific impermissive decision rule, different agents could be bound to distinct incompatible scoring rules or distinct incompatible decision rules.
- 8.
What we here call a “carving” is sometimes called a partition elsewhere. The reason for this terminological choice is that, in the literature from which our paper draws, “carving” is a much more popular term.
- 9.
Decidability should be understood in its most basic sense: given an input (evidential carving), there is always an effective method for returning an output (credence assignment).
- 10.
- 11.
See [13], Sect. 4.3] for formal definitions and the importance of calibration. Decomposition is crucial to Pettigrew’s account of accuracy: it allows him to exclude uncalibrated divergence measures. Roughly, an agent has a calibrated credence X in P if, relative to a body of evidence, X is the proportion of all propositions of a certain type (P) that appear to be true. This is the distinctive feature of strictly proper scoring rules: an agent’s expected score is maximized by assigning calibrated credences in P. For example, the following improper scoring rule respects Alethic vindication, Perfectionism, Divergence Additivity and Divergence Continuity but violates Decomposition:
-
1.
Relative to an agent’s credence X in P, if P is true, then the agent’s score is \((1-X);\)
-
2.
Relative to an agent’s credence X in P, if P is false, then the agent’s score is \(|0-X|\).
See also [13, pp. 37–40, 48].
-
1.
- 12.
Some authors have suggested that there are no distinct consistency requirements of rationality. Specifically, it is possible that process requirements of rationality, which govern how rational agents form and revise beliefs, secure consistency [11]. What matters in this paper is that inconsistent agents violated at least one rationality requirement. See also [4, sect. 9.2] on consistency requirements.
- 13.
We simply calculated the derivative of this function to find its optimum. The derivative of the function \(f(X) = 0.8 \cdot ((1-X)^{3/2}) + 0.2 \cdot ((0-X)^{3/2})\) is equal to 0 when \(X \approx 0.941\).
- 14.
The derivative of the function \(f(X) = 0.8 \cdot ((1-X)^3) + 0.2 \cdot ((0-X)^3)\) is equal to 0 when \(X \approx 0.666\).
- 15.
While we focused on two instances of improper scoring rules, the same problem would arise with any improper scoring rule. The fact that a scoring rule is not uniquely optimized when \(Z = X\) explains why an agent could end up with an inconsistent combination of attitudes. Only improper scoring rules are not uniquely optimized when \(Z = X\).
References
Aumann, R.J.: Agreeing to disagree. Annals Stat. 4, 1236–1239 (1976)
Ballantyne, N., Coffman, E.J.: Uniqueness, evidence, and rationality. Philos. Impr. 11(18), 1–13 (2011)
Brier, G.W.: Verification of forecasts expressed in terms of probability. Mon. Weather Rev. 78(1), 1–3 (1950)
Broome, J.: Rationality Through Reasoning. Wiley, Oxford (2013)
Christensen, D.: Conciliation, Uniqueness and Rational Toxicity. Noûs. doi:10.1111/nous.12077 (2014)
Horowitz, S.: Immoderately Rational. Philos. Stud. 167(1), 41–56 (2014)
Horowitz, S.: Laws of credence and laws of choice. Episteme 14(1), 31–37 (2017). doi:10.1017/epi.2016.49
Kelly, T.: The Epistemic Significance of Disagreement. In: Gendler, T.S., Hawthorne, J. (eds.) Oxford Studies in Epistemology, pp. 167–96. Clarendon Press, Oxford (2005)
Kelly, T.: Peer disagreement and higher-order evidence. In: Warfield, T., Feldman, R. (eds.) Disagreement, pp. 111–72. Oxford University Press, Oxford (2010)
Kelly, T.: Evidence can be permissive. In: Steup, M., Turri, J., Sosa, E. (eds.) Contemporary Debates in Epistemology, pp. 298–312. Wiley, Boston (2014)
Kolodny, N.: State or process requirements? Mind 116(462), 372–385 (2007)
Meacham, C.G.: Impermissive Bayesianism. Erkenntnis 79(6), 1185–1217 (2014)
Pettigrew, R.: Accuracy and the Laws of Credence. Oxford University Press, Oxford (2016a)
Pettigrew, R.: Accuracy, risk, and the principle of indifference. Philos. Phenomenol. Res. 92(1), 35–59 (2016b). doi:10.1111/phpr.12097
Predd, J.B., Seiringer, R., Lieb, E.H., Osherson, D.N., Vincent Poor, H., Kulkarni, S.R.: Probabilistic coherence and proper scoring rules. IEEE 55(10), 4786–4792 (2009)
Sharadin, N.: A paritial defense of permissivism. Ratio 30, 57–71 (2015). http://onlinelibrary.wiley.com/doi/10.1111/rati.12115/full
Schoenfield, M.: Permission to believe: why permissivism is true and what it tells us about irrelevant influences on belief. Noûs. 48(2), 193–218 (2014)
Schoenfield, M.: Bridging rationality and accuracy. J. Philos. 112(12), 633–57 (2015). doi:10.5840/jphil20151121242
Smithson, R.: The principle of indifference and inductive scepticism. Brit. J. Phil. Sci. 68, 253–272 (2017)
White, R.: Epistemic Permissiveness. Philos. Perspect. 19(1), 445–459 (2005)
White, R.: Evidence cannot be permissive. In: Steup, M., Turri, J., Sosa, E. (eds.) Contemporary Debates in Epistemology, pp. 312–23. Wiley, Boston (2014)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer-Verlag GmbH Germany
About this paper
Cite this paper
Daoust, MK., Montminy, D. (2017). Testing Minimax for Rational Ignorant Agents. In: Baltag, A., Seligman, J., Yamada, T. (eds) Logic, Rationality, and Interaction. LORI 2017. Lecture Notes in Computer Science(), vol 10455. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55665-8_37
Download citation
DOI: https://doi.org/10.1007/978-3-662-55665-8_37
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-55664-1
Online ISBN: 978-3-662-55665-8
eBook Packages: Computer ScienceComputer Science (R0)
