Minimax and the value of information

Abstract

In his discussion of minimax decision rules, Savage (The foundations of statistics, Dover Publications Inc., Mineola 1954, p. 170) presents an example purporting to show that minimax applied to negative expected utility (referred to by Savage as “negative income”) is an inadequate decision criterion for statistics; he suggests the application of a minimax regret rule instead. The crux of Savage’s objection is the possibility that a decision maker would choose to ignore even “extensive” information. More recently, Parmigiani (Theor Decis 33:241–252, 1992) has suggested that minimax regret suffers from the same flaw. He demonstrates the existence of “relevant” experiments that a minimax regret agent would never pay a positive cost to observe. On closer inspection, I find that minimax regret is more resilient to this critique than would first appear. In particular, there are cases in which no experiment has any value to an agent employing the minimax negative income rule, while we may always devise a hypothetical experiment for which a minimax regret agent would pay. The force of Parmigiani’s critique is further blunted by the observation that “relevant” experiments exist for which a Bayesian agent would never pay. I conclude with a discussion of pessimism in the context of minimax decision rules.

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Notes

  1. 1.

    Manski (2004, 2007), Stoye (2009, 2012)

  2. 2.

    Consistency in this context means that the estimator will converge in probability to the true parameter value. A pair comprised of a prior distribution on the parameter space, and a true value of the parameter, is consistent if that particular prior will converge in probability to that particular parameter value as the amount of available data increases. Freedman (1965) gives an example with countably many parameters where the set of consistent prior-parameter pairs is of category 1. More generally, as a consequence of well-known results in measure theory (see Parasarathy 1967, p. 29) and topology (see Schaefer 1966, p. 23), all measures on complete, separable metric spaces are tight, and hence put probability one on a countable union of compact sets. Moreover, all locally compact Hausdorff topological vector spaces are finite dimensional. This implies that all measures on an infinite-dimensional Hausdorff topological vector space assign probability one to a category 1 (“meagre”) set. Therefore, with infinitely many parameters, a Bayesian estimator is inconsistent outside an insignificant portion of the parameter space.

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Acknowledgments

I am grateful to Roy Radner for calling my attention to this problem as well as for numerous fruitful discussions. I also thank Jörg Stoye for pointing me to useful references and providing comments on an earlier version. Any errors are mine alone.

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Correspondence to Evan Sadler.

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Sadler, E. Minimax and the value of information. Theory Decis 78, 575–586 (2015). https://doi.org/10.1007/s11238-014-9442-3

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Keywords

  • Minimax regret
  • Ultrapessimism