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2007b)How Simplicity Helps You Find the Truth Without Pointing at it
 Philosophy of Mathematics and Induction
"... It seems that a fixed bias toward simplicity should help one find the truth, since scientific theorizing is guided by such a bias. But it also seems that a fixed bias toward simplicity cannot indicate or point at the truth, since an indicator has to be sensitive to what it indicates. I argue that bo ..."
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It seems that a fixed bias toward simplicity should help one find the truth, since scientific theorizing is guided by such a bias. But it also seems that a fixed bias toward simplicity cannot indicate or point at the truth, since an indicator has to be sensitive to what it indicates. I argue that both views are correct. It is demonstrated, for a broad range of cases, that the Ockham strategy of favoring the simplest hypothesis, together with the strategy of never dropping the simplest hypothesis until it is no longer simplest, uniquely minimizes reversals of opinion and the times at which the reversals occur prior to convergence to the truth. Thus, simplicity guides one down the straightest path to the truth, even though that path may involve twists and turns along the way. The proof does not appeal to prior probabilities biased toward simplicity. Instead, it is based upon minimization of worstcase cost bounds over complexity classes of possibilities. 0.1 The Simplicity Puzzle There are infinitely many alternative hypotheses consistent with any finite amount of experience, so how is one entitled to choose among them? Scientists boldly respond with appeals to “Ockham’s razor”, which selects the “simplest ” hypothesis among them,
Ockham’s Razor, Empirical Complexity, and Truthfinding Efficiency
 THEORETICAL COMPUTER SCIENCE
, 2007
"... The nature of empirical simplicity and its relationship to scientific truth are longstanding puzzles. In this paper, empirical simplicity is explicated in terms of empirical effects, which are defined in terms of the structure of the inference problem addressed. Problem instances are classified acc ..."
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Cited by 7 (7 self)
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The nature of empirical simplicity and its relationship to scientific truth are longstanding puzzles. In this paper, empirical simplicity is explicated in terms of empirical effects, which are defined in terms of the structure of the inference problem addressed. Problem instances are classified according to the number of empirical effects they present. Simple answers are satisfied by simple worlds. An efficient solution achieves optimum worstcase cost over each complexity class with respect to such costs such as the number of retractions or errors prior to convergence and elapsed time to convergence. It is shown that always choosing the simplest theory compatible with experience and hanging onto it while it remains simplest is both necessary and sufficient for efficiency.
Mind change efficient learning
 Info. & Comp
, 2005
"... Abstract. This paper studies efficient learning with respect to mind changes. Our starting point is the idea that a learner that is efficient with respect to mind changes minimizes mind changes not only globally in the entire learning problem, but also locally in subproblems after receiving some evi ..."
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Abstract. This paper studies efficient learning with respect to mind changes. Our starting point is the idea that a learner that is efficient with respect to mind changes minimizes mind changes not only globally in the entire learning problem, but also locally in subproblems after receiving some evidence. Formalizing this idea leads to the notion of uniform mind change optimality. We characterize the structure of language classes that can be identified with at most α mind changes by some learner (not necessarily effective): A language class L is identifiable with α mind changes iff the accumulation order of L is at most α. Accumulation order is a classic concept from pointset topology. To aid the construction of learning algorithms, we show that the characteristic property of uniformly mind change optimal learners is that they output conjectures (languages) with maximal accumulation order. We illustrate the theory by describing mind change optimal learners for various problems such as identifying linear subspaces and onevariable patterns. 1
Ockham’s Razor, Truth, and Information
, 2007
"... In science, one faces the problem of selecting the true theory from a range of alternative theories. The typical response is to select the simplest theory compatible with available evidence, on the authority of “Ockham’s Razor”. But how can a fixed bias toward simplicity help one find possibly compl ..."
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Cited by 2 (0 self)
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In science, one faces the problem of selecting the true theory from a range of alternative theories. The typical response is to select the simplest theory compatible with available evidence, on the authority of “Ockham’s Razor”. But how can a fixed bias toward simplicity help one find possibly complex truths? A short survey of standard answers to this question reveals them to be either wishful, circular, or irrelevant. A new explanation is presented, based on minimizing the reversals of opinion prior to convergence to the truth. According to this alternative approach, Ockham’s razor does not inform one which theory is true but is, nonetheless, the uniquely most efficient strategy for arriving at the true theory, where efficiency is a matter of minimizing reversals of opinion prior to finding the true theory. 1
Simplicity, Truth, and the Unending Game of Science
, 2005
"... This paper presents a new explanation of how preferring the simplest theory compatible with experience assists one in finding the true answer to a scientific question when the answers are theories or models. Science is portrayed as an infinite game between science and nature. Simplicity is a structu ..."
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This paper presents a new explanation of how preferring the simplest theory compatible with experience assists one in finding the true answer to a scientific question when the answers are theories or models. Science is portrayed as an infinite game between science and nature. Simplicity is a structural invariant reflecting sequences of theory choices nature could force the scientist to produce. It is demonstrated that among the methods that converge to the truth in an empirical problem, the ones that do so with a minimum number of reversals of opinion prior to convergence are exactly the ones that prefer simple theories. The idea explains not only simplicity tastes in model selection, but aspects of theory testing and the unwillingness of natural science to break symmetries without a reason. In natural science, one typically faces a situation in which several (or even infinitely many) available theories are compatible with experience. Standard practice is to choose the simplest theory among them and to cite “Ockham’s razor ” as the excuse (figure
(will be inserted by the editor) Ockham Efficiency Theorem for Stochastic Empirical Methods
, 2010
"... the date of receipt and acceptance should be inserted later Abstract Ockham’s razor is the principle that, all other things being equal, scientists ought to prefer simpler theories. In recent years, philosophers have argued that simpler theories make better predictions, possess theoretical virtues l ..."
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the date of receipt and acceptance should be inserted later Abstract Ockham’s razor is the principle that, all other things being equal, scientists ought to prefer simpler theories. In recent years, philosophers have argued that simpler theories make better predictions, possess theoretical virtues like explanatory power, and have other pragmatic virtues like computational tractability. However, such arguments fail to explain how and why a preference for simplicity can help one find true theories in scientific inquiry, unless one already assumes that the truth is simple. One new solution to that problem is the Ockham efficiency theorem (Kelly 2002, 2004, 2007ad, Kelly and Glymour 2004), which states that scientists who heed Ockham’s razor retract their opinions less often and sooner than do their nonOckham competitors. The theorem neglects, however, to consider competitors following random (“mixed”) strategies and in many applications random strategies are known to achieve better worstcase loss than deterministic strategies. In this paper, we describe two ways to extend the result to a very general class of random, empirical strategies. The first extension concerns expected retractions, retraction times, and errors and the
Efficiency
, 2007
"... The nature of empirical simplicity and its relationship to scientific truth are longstanding puzzles. In this paper, empirical simplicity is explicated in terms of empirical effects, which are defined in terms of the structure of the inference problem addressed. Problem instances are classified acc ..."
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The nature of empirical simplicity and its relationship to scientific truth are longstanding puzzles. In this paper, empirical simplicity is explicated in terms of empirical effects, which are defined in terms of the structure of the inference problem addressed. Problem instances are classified according to the number of empirical effects they present. Simple answers are satisfied by simple worlds. An efficient solution achieves optimum worstcase cost over each complexity class with respect to such costs such as the number of retractions or errors prior to convergence and elapsed time to convergence. It is shown that always choosing the simplest theory compatible with experience and hanging onto it while it remains simplest is both necessary and sufficient for efficiency. 1 The Simplicity Puzzle Machine learning, statistics, and the philosophy of science all recommend the selection of simple theories or models on the basis of empirical data, where simplicity has something to do with minimizing independent entities, principles, causes, or equational
Noname manuscript No. (will be inserted by the editor) Ockham Efficiency Theorem for Stochastic Empirical Methods
"... the date of receipt and acceptance should be inserted later Abstract Ockham’s razor is the principle that, all other things being equal, scientists ought to prefer simpler theories. In recent years, philosophers have argued that simpler theories make better predictions, possess theoretical virtues l ..."
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the date of receipt and acceptance should be inserted later Abstract Ockham’s razor is the principle that, all other things being equal, scientists ought to prefer simpler theories. In recent years, philosophers have argued that simpler theories make better predictions, possess theoretical virtues like explanatory power, and have other pragmatic virtues like computational tractability. However, such arguments fail to explain how and why a preference for simplicity can help one find true theories in scientific inquiry, unless one already assumes that the truth is simple. One new solution to that problem is the Ockham efficiency theorem (Kelly 2002, 2004, 2007ad, Kelly and Glymour 2004), which states that scientists who heed Ockham’s razor retract their opinions less often and sooner than do their nonOckham competitors. The theorem neglects, however, to consider competitors following random (“mixed”) strategies and in many applications random strategies are known to achieve better worstcase loss than deterministic strategies. In this paper, we describe two ways to extend the result to a very general class of random, empirical strategies. The first extension concerns expected retractions, retraction times, and errors and the second extension concerns retractions in chance, times of retractions in chance, and chances of errors.
Argument, Inquiry, and the Unity of Science
, 2008
"... Ockham’s razor impels scientists to seek ever greater unity in nature. That seems to saddle science with a metaphysical presupposition of simplicity that might be false. The objection is apt if scientific method is understood as a system of inductive logic or proof, for then the unity of science mus ..."
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Ockham’s razor impels scientists to seek ever greater unity in nature. That seems to saddle science with a metaphysical presupposition of simplicity that might be false. The objection is apt if scientific method is understood as a system of inductive logic or proof, for then the unity of science must, somehow, function as an unjustified premise in scientific arguments. But if science is understood, instead, primarily as a process of discovery that aims at finding the truth as efficiently as possible, the unity of science can be understood as an optimally truthconducive heuristic rather than as a metaphysical presupposition. Optimal truth conduciveness is what epistemic justification is for. Therefore, Ockham’s razor is justified as a scientific heuristic even though it might be false. 1 Ockam’s Razor and the Unity of Science In the Preface to his De Revolutionibus, Nikolaus Copernicus did not cite any new or crucial experiments in favor of his heliocentric astronomical hypothesis: his argument was based squarely on what he called the harmonies of his system. What he had in mind is the fact that his theory is severely tested by data that Ptolemy’s theory merely accommodates; for example, heliocentrism entails that planetary retrograde motion must happen either at solar conjunction or at solar opposition, whereas in Ptolemy’s theory retrograde motion is entirely independent of solar position. Fresnel’s argument for the wave theory of light centered on the ability of the theory to provide a unified explanation of diffraction bands around shadows and of the rings that appear when lenses are pressed together, which are qualitatively completely different. Each phenomenon allows one to derive the wave lengths of the various colors of light, which yields a sharp, testable prediction with respect to the other phenomena. Universal gravitation allowed Newton to estimate the gravitational constant from terrestrial pendula and then test the theory against the moon’s orbit. Maxwell’s electromagnetic equations unified magnetic and electrical forces. Darwin’s theory of evolution uses common ancestry to explain homologies or similarities of structure across diverse environments.
Ockham’s Razor, Hume’s Problem, Ellsberg’s Paradox, Dilation, and Optimal Truth Conduciveness
, 2008
"... Classical Bayesianism represents ignorance, if at all, by flatness of prior probabilities. Such probabilities are an essential part of the standard Bayesian explanation of Ockham’s razor. But flatness as a model of ignorance is called into question by Ellsberg’s paradox, which has led to the conside ..."
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Classical Bayesianism represents ignorance, if at all, by flatness of prior probabilities. Such probabilities are an essential part of the standard Bayesian explanation of Ockham’s razor. But flatness as a model of ignorance is called into question by Ellsberg’s paradox, which has led to the consideration of incoherent or inexact degrees of belief, both of which undermine the usual explanation of Ockham’s razor. An alternative explanation of Ockham’s razor is presented, according to which always favoring the uniquely simplest theory compatible with experience keeps one on the shortest or most direct path to the truth. It turns out that minimization of total distance to the truth implies coherent degrees of belief strongly biased toward simplicity. If one focuses on retractions or drops in credence, then a more reasonably moderate bias toward simplicity results but optimal efficiency then demands either incoherence or inexact probabilities, both of which are solutions to Ellsberg’s paradox. Finally, it turns out that dilation, or increasing imprecision in light of new information, is necessary if agents with inexact probabilities are to minimize total retractions. So, in place of paradox and tension, one obtains a unified perspective on Ockham’s razor, Ellsberg’s paradox, dilation, and the justification of inductive inference. 1