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Cosmological Model Selection and Akaike's Criterion

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Cosmological Model Selection and Akaike's Criterion Cosmological Model Selection and Akaike’s Criterion A thesis presented to the faculty of the College of Arts and Sciences of Ohio University In partial fulfillment of the requirements for the degree of Master of Arts Christopher S. Arledge August 2015 © 2015 Christopher S. Arledge. All Rights Reserved. 2 This thesis titled Cosmological Model Selection and Akaike’s Criterion by CHRISTOPHER S. ARLEDGE has been approved for the Department of Philosophy and the College of Arts and Sciences by Philip Ehrlich Professor of Philosophy Robert Frank Dean, College of Arts and Sciences 3 ABSTRACT ARLEDGE, CHRISTOPHER S., M.A., August 2015, Philosophy Cosmological Model Selection and Akaike’s Criterion Director of Thesis: Philip Ehrlich Contemporary cosmology is teeming with model underdetermination and cosmologists are looking for methods with which to relieve some of this underdetermination. One such method that has found its way into cosmology in recent years is the Akaike Information Criterion (AIC). The criterion is meant to select the model that loses the least amount of information in its approximation of the data, and furthermore AIC shows a preference for simplicity by containing a penalty term that penalizes models with excessive complexity. The principle aim of this paper is to investigate some of the strengths and weaknesses of AIC against two philosophical backdrops in order to determine its usefulness in cosmological model selection. The backdrops or positions against which AIC will be assessed are I) realist and II) antirealist. It will be argued that on both of these positions there is at least one feature of AIC that proves problematic for the satisfaction of the aims of the position. 4 ACKNOWLEDGEMENTS I would like to express my gratitude to Philip Ehrlich for his invaluable help during the composition of this thesis. I’d also like to thank Yoichi Ishida for his helpful comments. I would like to thank Jordan Shonberg and Ryan Ross for their help in making the thesis more readable. Finally I would like to extend a special thanks to John Norton for his willingness to be on the committee and for his insightful comments and criticisms. 5 TABLE OF CONTENTS Page Abstract……………………………………………………………………………….…..3 Acknowledgments……………………………………………….……………...………..4 1. Introduction…………………………….…………………………….……..………....6 2. Akaike Information Criterion……………………………..…….......................….......13 3. Philosophical Positions……………….……………………………………….………15 4. Limiting Features of AIC….……………..…………..…………………………..….....20 5. Conclusion…………………………………………………………………………….31 References………………………………………………………………………………..33 6 1. INTRODUCTION Contemporary physical cosmology is rife with underdetermination. What underdetermination amounts to is the claim that for some set of empirical data x there are multiple theories that can each provide a good account of x and yet each theory is equally well supported on the basis of x.1 Underdetermination is often discussed in the context of scientific theories. But cosmologists are faced with a slightly different sort of underdetermination, namely underdetermination of cosmological models. In model underdetermination, it is the various models that are built out of the foundational theories that are underdetermined and not the foundational theories themselves. So in cosmology the foundational theories of General Relativity (GR) and Quantum Mechanics (QM) are taken for granted and it is the models constructed out of these theories that face the challenge of underdetermination (Butterfield 2012, 2014). A prime example of model underdetermination in cosmology is that of dark energy modeling (which models the acceleration of the expansion factor of the universe). Presently there are no less than nine mutually incompatible dark energy models in competition with one another.2 The available evidence is insufficient to offer an empirical distinction between the models, though nothing inherent in the models inhibits future evidence from providing an empirical distinction. Another example of cosmological model underdetermination is one in which relativistic dark matter models and modified gravity models compete for !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 1 The term “account” expresses the ability of a theory T to save the phenomena with regards to a particular data set x. 2 Several of these models are already considered to be less viable than others. For instance the cosmological constant model is considered much more viable than the Dvali-Gabadadze-Porrati model, which is a model formulated on brane-world assumptions (cf. Li et al. 2010). 7 primacy in accounting for the rotation curves of spiral galaxies and other related phenomena.3 The extent to which cosmological models are underdetermined depends on the conception of underdetermination invoked. One conception of underdetermination is that of Pierre Duhem (1954), who advocates a kind of holist underdetermination. On this view a hypothesis H cannot be tested in isolation since there is always a body of auxiliary hypotheses Hn that surround H. Therefore when an experiment fails to bear out the predictions of H it need not be the case that H is falsified since it could always be one of the auxiliary hypotheses that is the troublemaker. Consider an experiment in which a telescope is used to test some prediction made by an astronomical theory. If the prediction is not born out, it does not follow that the astronomical theory has been falsified, since it could be the optical theory on which the telescope is built or some other auxiliary hypothesis that is actually the problem. Hence for any given experiment, it always remains underdetermined as to which hypothesis has actually been falsified. On another conception theories or models are underdetermined based on the evidence that is currently available, meaning that none of the present evidence can better support one of the competing theories over another. However, this conception of underdetermination does not preclude the possibility of future evidence providing better support to one of the competing theories. The theories are therefore underdetermined in practice. Proponents of this conception of underdetermination include Larry Lauden and !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 3 Of course, the juxtaposition of relativistic dark matter models and modified gravity models may ultimately result in theory underdetermination as the modified gravity models draw GR into question. Since the paper is concerned with model selection, however, the underdetermination of the foundational theories underlying these models will not be treated. 8 Jarrett Leplin (1991). Laudan and Leplin argue that because our experimental methods and our extra-empirical assumptions change with time, it is unwarranted to conclude that any two theories that are said to be empirically equivalent at time T will remain equivalent at some future time T1 (Stanford, 2013). Hence two theories might appear to be empirical equivalents at present, but in the future may be shown to be empirically disparate. The two conceptions of underdetermination presented above make universal claims that may be seen as overzealous. Accordingly on a third conception, the underdetermination of theories or models is treated on a case-by-case basis. Some theories or models will have empirical equivalents that cannot be distinguished by any possible amount of evidence. Bas van Fraassen (1980, 46-69) contrasts formulations of Newton’s theory that differ only in regards to the velocity of the solar system with regard to absolute space. Since any given constant velocity of the solar system with respect to absolute space would be observationally indistinguishable, no possible body of evidence will be able to resolve this underdetermination. On the other hand, some theories or models will be underdetermined with respect to the currently available data. Future data collection may show one theory or model accounts for the data better than its competitor(s). A fairly recent example of this is the competition between the big bang and the steady state models of the universe in the early 20th century. Initially both models accounted for the observed data (e.g. Hubble’s law). However in the 1960s, the discovery of the Cosmic Microwave Background radiation (CMB), which was predicted by the big- bang model, showed that the steady-state model could no longer account for the data 9 when the CMB is included. Prior to the 1960s the two models were considered to be empirically equivalent, but posterior to 1960 the models were shown to be empirically inequivalent, with greater support provided to the big-bang model. Whether cosmological model underdetermination is of the second or the third kind, the point is clear: cosmologists need a method (or methods) to resolve some of the underdetermination. Various proposals have been made ranging from parameter fitting to Bayesian Inference (BI) (Mukherjee and Parkinson 2008; Wandelt et al. 2013; Watkinson et al. 2012; Weinberg 2013).4 In recent years, however, cosmologists have begun to make use of a model selection criterion known as the Akaike Information Criterion (AIC) (Biesiada 2007; Godłowski and Szydłowski 2005; Li, et al. 2010; Szydłowski, et al. 2006, Tan and Biswas 2012). AIC differs from parameter estimation and BI in that it is an information-theoretic selection criterion. This means that AIC selects for models that lose the least amount of information in the approximation of the generating model (that is the data-generating
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