Abstract
There are several methods for model selection in cosmology which have at least two major goals, that of finding the correct model or predicting well. In this work, we discuss through a study of well-known model selection methods like Akaike information criterion (AIC), Bayesian information criterion (BIC), deviance information criterion (DIC) and Bayesian evidence, how these different goals are pursued in each paradigm. We also apply another method for model selection which is less seen in cosmological literature, the cross-validation method. Using these methods, we will compare two different scenarios in cosmology: \(\Lambda \)CDM model and dynamical dark energy. We show that each of the methods tends to different results in model selection. While BIC and Bayesian evidence overrule the dynamical dark energy scenarios with 2 or 3 extra degree of freedom, the DIC and cross-validation method prefer these dynamical models to \(\Lambda \)CDM model. Assuming the numerical results of different analysis and combining cosmological and statistical aspects of the subject, we propose cross-validation as an interesting method for model selection in cosmology that can lead to different results in comparison with usual methods of model selection.
Similar content being viewed by others
Data Availability Statement
This manuscript has associated data in a data repository [Authors’ comment: All observational data used in this paper are available upon request by contacting the corresponding author.]
References
A.G. Riess, A.V. Filippenko, P. Challis et al., AJ 116, 1009 (1998)
S. Perlmutter, G. Aldering, G. Goldhaber et al., ApJ 517, 565 (1999)
S.M. Carroll, V. Duvvuri, M. Trodden, M.S. Turner, Phys. Rev. D 70, 043528 (2004)
M. Kowalski, D. Rubin, G. Aldering et al., ApJ 686, 749 (2008)
S. Nojiri, S.D. Odintsov, Phys. Rep. arXiv:1011.0544 [gr-qc])
C. Bennett et al. (WMAP Collaboration) ApJS 148, 1 (2003)
H.V. Peiris et al. (WMAP), Astrophys. J. Suppl. 148, 213 (2003), arXiv:astro-ph/0302225 [astro-ph]
D. Spergel et al. (WMAP Collaboration), ApJS. 148, 175 (2003)
S. Weinberg, Rev. Mod. Phys. 61, 1 (1989)
T. Padmanabhan, Phys. Rep. 380, 235 (2003)
E.J. Copeland, M. Sami, S. Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2006)
T. Delubac et al. (BOSS), Astron. Astrophys. arXiv:1404.1801 [astro-ph.CO])
E. Macaulay, I.K. Wehus, H.K. Eriksen, Phys. Rev. Lett. arXiv:1303.6583 [astro-ph.CO])
W.L. Freedman, Nat. Astron. arXiv:1706.02739 [astro-ph.CO])
N. Aghanim et al. (Planck) (2018). arXiv:1807.06209 [astro-ph.CO]
A.G. Riess, S. Casertano, W. Yuan, L.M. Macri, D. Scolnic, Astrophys. J. arXiv:1903.07603 [astro-ph.CO])
M. Rezaei, M. Malekjani, Phys. Rev. D 96, 063519 (2017)
T. Yang, A. Banerjee, E. Colgáin, (2019), arXiv:1911.01681 [astro-ph.CO]
N. Khadka, B. Ratra (2019). arXiv:1909.01400 [astro-ph.CO]
E. Lusso, E. Piedipalumbo, G. Risaliti, M. Paolillo, S. Bisogni, E. Nardini, L. Amati, Astron. Astrophys. arXiv:1907.07692 [astro-ph.CO])
M. Benetti, S. Capozziello, JCAP arXiv:1910.09975 [astro-ph.CO])
G. Veneziano, Nucl. Phys. B 159, 213 (1979)
J.K. Erickson, R. Caldwell, P.J. Steinhardt, C. Armendariz-Picon, V.F. Mukhanov, Phys. Rev. Lett. 88, 121301 (2002)
S. Thomas, Phys. Rev. Lett. 89, 081301 (2002)
C. Armendariz-Picon, V.F. Mukhanov, P.J. Steinhardt, Phys. Rev. D arXiv:0006373 [astroph])
R.R. Caldwell, Phys. Lett. B 545, 23 (2002)
T. Padmanabhan, Phys. Rev. D 66, 021301 (2002)
M. Gasperini, F.P.G. Veneziano, Phys. Rev. D 65, 023508 (2002)
E. Elizalde, S. Nojiri, S.D. Odintsov, Phys. Rev. D arXiv:0405034 [hep-th])
A. Gomez-Valent, J. Sola, Mon. Not. R. Astron. Soc. arXiv:1412.3785 [astro-ph.CO])
Lloyd Knox, Marius Millea, Phys. Rev. D 101(4), 043533 (2020)
H. Akaike, IEEE Trans. Autom. Control 19, 716 (1974)
G. Schwarz, Ann. Stat. 6, 461 (1978)
D.J. Spiegelhalter, N.G. Best, B.P. Carlin, A. Van Der Linde, J. R. Stat. Soc. B64(4), 583 (2002)
Andrew R. Liddle, Mon. Not. R. Astron. Soc. 377, L74–L78 (2007)
M. Trashorras, S. Nesseris, J.G. Bellido, Phys. Rev. D 94, 063511 (2016)
M. Malekjani, S. Basilakos, Z. Davari, A. Mehrabi, M. Rezaei, Mon. Not. R. Astron. Soc. arXiv:1609.01998 [astro-ph.CO])
M. Rezaei, M. Malekjani, S. Basilakos, A. Mehrabi, D.F. Mota, Astrophys. J. arXiv:1706.02537 [astro-ph.CO])
M. Malekjani, M. Rezaei, I.A. Akhlaghi, Phys. Rev. D arXiv:1809.08792 [gr-qc])
M. Rezaei, Mon. Not. R. Astron. Soc. arXiv:1902.04776 [gr-qc])
W. Lin, K.J. Mack, L. Hou (2019). arXiv:1910.02978 [astro-ph.CO]
M. Rezaei, M. Malekjani, J. Sola, Phys. Rev. D arXiv:1905.00100 [gr-qc])
M. Rezaei, T. Naderi, M. Malekjani, A. Mehrabi, Eur. Phys. J. C arXiv:2004.08168 [astro- ph.CO])
M. Rezaei, S.P. Ojaghi, M. Malekjani, Astrophys. J. arXiv:2008.03092 [astro-ph.CO])
R.D. Cousins, Phys. Rev. Lett. arXiv:0807.1330 [hep-ph])
R.D. Cousins, Synthese arXiv:1310.3791 [physics.data-an])
G. Efstathiou, Mon. Not. R. Astron. Soc. arXiv:0802.3185 [astro-ph])
S. Gariazzo, EPJC 80, 552 (2020). arXiv:1910.06646
B. Dabbs, B. Junker, arXiv preprint arXiv:1605.03000 (2016)
J. Aldrich, Int. Stat. Rev. 66(1), 61 (2007)
W. Ralph, The combinatorics of occam’s razor (2015). arXiv:1504.07441 [math.CO]
R. Shibata, Biometrika 71, 43 (1984)
J. Shao, Stat. Sin. 7, 221 (1997)
R. Dutta, M. Bogdan, J.K. Ghosh, Model selection and multiple testing—a Bayesian and empirical Bayes overview and some new results (2015), arXiv:1510.00547 [math.ST]
A. Kurek, M. Szydlowski, Astrophys. J. arXiv:astro-ph/0702484 [astro-ph])
F. Arevalo, A. Cid, J. Moya, Eur. Phys. J. C arXiv:1610.09330 [astro-ph.CO])
R. Trotta (2017) arXiv:1701.01467 [astro-ph.CO]
M.C. March, G.D. Starkman, R. Trotta, P.M. Vaudrevange, Mon. Not. R. Astron. Soc. arXiv:1005.3655 [astro-ph.CO])
S. Wager, Cross-validation, risk estimation, and model selection (2019). arXiv:1909.11696 [stat.ME]
B. Ghojogh, M. Crowley, The theory behind overfitting, cross validation, regularization, bagging, and boosting: tutorial (2019). arXiv:1905.12787 [stat.ML]
S. Arlot, A. Celisse, A survey of cross-validation procedures for model selection. Stat. Surv. 4, 40–79 (2010)
A. Vehtari, A. Gelman, J. Gabry, Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC, arXiv:1507.04544
R. Kohavi, A study of cross-validation and bootstrap for accuracy estimation and model selection, in Proceedings of the 14th International Joint Conference on Artificial Intelligence, Vol. 2, Canada (1995)
Zhang et al., Cross-validation for selecting a model selection procedure. J. Econom. 187(1), 95 (2015)
A. Mehrabi, S. Basilakos, F. Pace, MNRAS arXiv:1504.01262 [astro-ph.CO])
I. Maor, R. Brustein, P.J. Steinhardt, Phys. Rev. Lett. 86, 6 (2001). [Erratum: Phys. Rev. Lett.87,049901(2001)], arXiv:astro-ph/0007297 [astro-ph]
A.G. Riess et al. (Supernova Search Team), ApJ 607, 665 (2004)
B.A. Bassett, M. Brownstone, A. Cardoso, M. Cortes, Y. Fantaye, R. Hlozek, J. Kotze, P. Okouma, JCAP arXiv:0709.0526 [astro-ph])
G. Efstathiou, Mon. Not. R. Astron. Soc. arXiv:astro-ph/9904356 [astro-ph])
H.K. Jassal, J.S. Bagla, T. Padmanabhan, Mon. Not. R. Astron. Soc. arXiv:astroph/0404378 [astro-ph])
E.M. Barboza, J.S. Alcaniz, Z.H. Zhu, R. Silva, Phys. Rev. D arXiv:0905.4052 [astro-ph.CO])
H. Pade, Ann. Sci. Ecole Norm. Sup. 9(3), 1 (1892)
A. Baker, P. Graves-Morris, Pade Approximants (Cambridge University Press, Cambridge, 1996).
M. Adachi, M. Kasai, Prog. Theor. Phys. arXiv:1111.6396 [astro-ph.CO])
M. Rezaei, Mon. Not. R. Astron. Soc. arXiv:1904.02785 [gr-qc])
D.M. Scolnic et al., Astrophys. J. arXiv:1710.00845 [astro-ph.CO])
O. Farooq, F.R. Madiyar, S. Crandall, B. Ratra, Astrophys. J. arXiv:1607.03537 [astro-ph.CO])
S. Alam et al., BOSS. Mon. Not. R. Astron. Soc. 470, 2617 (2017)
J.E. Bautista et al., Astron. Astrophys. 603, A12 (2017)
F. Beutler, C. Blake, M. Colless, D.H. Jones, L.S. Smith et al., MNRAS 416, 3017 (2011)
A.J. Ross et al., Mon. Not. R. Astron. Soc. 449, 835 (2015)
H. Gil-Marn et al., Mon. Not. R. Astron. Soc. 477(2), 1604 (2018)
T.M.C. Abbott et al. (DES), Mon. Not. R. Astron. Soc. 483, 4866 (2019)
A.R. Liddle, Information criteria for astrophysical model selection. Mon. Not. R. Astron. Soc. 377, L74 (2007)
F.K. Anagnostopoulos, S. Basilakos, E.N. Saridakis, Observational constraints on Myrzakulov gravity, arXiv:2012.06524 [gr-qc]
H. Jeffreys, Theory of Probability, ed. by H. Jeffreys (Oxford University Press, Cambridge, 1961)
S. Nesseris, J. Garcia-Bellido, JCAP 1308, 036 (2013)
R.E. Kass, A.E. Raftery, J. Am. Stat. Assoc. 90, 773 (1995)
S. Pan, W. Yang, E. Di Valentino, A. Shafieloo, S. Chakraborty (2019). arXiv:1907.12551 [astro-ph.CO]
Acknowledgements
The authors gratefully thank the Referee for the constructive comments and recommendations which definitely help to improve the readability and quality of the paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rezaei, M., Malekjani, M. Comparison between different methods of model selection in cosmology. Eur. Phys. J. Plus 136, 219 (2021). https://doi.org/10.1140/epjp/s13360-021-01200-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/s13360-021-01200-w