# Fine-tuning in the context of Bayesian theory testing

## Abstract

Fine-tuning in physics and cosmology is often used as evidence that a theory is incomplete. For example, the parameters of the standard model of particle physics are “unnaturally” small (in various technical senses), which has driven much of the search for physics beyond the standard model. Of particular interest is the fine-tuning of the universe for life, which suggests that our universe’s ability to create physical life forms is improbable and in need of explanation, perhaps by a multiverse. This claim has been challenged on the grounds that the relevant probability measure cannot be justified because it cannot be normalized, and so small probabilities cannot be inferred. We show how fine-tuning can be formulated within the context of Bayesian theory testing (or *model selection*) in the physical sciences. The normalizability problem is seen to be a general problem for testing any theory with free parameters, and not a unique problem for fine-tuning. Physical theories in fact avoid such problems in one of two ways. Dimensional parameters are bounded by the Planck scale, avoiding troublesome infinities, and we are not compelled to assume that dimensionless parameters are distributed uniformly, which avoids non-normalizability.

## Keywords

Probability Bayesian Fine-tuning## Notes

### Acknowledgments

Supported by a grant from the John Templeton Foundation. This publication was made possible through the support of a grant from the John Templeton Foundation. The opinions expressed in this publication are those of the author and do not necessarily reflect the views of the John Templeton Foundation.

## References

- Albert, D.Z. (2015).
*After physics*. Cambridge: Harvard University Press.CrossRefGoogle Scholar - Barnes, L. (2012). The fine-tuning Of the universe for intelligent life.
*Publications of the Astronomical Society of Australia*,*29*, 529–564.CrossRefGoogle Scholar - Barrow, J.D., & Tipler, F.J. (1986).
*The anthropic cosmological principle*. Oxford: Clarendon Press.Google Scholar - Carr, B.J., & Rees, M.J. (1979). The anthropic principle and the structure of the physical world.
*Nature*,*278*, 605–612.CrossRefGoogle Scholar - Carter, B. (1974). Large number coincidences and the anthropic principle in cosmology. In Longair, M.S. (Ed.)
*Confrontation of cosmological theories with observational data*(pp. 291–298). Dordrecht: D. Reidel.Google Scholar - Caticha, A. (2009). Quantifying rational belief.
*AIP Conference Proceedings*,*1193*, 60–8.CrossRefGoogle Scholar - Collins, R. (2009). The teleological argument: An exploration of the fine-tuning of the universe. In Craig, W.L., & Moreland, J.P. (Eds.)
*The Blackwell companion to natural theology*. Oxford: Blackwell Publishing.Google Scholar - Colyvan, M., Garfield, J., & Priest, G. (2005). Problems with the argument from fine tuning.
*Synthese*,*145*, 325–38.CrossRefGoogle Scholar - Cox, R.T. (1946). Probability, frequency and reasonable expectation.
*American Journal of Physics*,*17*, 1–13.CrossRefGoogle Scholar - Davies, P.C.W. (1983). The anthropic principle.
*Progress in Particle and Nuclear Physics*,*10*, 1–38.CrossRefGoogle Scholar - Dine, M. (2015). Naturalness under stress. arXiv:1501.01035.
- Donoghue, J.F. (2007). The fine-tuning problems of particle physics and anthropic mechanisms. In Carr, B.J. (Ed.)
*Universe or multiverse?*(pp. 231–246). Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Glymour, C. (1980).
*Theory and evidence*. Princeton: Princeton University Press.Google Scholar - Halvorson, H. (2014). A probability problem in the fine-tuning argument, Preprint: http://philsci-archive.pitt.edu/11004/.
- Hogan, C.J. (2000). Why the universe is just so.
*Reviews of Modern Physics*,*72*, 1149–1161.CrossRefGoogle Scholar - Jaynes, E. (2003).
*Probability theory: the logic of science*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Jeffreys, H. (1946). An invariant form for the prior probability in estimation problems.
*Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences*,*186*(1007), 453–461.CrossRefGoogle Scholar - Klee, R. (2002). The revenge of Pythagoras: How a mathematical sharp practice undermines the contemporary design argument in astrophysical cosmology.
*The British Journal for the Philosophy of Science*,*53*, 331–354.CrossRefGoogle Scholar - Knuth, K.H., & Skilling, J. (2012). Foundations of inference.
*Axioms*,*1*(1), 38–73.CrossRefGoogle Scholar - Kolmogorov, A. (1933).
*Foundations of the theory of probability*. Berlin: Julius Springer.Google Scholar - Lewis, G.F., & Barnes, L.A. (2016).
*A fortunate universe: Life in a finely tuned cosmos*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Linde, A. (2015). A brief history of the multiverse. arXiv:1512.01203.
- Lineweaver, C.H., & Egan, C.A. (2007). The cosmic coincidence as a temporal selection effect produced by the age distribution of terrestrial planets in the universe.
*The Astrophysical Journal*,*671*, 853–860.CrossRefGoogle Scholar - Loewer, B. (2004). David Lewis humean theory of objective chance.
*Philosophy of Science*,*71*(5), 1115–25.CrossRefGoogle Scholar - McGrew, T., McGrew, L., & Vestrup, E. (2001). Probabilities and the fine-tuning argument: A sceptical view.
*Mind*,*110*, 1027–37.CrossRefGoogle Scholar - Monton, B. (2006). God, fine-tuning, and the problem of old evidence.
*The British Journal for the Philosophy of Science*,*57*, 405–424.CrossRefGoogle Scholar - Ramsey, F.P. (1926). Truth and probability. In Braithwaite, R.B. (Ed.)
*The foundations of mathematics and other logical essays*(pp. 156–198). London: Kegan, Paul, Trench, Trubner & Co.Google Scholar - Roberts, J.T. (2011). Fine-tuning and the infrared bull’s-eye.
*Philosophical Studies*,*160*, 287–303.CrossRefGoogle Scholar - Schellekens, A.N. (2013). Life at the interface of particle physics and string theory.
*Reviews of Modern Physics*,*85*, 1491–1540.CrossRefGoogle Scholar - Silk, J. (1977). Cosmogony and the magnitude of the dimensionless gravitational coupling constant.
*Nature*,*265*, 710–711.CrossRefGoogle Scholar - Skilling, J. (2014). Foundations and algorithms. In Hobson, M., et al. (Eds.)
*Bayesian Methods in Cosmology*. Cambridge: Cambridge University Press.Google Scholar - Swinburne, R. (2004).
*The existence of god*. Oxford: Clarendon Press.CrossRefGoogle Scholar - ’t Hooft, G. (1980). Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking. In ’t Hooft, G., et al. (Eds.)
*Proceedings of recent developments in Gauge theories of 1979 Cargese Institute*. New York: Plenum.Google Scholar - Tegmark, M., Aguirre, A., Rees, M.J., & Wilczek, F. (2006). Dimensionless constants, cosmology, and other dark matters.
*Physical Review D*,*73*, 023505.CrossRefGoogle Scholar - Weinberg, S. (1989). The cosmological constant problem.
*Reviews of Modern Physics*,*61*, 1–23.CrossRefGoogle Scholar - Wilson, K. (1979). Private communication, cited in L. Susskind.
*Physical Review D*,*2619*(1979), 20.Google Scholar - Yang, R., & Berger, J.O. (1997). A catalogue of noninformative priors, Institute of Statistics and Decision Sciences Discussion Paper, Duke University (http://www.stats.org.uk/priors/noninformative/YangBerger1998.pdf).