Climatic Change

, Volume 113, Issue 2, pp 163–179 | Cite as

Solution to the paradox of climate sensitivity

  • Salvador PueyoEmail author


Most countries endorse a limit of either 2°C or 1.5°C global warming above pre-industrial levels. However, for several reasons, there is still a significant uncertainty in the climate sensitivity parameter, which relates greenhouse gas concentration (or other forcings) to steady-state temperature. One key source of uncertainty is the disagreement about the appropriate prior for Bayesian estimation. A common choice is the uniform distribution, often thought to contain no information. However, when used to estimate sensitivity it leads to paradoxical results, which have been interpreted as revealing an inherent indeterminacy in the prior of choice. If this were the case, part of the uncertainty would be irreducible. Here I develop an objective Bayesian approach to this problem. I show that both Jaynes’ invariant groups criterion and a new criterion based on information theory lead to the conclusion that there is a uniquely defined non-informative prior of climate sensitivity, which is distinct from the uniform and solves the paradox. This prior distribution is the log-uniform. Furthermore, this result is supported empirically by the observation that other comparable non-equilibrium parameters display a scale-invariant, log-uniform-like frequency distribution. Rather than advocating a direct use of this prior, I recommend to refine it with a limited use of expert elicitation or other methods. A sound prior is a key ingredient in the process to reach a consensus low-uncertainty estimate of climate sensitivity to inform climate policy.


Electronic Supplementary Material Probability Density Function Mutual Information Prior Distribution Climate Sensitivity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



I am grateful for the useful comments by J. Ballester, X. Rodó, M. Oppenheimer, G. Yohe and three anonymous referees. I thank Bill Shipley for calling my attention on the paper by Baker and Christakos (2007).

Supplementary material

10584_2011_328_MOESM1_ESM.pdf (243 kb)
ESM 1 (PDF 242 kb)


  1. Allen MR, Frame DJ (2007) Call off the quest. Science 318:582–583CrossRefGoogle Scholar
  2. Andronova NG, Schlesinger ME (2001) Objective estimation of the probability density function for climate sensitivity. J Geophys Res-Atmos 106:22605–22611CrossRefGoogle Scholar
  3. Annan JD, Hargreaves JC (2006) Using multiple observationally-based constraints to estimate climate sensitivity. Geophys Res Lett 33:L06704CrossRefGoogle Scholar
  4. Annan JD, Hargreaves JC (2011) On the generation and interpretation of probabilistic estimates of climate sensitivity. Clim Chang 104:423–436CrossRefGoogle Scholar
  5. Arrhenius S (1896) On the influence of carbonic acid in the air upon the temperature of the ground. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 4:237–277Google Scholar
  6. Baker R, Christakos G (2007) Revisiting prior distributions, part I: priors based on a physical invariance principle. Stoch Environ Res Ris Assess 21:427–434CrossRefGoogle Scholar
  7. Bertrand J (1889) Calcul des Probabilités. Gauthier-Villars, ParisGoogle Scholar
  8. Birnbaum A (1962) On the foundations of statistical inference. J Am Stat Assoc 57:269–306Google Scholar
  9. Bony S, Colman R, Kattsov VM, Allan RP, Bretherton CS, Dufresne JL, Hall A, Hallegatte S, Holland MM, Ingram W, Randall DA, Soden BJ, Tselioudis G, Webb MJ (2006) How well do we understand and evaluate climate change feedback processes? J Climate 19:3445–3482CrossRefGoogle Scholar
  10. Charney JG, Arakawa A, Baker DJ, Bolin B, Dickinson RE, Goody RM, Leith CE, Stommel HM, Wunsch CI (1979) Carbon dioxide and climate: a scientific assessment. National Academy of Sciences, Washington DCGoogle Scholar
  11. Crucifix M (2006) Does the last glacial maximum constrain climate sensitivity? Geophys Res Lett 33:L18701CrossRefGoogle Scholar
  12. Fienberg SE (2006) When did Bayesian inference become “Bayesian”? Bayesian Anal 1:1–40CrossRefGoogle Scholar
  13. Forest CE, Stone PH, Sokolov AP, Allen MR, Webster MD (2002) Quantifying uncertainties in climate system properties with the use of recent climate observations. Science 295:113–117CrossRefGoogle Scholar
  14. Forest CE, Stone PH, Sokolov AP (2006) Estimated PDFs of climate system properties including natural and anthropogenic forcings. Geophys Res Lett 33:L01705CrossRefGoogle Scholar
  15. Forster PMF, Gregory JM (2006) The climate sensitivity and its components diagnosed from Earth radiation budget data. J Climate 19:39–52CrossRefGoogle Scholar
  16. Frame DJ, Booth BBB, Kettleborough JA, Stainforth DA, Gregory JM, Collins M, Allen MR (2005) Constraining climate forecasts: the role of prior assumptions. Geophys Res Lett 32:L09702CrossRefGoogle Scholar
  17. Frame DJ, Faull NE, Joshi MM, Allen MR (2007) Probabilistic climate forecasts and inductive problems. Phil Trans R Soc A 365:1971–1992CrossRefGoogle Scholar
  18. Frame DJ, Aina T, Christensen CM, Faull NE, Knight SHE, Piani C, Rosier SM, Yamazaki K, Yamazaki Y, Allen MR (2009) The BBC climate change experiment: design of the coupled model ensemble. Phil Trans R Soc A 367:855–870CrossRefGoogle Scholar
  19. Garthwaite PH, Kadane JB, O’Hagan A (2005) Statistical methods for eliciting probability distributions. J Am Stat Assoc 100:680–700CrossRefGoogle Scholar
  20. Grandison S, Morris RJ (2008) Biological pathway kinetic rate constants are scale-invariant. Bioinformatics 24:741–743CrossRefGoogle Scholar
  21. Hansen J, Russell G, Lacis A, Fung I, Rind D (1985) Climate response times: dependence on climate sensitivity and ocean mixing. Science 229:857–859CrossRefGoogle Scholar
  22. Hansen J, Sato M, Kharecha P, Beerling D, Berner R, Masson-Delmotte V, Pagani M, Raymo M, Royer DL, Zachos JC (2008) Target atmospheric CO2: where should humanity aim? Open Atmos Sci 2:217–231CrossRefGoogle Scholar
  23. Hegerl GC, Crowley T, Hyde WT, Frame D (2006) Constraints on climate sensitivity from temperature reconstructions of the past seven centuries. Nature 440:1029–1032CrossRefGoogle Scholar
  24. Hegerl GC, Zwiers FW, Braconnot P, Gillett NP, Luo Y, Marengo Orsini JA, Nicholls N, Penner JE, Stott PA (2007) Understanding and attributing climate change. In: Solomon S, Qin D, Manning M, Chen Z, Marquis M, Averyt KB, Tignor M, Miller HL (eds) Climate change 2007: the physical science basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press, Cambridge, pp 663–745Google Scholar
  25. Henriksson SV, Arjas E, Laine M, Tamminen J, Laaksonen A (2010) Comment on “Using multiple observationally-based constraints to estimate climate sensitivity” by J. D. Annan and J. C. Hargreaves, Geophys. Res. Lett., 2006. Clim Past 6:411–414CrossRefGoogle Scholar
  26. Jaynes ET (1957) Information theory and statistical mechanics. Phys Rev 106:620–630CrossRefGoogle Scholar
  27. Jaynes ET (1968) Prior probabilities. IEEE T Syst Sci Cyb SSC-4:227–241CrossRefGoogle Scholar
  28. Jaynes ET (1973) The well-posed problem. Found Phys 3:477–493CrossRefGoogle Scholar
  29. Jaynes ET (2003) Probability theory: the logic of science. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  30. Jeffreys H (1939) Theory of probability. Clarendon, OxfordGoogle Scholar
  31. Jeffreys H (1946) An invariant form for the prior probability in estimation problems. P Roy Soc Lond A 186:453–461CrossRefGoogle Scholar
  32. Jewson S, Rowlands D, Allen M (2009) A new method for making objective probabilistic climate forecasts from numerical climate models based on Jeffreys’ prior. arXiv:0908.4207Google Scholar
  33. Jewson S, Rowlands D, Allen M (2010a) Objective probabilistic forecasts of future climate based on Jeffreys’ prior: The case of correlated observables. arXiv:1005.2354Google Scholar
  34. Jewson S, Rowlands D, Allen M (2010b) Objective climate model predictions using Jeffreys’ prior: The general multivariate normal case. arXiv:1005.3907Google Scholar
  35. Kass RE, Wasserman L (1996) The selection of prior distributions by formal rules. J Am Stat Assoc 91:1343–1370, Correction in J Am Stat Assoc 93:412Google Scholar
  36. Knutti R, Hegerl GC (2008) The equilibrium sensitivity of the Earth’s temperature to radiation changes. Nat Geosci 1:735–743CrossRefGoogle Scholar
  37. Knutti R, Stocker TF, Joos F, Plattner GK (2002) Constraints on radiative forcing and future climate change from observations and climate model ensembles. Nature 416:719–723CrossRefGoogle Scholar
  38. Lebon G, Jou D, Casas-Vázquez J (2008) Understanding non-equilibrium thermodynamics. Springer, BerlinCrossRefGoogle Scholar
  39. Lemoine DM (2010) Climate sensitivity distributions depend on the possibility that models share biases. J Climate 23:4395–4415, Correction in J Climate 24:962–962CrossRefGoogle Scholar
  40. Lide DR (ed) (2009) CRC handbook of chemistry and physics, 90th ed. CRC Press, Boca RatonGoogle Scholar
  41. Matthews HD, Gillett NP, Stott PA, Zickfeld K (2009) The proportionality of global warming to cumulative carbon emissions. Nature 459:829–832CrossRefGoogle Scholar
  42. Meinshausen M, Meinshausen N, Hare W, Raper SCB, Frieler K, Knutti R, Frame DJ, Allen MR (2009) Greenhouse-gas emission targets for limiting global warming to 2°C. Nature 458:1158–1162CrossRefGoogle Scholar
  43. Murphy J, Sexton DMH, Barnett DN, Jones GS, Webb MJ, Collins M, Stainforth DA (2004) Quantification of modelling uncertainties in a large ensemble of climate change simulations. Nature 430:768–772CrossRefGoogle Scholar
  44. National Research Council (2010) Climate stabilization targets: emissions, concentrations, and impacts over decades to millennia. National Academies Press, Washington DCGoogle Scholar
  45. Oppenheimer M, O’Neill BC, Webster M (2008) Negative learning. Clim Chang 89:155–172CrossRefGoogle Scholar
  46. Piani C, Frame DJ, Stainforth DA, Allen MR (2005) Constraints on climate change from a multi-thousand member ensemble of simulations. Geophys Res Lett 32:L23825CrossRefGoogle Scholar
  47. Preston FW (1948) The commonness, and rarity, of species. Ecology 29:254–283CrossRefGoogle Scholar
  48. Pueyo S (2007) Self-organised criticality and the response of wildland fires to climate change. Clim Chang 82:131–161CrossRefGoogle Scholar
  49. Pueyo S, He F, Zillio T (2007) The maximum entropy formalism and the idiosyncratic theory of biodiversity. Ecol Lett 10:1017–1028CrossRefGoogle Scholar
  50. Ramaswamy V, Boucher O, Haigh J, Hauglustaine D, Haywood J, Myhre G, Nakajima T, Shi GY, Solomon S (2001) Radiative forcing of climate change. In: Houghton JT, Ding Y, Griggs DJ, Noguer M, van der Linden PJ, Dai X, Maskell K, Johnson CA (eds) Climate change 2001: the scientific basis. Contribution of Working Group I to the Third Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press, Cambridge, pp 349–416Google Scholar
  51. Roe GH, Baker MB (2007) Why is climate sensitivity so unpredictable? Science 318:629–632CrossRefGoogle Scholar
  52. Roe GH, Baker MB (2011) Comment on “Another look at climate sensitivity” by Zaliapin and Ghil (2010). Nonlin Process Geophys 18:125–127CrossRefGoogle Scholar
  53. Rogelj J, Nabel J, Chen C, Hare W, Markmann K, Meinshausen M, Schaeffer M, Macey K, Höhne N (2010) Copenhagen Accord pledges are paltry. Nature 464:1126–1128CrossRefGoogle Scholar
  54. Rougier J (2007) Probabilistic inference for future climate using an ensemble of climate model evaluations. Clim Chang 81:247–264CrossRefGoogle Scholar
  55. Rougier J, Sexton DMH (2007) Inference in ensemble experiments. Philos T Roy Soc A 365:2133–2143CrossRefGoogle Scholar
  56. Royer DL, Berner RA, Park J (2007) Climate sensitivity constrained by CO2 concentrations over the past 420 million years. Nature 446:530–532CrossRefGoogle Scholar
  57. Sanderson BM, Shell KM, Ingram W (2010) Climate feedbacks determined using radiative kernels in a multi-thousand member ensemble of AOGCMs. Clim Dyn 35:1219–1236CrossRefGoogle Scholar
  58. Sansó B, Forest CE (2009) Statistical calibration of climate system properties. Appl Statist 58:485–503Google Scholar
  59. Schneider SH, Mastrandrea MD (2005) Probabilistic assessment of “dangerous” climate change and emissions pathways. P Natl Acad Sci USA 102:15728–15735CrossRefGoogle Scholar
  60. Shannon CE (1948) A mathematical theory of communication. AT&T Tech J 27(379–423):623–656Google Scholar
  61. Tomassini L, Reichert P, Knutti R, Stocker TF, Borsuk ME (2007) Robust Bayesian uncertainty analysis of climate system properties using Markov chain Monte Carlo methods. J Climate 20:1239–1254CrossRefGoogle Scholar
  62. Tversky A, Koehler DJ (1994) Support theory: a nonextensional representation of subjective probability. Psychol Rev 101:547–567CrossRefGoogle Scholar
  63. UNFCCC (2011) Decision 1/CP.16. United Nations Framework Convention on Climate ChangeGoogle Scholar
  64. Urban NM, Keller K (2009) Complementary observational constraints on climate sensitivity. Geophys Res Lett 36:L04708CrossRefGoogle Scholar
  65. van der Sluijs J, van Eijndhoven J, Shackley S, Wynne B (1998) Anchoring devices in science for policy: the case of consensus around climate sensitivity. Soc Studies Sci 28:291–323CrossRefGoogle Scholar
  66. Webster MD, Sokolov AP (2000) A methodology for quantifying uncertainty in climate projections. Clim Chang 46:417–446CrossRefGoogle Scholar
  67. Weitzman ML (2009) On modeling and interpreting the economics of catastrophic climate change. Rev Econ Stat 91:1–19CrossRefGoogle Scholar
  68. Zaliapin I, Ghil M (2010) Another look at climate sensitivity. Nonlin Process Geophys 17:113–122CrossRefGoogle Scholar
  69. Zaliapin I, Ghil M (2011) Reply to Roe and Baker’s comment on “Another look at climate sensitivity” by Zaliapin and Ghil (2010). Nonlin Process Geophys 18:129–131CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Institut Català de Ciències del Clima (IC3)BarcelonaSpain

Personalised recommendations