Population Ecology

, Volume 58, Issue 1, pp 45–52 | Cite as

Bayes estimates as an approximation to maximum likelihood estimates

Special Feature: Original Article Bayesian, Fisherian, error, and evidential statistical approaches for population ecology

Abstract

Ronald A. Fisher, who is the founder of maximum likelihood estimation (ML estimation), criticized the Bayes estimation of using a uniform prior distribution, because we can create estimates arbitrarily if we use Bayes estimation by changing the transformation used before the analysis. Thus, the Bayes estimates lack the scientific objectivity, especially when the amount of data is small. However, we can use the Bayes estimates as an approximation to the objective ML estimates if we use an appropriate transformation that makes the posterior distribution close to a normal distribution. One-to-one correspondence exists between a uniform prior distribution under a transformed scale and a non-uniform prior distribution under the original scale. For this reason, the Bayes estimation of ML estimates is essentially identical to the estimation using Jeffreys prior.

Keywords

Empirical Jeffreys prior Posterior distribution Sika deer population Skewness State-space model Transformation 

Supplementary material

10144_2015_526_MOESM1_ESM.pdf (87 kb)
Supplementary material 1 (PDF 86 kb)

References

  1. Aranda-Ordaz FJ (1981) On two families of transformations to additivity for binary response data. Biometrika 68:357–363CrossRefGoogle Scholar
  2. Bartlett MS (1947) The use of transformations. Biometrics 3:39–52PubMedCrossRefGoogle Scholar
  3. Bayes T (1763) An essay towards solving a problem in the doctrine of chances. Philos Trans R Soc 53:370–418CrossRefGoogle Scholar
  4. Beall G (1942) The transformation of data from entomological field experiments so that the analysis of variance becomes applicable. Biometrika 32:243–262CrossRefGoogle Scholar
  5. Box GEP, Tiao GC (1973) Bayesian inference in statistical analysis. Wiley, New YorkGoogle Scholar
  6. Clark JS (2005) Why environmental scientists are becoming Bayesians. Ecol Lett 8:2–14CrossRefGoogle Scholar
  7. Clark JS (2007) Models for ecological data: an introduction. Princeton University Press, PrincetonGoogle Scholar
  8. de Valpine P (2003) Better inferences from population-dynamics experiments using Monte Carlo state-space likelihood methods. Ecology 84:3064–3077CrossRefGoogle Scholar
  9. de Valpine P (2004) Monte Carlo state-space likelihoods by weighted posterior kernel density estimation. J Am Stat Assoc 99:523–535CrossRefGoogle Scholar
  10. Dennis B (2004) Statistics and the scientific method in ecology (with commentary). In: Taper ML, Lele SR (eds) The nature of scientific evidence: statistical, philosophical, and empirical considerations. University of Chicago Press, Chicago, pp 327–378CrossRefGoogle Scholar
  11. Efron B (1998) R. A. Fisher in the 21st century. Stat Sci 13:95–114CrossRefGoogle Scholar
  12. Ellison AM (2004) Bayesian inference in ecology. Ecol Lett 7:509–520CrossRefGoogle Scholar
  13. Fisher RA (1922) On the mathematical foundations of theoretical statistics. Philos Trans R Soc A Math Phys Sci 222:309–368CrossRefGoogle Scholar
  14. Fisher RA (1973) Statistical methods and scientific inference, 3rd edn. Hafner Press, New YorkGoogle Scholar
  15. Guerrero VM, Johnson RA (1982) Use of the Box-Cox transformation with binary response models. Biometrika 69:309–314CrossRefGoogle Scholar
  16. Hubbard R, Bayarri MJ (2003) Confusion over measures of evidence (p’s) versus errors (α’s) in classical statistical testing. Am Stat 57:171–178CrossRefGoogle Scholar
  17. Jeffreys H (1946) An invariant form for the prior probability in estimation problems. Proc R Soc A Math Phys Sci 186:453–461CrossRefGoogle Scholar
  18. Jeffreys H (1961) Theory of probability, 3rd edn. Oxford University Press, OxfordGoogle Scholar
  19. Kaji K, Okada H, Yamanaka M, Matsuda H, Yabe T (2004) Irruption of a colonizing sika deer population. J Wildl Manage 68:889–899CrossRefGoogle Scholar
  20. Kolmogorov AN (1933) Foundations of the theory of probability (Translated from the 1st German edition of 1933 by N. Morrison, 1956), 2nd English edn. Chelsea Publishing Company, New YorkGoogle Scholar
  21. Laplace PS (1825) A philosophical essay on probabilities (Translated from the fifth French edition of 1825 by Andrew I. Dale, 1995). Springer, New YorkGoogle Scholar
  22. Lele SR, Dennis B, Lutscher F (2007) Data cloning: easy maximum likelihood estimation for complex ecological models using Bayesian Markov chain Monte Carlo methods. Ecol Lett 10:551–563PubMedCrossRefGoogle Scholar
  23. Lele SR, Nadeem K, Schmuland B (2010) Estimability and likelihood inference for generalized linear mixed models using data cloning. J Am Stat Assoc 105:1617–1625CrossRefGoogle Scholar
  24. Marsaglia G (1984) The exact-approximation method for generating random variables in a computer. J Am Stat Assoc 79:218–221CrossRefGoogle Scholar
  25. Neyman J (1934) On the two different aspects of the representative method. J R Stat Soc 97:558–606CrossRefGoogle Scholar
  26. Neyman J (1935) On the problem of confidence intervals. Ann Math Stat 6:111–116CrossRefGoogle Scholar
  27. Ratkowsky DA (1983) Nonlinear regression modeling: a unified practical approach. Dekker, New YorkGoogle Scholar
  28. Ratkowsky DA (1990) Handbook of nonlinear regression models. Dekker, New YorkGoogle Scholar
  29. Salsburg D (2001) The lady tasting tea: how statistics revolutionized science in the twentieth century. Owl Books, New YorkGoogle Scholar
  30. Shibamura R (2004) Statistical theory of R. A. Fisher. Kyushu University Press, Fukuoka (in Japanese)Google Scholar
  31. Sólymos P (2010) dclone: data cloning in R. R J 2:29–37Google Scholar
  32. Spiegelhalter DJ, Thomas A, Best N, Lunn D (2003) WinBUGS user manual, version 1.4. MRC Biostatistics Unit, CambridgeGoogle Scholar
  33. Uno H, Kaji K, Saitoh T, Matsuda H, Hirakawa H, Yamamura K, Tamada K (2006) Evaluation of relative density indices for sika deer in eastern Hokkaido, Japan. Ecol Res 21:624–632CrossRefGoogle Scholar
  34. Walker AM (1969) On the asymptotic behaviour of posterior distributions. J R Stat Soc B 31:80–88Google Scholar
  35. Wood SN (2010) Statistical inference for noisy nonlinear ecological dynamic systems. Nature 466:1102–1104PubMedCrossRefGoogle Scholar
  36. Yamamura K (1999) Transformation using (x + 0.5) to stabilize the variance of populations. Res Popul Ecol 41:229–234CrossRefGoogle Scholar
  37. Yamamura K (2014) Estimation of the predictive ability of ecological models. Commun Stat Simul Comput. doi:10.1080/03610918.2014.889161 Google Scholar
  38. Yamamura K, Matsuda H, Yokomizo H, Kaji K, Uno H, Tamada K, Kurumada T, Saitoh T, Hirakawa H (2008) Harvest-based Bayesian estimation of sika deer populations using state-space models. Popul Ecol 50:131–144CrossRefGoogle Scholar

Copyright information

© The Society of Population Ecology and Springer Japan 2015

Authors and Affiliations

  1. 1.National Institute for Agro-Environmental SciencesTsukubaJapan

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