Hostname: page-component-745bb68f8f-mzp66 Total loading time: 0 Render date: 2025-01-08T13:51:32.606Z Has data issue: false hasContentIssue false
  • English
  • Français

Principles of Statistical Inference: Likelihood and the Bayesian Paradigm

Published online by Cambridge University Press:  21 July 2017

Steve C. Wang
Steve C. Wang*
Affiliation:
Department of Geological and Environmental Sciences, Stanford University, 450 Serra Mall, Building 320, Stanford, CA 94306 Department of Mathematics and Statistics, Swarthmore College, 500 College Ave, Swarthmore, PA 19081
Get access

Abstract

We review two foundations of statistical inference, the theory of likelihood and the Bayesian paradigm. We begin by applying principles of likelihood to generate point estimators (maximum likelihood estimators) and hypothesis tests (likelihood ratio tests). We then describe the Bayesian approach, focusing on two controversial aspects: the use of prior information and subjective probability. We illustrate these analyses using simple examples.

Type
General Toolkit
Information
The Paleontological Society Papers , Volume 16: Quantitative Methods in Paleobiology , October 2010 , pp. 1 - 18
Copyright
Copyright © 2010 by the Paleontological Society 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

ABC News (2001). Testing police honesty: a PrimeTime investigation with lost wallets. PrimeTime [television program]. Originally broadcast May 17, 2001. Summarized at http://abcnews.go.com/Primetime/story?id=132229.Google Scholar
Alroy, J. 1998. Cope's rule and the dynamics of body mass evolution in North American fossil mammals. Science 28:731–734.Google Scholar
Berger, J. 2006. The case for objective Bayesian analysis. Bayesian Analysis 1:385–402.Google Scholar
Casella, G., and Berger, R. L. 2002. Statistical inference, 2nd ed. Duxbury, Pacific Grove, CA, 660 p.Google Scholar
Foote, M. 2005. Pulsed origination and extinction in the marine realm. Paleobiology 31:6–20.Google Scholar
Foote, M. 2007. Extinction and quiescence in marine animal genera. Paleobiology 33:262–273.Google Scholar
Gelfand, A. E., and Smith, A. F. M. 1990. Sampling-based approaches to calculating marginal densities. Journal of the American Statistical Association 85:398–409.CrossRefGoogle Scholar
Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. 2003. Bayesian Data Analysis, 2ed. Chapman and Hall, London, 696 p.Google Scholar
Geman, S., and Geman, D. 1984. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence 6:721–741.Google Scholar
Hammer, Ø., and Harper, D. A. T. 2005. Paleontological Data Analysis. Blackwell, Oxford, 351 p.Google Scholar
Hannisdal, B. 2007. Inferring phenotypic evolution in the fossil record by Bayesian inversion. Paleobiology 33:98–115.Google Scholar
Harper, D. A. T. (ed.) 1999. Numerical Palaeobiology: Computer-based Modelling and Analysis of Fossils and their Distributions. Wiley, Chichester, 468 p.Google Scholar
Hastings, W. K. 1970. Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57:97–109.Google Scholar
Hunt, G. 2006. Fitting and comparing models of phyletic evolution: random walks and beyond. Paleobiology 32:578–601.Google Scholar
Hunt, G. 2007. The relative importance of directional change, random walks, and stasis in the evolution of fossil lineages. Proceedings of the National Academy of Sciences USA 104:18404–18408.Google Scholar
Hunt, G. 2008. Gradual or pulsed evolution: when should punctuational explanations be preferred? Paleobiology 34:360–377.Google Scholar
Jeffreys, H. 1961. Theory of Probability, 3rd ed. Oxford University Press, Oxford, 470 p.Google Scholar
Kass, R. E. and Wasserman, L. 1996. The selection of prior distributions by formal rules. Journal of the American Statistical Association 91:1343–1370.Google Scholar
Kelley, P. H., and Hansen, T. A. 1993. Evolution of the naticid gastropod predator-prey system: An evaluation of the hypothesis of escalation: Palaios 8:358–375.CrossRefGoogle Scholar
Macellari, C. E. 1986. Late Campanian-Maastrichtian ammonite fauna from Seymour Island (Antarctic Peninsula). Journal of Paleontology 60 (supplement).Google Scholar
Manly, B. F. J. 2006. Randomization, Bootstrap and Monte Carlo Methods in Biology, 3ed. Chapman and Hall, London. 480 p.Google Scholar
McConway, K. J., and Sims, H. J. 2004. A likelihood-based method for testing for non-stochastic variation of diversification rates in phylogenies. Evolution 58:12–23.Google Scholar
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E. 1953. Equations of state calculations by fast computing machines. Journal of Chemical Physics 21:1087–1092.Google Scholar
Puolamäki, K., Fortelius, M., and Mannila, H. 2006. Seriation in paleontological data using Markov Chain Monte Carlo methods. PLoS Computational Biology 2:e6.Google Scholar
Sims, H. J., and McConway, K. J. 2003. Non-stochastic variation of species-level diversification rates within angiosperms. Evolution 57:460–479.Google Scholar
Sokal, R. R., and Rpjlf, F. J. 1995. Biometry, 3rd ed. W. H. Freeman, San Francisco, 887 p.Google Scholar
Solow, A. R. 1996. A test for a common upper endpoint in fossil taxa. Paleobiology 22:406–410.CrossRefGoogle Scholar
Solow, A. R., and Smith, W. K. 1997. On fossil preservation and the stratigraphic ranges of taxa. Paleobiology 23:271–277.Google Scholar
Solow, A. R., and Smith, W. K. 2000. Testing for a mass extinction without selecting taxa. Paleobiology 26:647–650.Google Scholar
Solow, A. R., and Smith, W. K. 2010. A test for Cope's rule. Evolution 64:583–586.Google Scholar
Solow, A. R., Roberts, D. L., and Robbirt, K. M. 2006. On the Pleistocene extinctions of Alaskan mammoths and horses. Proceedings of the National Academy of Sciences USA 103:7351–7353.Google Scholar
Strauss, D., and Sadler, P. M. 1989. Classical confidence intervals and Bayesian probability estimates for ends of local taxon ranges. Mathematical Geology 21:411–427.Google Scholar
Wagner, P. J. 1998. A likelihood approach for estimating phylogenetic relationships among fossil taxa. Paleobiology 24:430–449.Google Scholar
Wagner, P. J. 2000. Likelihood tests of hypothesized durations: testing for and accommodating biasing factors. Paleobiology 26:431–449.Google Scholar
Wang, S. C., Chudzicki, D. J., and Everson, P. J. 2009. Optimal estimators of the position of a mass extinction when recovery potential is uniform. Paleobiology 35:447–459.CrossRefGoogle Scholar
Wang, S. C. and Everson, P. J. 2007: Confidence intervals for pulsed mass extinction events. Paleobiology 33:324–336.CrossRefGoogle Scholar
Wang, S. C., Roopnarine, P. D., Angielczyk, K. D., and Karcher, M. D. 2006. Modeling terrestrial food web collapse in the end-Permian extinction. Geological Society of America Abstracts with Programs 38:171.Google Scholar