Insights Into the Latent Multinomial Model Through Mark-Resight Data on Female Grizzly Bears With Cubs-of-the-Year

  • Megan D. Higgs
  • William A. Link
  • Gary C. White
  • Mark A. Haroldson
  • Daniel D. Bjornlie


Mark-resight designs for estimation of population abundance are common and attractive to researchers. However, inference from such designs is very limited when faced with sparse data, either from a low number of marked animals, a low probability of detection, or both. In the Greater Yellowstone Ecosystem, yearly mark-resight data are collected for female grizzly bears with cubs-of-the-year (FCOY), and inference suffers from both limitations. To overcome difficulties due to sparseness, we assume homogeneity in sighting probabilities over 16 years of bi-annual aerial surveys. We model counts of marked and unmarked animals as multinomial random variables, using the capture frequencies of marked animals for inference about the latent multinomial frequencies for unmarked animals. We discuss undesirable behavior of the commonly used discrete uniform prior distribution on the population size parameter and provide OpenBUGS code for fitting such models. The application provides valuable insights into subtleties of implementing Bayesian inference for latent multinomial models. We tie the discussion to our application, though the insights are broadly useful for applications of the latent multinomial model.

Key Words

Bayesian Discrete uniform Greater Yellowstone Ecosystem (GYE) Mark-recapture Population size 


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  1. Anderson, C., and Lindzey, F. G. (1996), “Moose Sightability Model Developed from Helicopter Surveys,” Wildlife Society Bulletin, 24, 247–259. Google Scholar
  2. Berger, J., Bernardo, J., and Sun, D. (2008), “Reference priors for discrete parameter spaces,” Technical report, Department of Statistical Science, Duke University. Google Scholar
  3. Bonner, S. J., and Holmberg, J. (2013, in press), “Mark-recapture with multiple non-invasive marks,” Biometrics. Google Scholar
  4. Bowden, D. C., and Kufeld, R. C. (1995), “Generalized Mark-Sight Population-Size Estimation Applied to Colorado Moose,” The Journal of Wildlife Management, 59, 840–851. CrossRefGoogle Scholar
  5. Chao, A. (1989), “Estimating Population Size for Sparse Data in Capture-Recapture Experiments,” Biometrics, 45, 427–438. MathSciNetCrossRefMATHGoogle Scholar
  6. Cherry, S., White, G. C., Keating, K. A., Haroldson, M. A., and Schwartz, C. C. (2007), “Evaluating Estimators for Numbers of Females with Cubs-of-the-Year in the Yellowstone Grizzly Bear Population,” Journal of Agricultural, Biological, and Environmental Statistics, 12, 195–215. MathSciNetCrossRefGoogle Scholar
  7. Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (2004), Bayesian Data Analysis (2nd ed.), Boca Raton: Chapman & Hall/CRC. MATHGoogle Scholar
  8. Harris, R. B., White, G. C., Schwartz, C. C., and Haroldson, M. A. (2007), “Population Growth of Yellowstone Grizzlies: Uncertainty, Correlation, and Future Monitoring,” Ursus, 18, 167–177. CrossRefGoogle Scholar
  9. Knight, R. R., Blanchard, B. M., and Eberhardt, L. L. (1995), “Appraising Status of the Yellowstone Grizzly Bear Population by Counting Females with Cubs-of-the-Year,” Wildlife Society Bulletin, 23, 245–248. Google Scholar
  10. Link, W. A. (2013, in press), “A cautionary note on the discrete uniform prior for the binomial N,” Ecology. Google Scholar
  11. Link, W. A., and Barker, R. J. (2010), Bayesian Inference with Ecological Applications, London: Elsevier. Google Scholar
  12. Link, W. A., Yoshizaki, J., Bailey, L. L., and Pollock, K. H. (2010), “Uncovering a Latent Multinomial: Analysis of Mark-Recapture Data with Misidentification,” Biometrics, 66, 178–185. MathSciNetCrossRefMATHGoogle Scholar
  13. Litt, A. R., and Steidl, R. J. (2009), “Improving Estimates of Abundance by Aggregating Sparse Capture-Recapture Data,” Journal of Agricultural, Biological, and Environmental Statistics, 15, 228–247. MathSciNetCrossRefGoogle Scholar
  14. Lunn, D., Jackson, C., Best, N., Thomas, A., and Spiegelhalter, D. (2013), The BUGS Book, Boca Raton: CRC Press, Taylor & Francis Group. MATHGoogle Scholar
  15. Lunn, D. J., Spiegelhalter, D. J., Thomas, A., and Best, N. (2009), “The BUGS Project: Evolution, Critique, and Future Directions,” Statistics in Medicine, 28, 3049–3067. MathSciNetCrossRefGoogle Scholar
  16. Lunn, D. J., Thomas, A., Best, N., and Spiegelhalter, D. (2000), “WinBUGS—a Bayesian Modelling Framework: Concepts, Structure, and Extensibility,” Statistics and Computing, 10, 325–337. CrossRefGoogle Scholar
  17. McClintock, B. T., Conn, P., Alonso, R., and Crooks, K. R. (2013), “Integrated Modeling of Bilateral Photo-Identification Data in Mark-Recapture Analyses,” Ecology. doi:10.1890/12-1613.1. Google Scholar
  18. McClintock, B. T., and Hoeting, J. A. (2010), “Bayesian Analysis of Abundance for Binomial Sighting Data with Unknown Number of Marked Individuals,” Environmental and Ecological Statistics, 17, 317–332. MathSciNetCrossRefGoogle Scholar
  19. McClintock, B. T., White, G. C., Antolin, M. F., and Tripp, D. W. (2009a), “Estimating Abundance Using Mark-Resight When Sampling Is with Replacement or the Number of Marked Individuals Is Unknown,” Biometrics, 65, 237–246. MathSciNetCrossRefMATHGoogle Scholar
  20. McClintock, B. T., White, G. C., Burnham, K. P., and Pryde, M. A. (2009b), “A Generalized Mixed Effects Model of Abundance for Mark-Resight Data When Sampling Is Without Replacement,” in Modeling Demographic Processes in Marked Populations, New York: Springer, pp. 271–289. CrossRefGoogle Scholar
  21. McKelvey, K. S., and Pearson, D. E. (2001), “Population Estimation with Sparse Data: the Role of Estimators Versus Indices Revisited,” Canadian Journal of Zoology, 79, 1754–1765. CrossRefGoogle Scholar
  22. Plummer, M. (2003), “JAGS: a program for analysis of Bayesian graphical models using Gibbs sampling.” Google Scholar
  23. R Development Core Team (2009), R: a Language and Environment for Statistical Computing, Vienna: R Foundation for Statistical Computing, URL ISBN 3-900051-07-0. Google Scholar
  24. Samuel, M. D., Garton, E. O., Schlegel, M. W., and Carson, R. G. (1987), “Visibility Bias During Aerial Surveys of Elk in North-Central Idaho,” The Journal of Wildlife Management, 51, 622–630. CrossRefGoogle Scholar
  25. Schwartz, C. C., Haroldson, M. A., Cherry, S., and Keating, K. A. (2008), “Evaluation of Rules to Distinguish Unique Female Grizzly Bears with Cubs in Yellowstone,” The Journal of Wildlife Management, 72, 543–554. CrossRefGoogle Scholar
  26. USFWS (1993), Grizzly bear recovery plan, U.S. Fish and Wildlife Service, Missoula, Montana, USA. Google Scholar
  27. White, G. C. (1996), “Noremark: Population Estimation from Mark-Resighting Surveys,” Wildlife Society Bulletin, 24, 50–52. Google Scholar
  28. — (2005), “Correcting Wildlife Counts Using Detection Probabilities,” Wildlife Research, 32, 211–216. CrossRefGoogle Scholar
  29. White, G. C., and Burnham, K. P. (1999), “Program Mark: Survival Estimation from Populations of Marked Animals,” Bird Study, 46, 120–139. CrossRefGoogle Scholar

Copyright information

© International Biometric Society 2013

Authors and Affiliations

  • Megan D. Higgs
    • 1
  • William A. Link
    • 2
  • Gary C. White
    • 3
  • Mark A. Haroldson
    • 4
  • Daniel D. Bjornlie
    • 5
  1. 1.Department of Mathematical SciencesMontana State UniversityBozemanUSA
  2. 2.Patuxent Wildlife Research CenterU.S. Geological SurveyLaurelUSA
  3. 3.Department of Fish, Wildlife, and Conservation BiologyColorado State UniversityFort CollinsUSA
  4. 4.Northern Rocky Mountain Science Center, Interagency Grizzly Bear Study TeamU.S. Geological SurveyBozemanUSA
  5. 5.Large Carnivore SectionWyoming Game and Fish DepartmentLanderUSA

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