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### Talk by Blaise Piédallu at ISEC 2014 on improving abundance estimates by using informative priors.

• 1. Using informative priors to improve parameters estimation in capture-recapture models Blaise Piédallu PhD Student Supervisors : Olivier Gimenez Pierre-Yves Quenette
• 2. Population of interest -Brown bears (Ursus arctos) in the French and Spanish Pyrénées -Population size : about 25 individuals in 2013 -Individuals are detected and identified through different methods (camera pictures, genetic sampling of hair and faeces) -2 population cores or « Regions » (Western and Central- Eastern), without communication (Western – 2 individuals in 2013 – and Central-Eastern – ~23 individuals in 2013)
• 3. Population of interest Objectives : - Estimate population size, test for difference in Regions - Since the population is small, use an informative prior to improve the precision
• 4. Dataset  Capture-Recapture data during 25 years (1989 – 2013)  A Jolly-Seber model is used to estimate population size - Capture-Recapture on n different occasions (here, n=25) - Open population: immigrations (births/reintroductions) and emigrations (deaths) State-Space Model : Detected (= 1) Not detected (= 0) p 1 - p 1 Not detected (= 0) Survived Time t Died f 1 - f Hidden information Observed information
• 5. Dataset  Capture-Recapture data during 25 years (1989 – 2013)  A Jolly-Seber model is used to estimate population size - Capture-Recapture on n different occasions (here, n=25) - Open population: immigrations (births/reintroductions) and emigrations (deaths) State-Space Model : Detected (= 1) Not detected (= 0) p 1 - p 1 Not detected (= 0) Survived Time t Died f 1 - f Hidden information Observed information
• 6. Model Selection The Bayesian computation was performed with the softwares -R- and JAGS. Tested models Survival Detection r . r + T + r.T r + T r T . r : « Region » effect T : Time effect . : no effect Survival probability : logit(phi[i,t]) <- alpha[1] + alpha[2]*cov.region[i] Detection probability : logit(p[i,t]) <- alpha[3] + alpha[4]*cov.region[i] + alpha[5]*t + alpha[6]*t*cov.region[i] + eps[i]
• 7. Model Selection Model selection by estimating posterior model probabilities (Kuo and Mallick, 1998) Survival probability : logit(phi[i,t]) <- alpha[1] + w[1]*alpha[2]*cov.region[i] Detection probability : logit(p[i,t]) <- alpha[3] + w[2]*alpha[4]*cov.region[i] + w[3]*alpha[5]*t + w[4]*alpha[6]*t*cov.region[i] + eps[i] Indicator variables - w ~ dbern(0.5) Multiplies every relevant parameter
• 8. Model Selection MCMC sampling Posterior model probability = Number of iterations using this model Total number of iterations
• 9. Model Selection Results : r . r + T + r.T 0,01027 0,05487 r + T 0,06833 0,36239 r 0,04008 0,18398 T 0,01216 0,05809 . 0,02383 0,12107 Survival Capture Model selected : Survival = f(.), Capture = f(r+T) Some models with no significance are ignored: the intersect of r and T only has a meaning if both r and T are used Theoretical number of models = 24 = 16
• 10. Estimated survival Estimated density with an uninformative prior U(0,1) f = 0.94 ± 0.015
• 11. Using informative priors Two priors for Survival were used for the next simulations : -A non informative prior U(0,1) -An informative prior B(a,b), with a and b chosen in order to get a mean of 0.9 and a standard deviation of 0.025
• 12. Using informative priors Entire dataset n = 25 years Both population cores No difference in population size estimation No difference in standard deviation
• 13. Using informative Priors - Splitting the dataset Dataset split n = 25 years Eastern population only No difference in population size estimation No difference in standard deviation
• 14. Using informative Priors - Splitting the dataset Dataset split n = 25 years Western population only No difference in population size estimation No difference in standard deviation
• 15. Using informative Priors - Splitting the dataset Dataset split n = 15 years (89-03) Both population cores Improvement in standard deviation by using informative priors
• 16. Using informative Priors - Splitting the dataset Dataset split n = 15 years (04-13) Both population cores Improvement in standard deviation by using informative priors Difference in population size estimation
• 17. Conclusion What can we say about informative priors ? -Even relatively small datasets may contain enough data in order to be used - In the case of the French Brown Bear, the information seems to come from the length of the study (over 25 years) -In the last 10 years, monitoring of the population has greatly increased – more people involved, improving the search for genetic samples in the Pyrénées -Informative priors are useful to create a more complex model including more parameters on a smaller timeframe
• 18. Conclusion What to do next ? Check if an informative prior has influences model selection Check the influence of an informative prior on a more complex model : -Add more age classes -Add gender Use the parameter estimates in order to check the influence of future reintroductions Perform a viability analysis of the population using the informative priors
• 19. The End From Pyrénée, written by Régis Loisel, drawn by Philippe Sternis THANK YOU FOR YOUR ATTENTION !
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