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 !