Dirichlet Process Prior in a Catch-Effort Hierarchical Model for Animal Abundance
description
Transcript of Dirichlet Process Prior in a Catch-Effort Hierarchical Model for Animal Abundance
Dirichlet Process Prior in a Catch-Effort Hierarchical Model for Animal Abundance
Carleton CollegePrasit DhakalJun Young Park
TPA (Turkey Permit Areas)
(1) Assume that the population of the wild turkey in region i is Ni
and the number of wild turkeys harvested in region i in period j is yij.
(2)pij is the removal probability of region i in period j, then the probability that an animal is removed during, but not before period j, is equal to 𝜋 𝑖𝑗=𝑝𝑖𝑗∏
𝑘=1
𝑗−1
(1−𝑝 𝑗𝑘)
(3)So the pmf of yij given Ni and would beMultinomial
Previous Model
Previous Model(4) Catch-effort Model: probability model for pij
(5) Abundance ModelOne way to model Ni would be using Poisson distribution, say
Where is the mean animal abundance in region i.
Where is the average density of animals per unit area in region i
• A new estimate of a parameter either comes from previously drawn values or from a baseline distribution G0
• Given a Dirichlet process DP(G0,α)
G0
otherwise
atandGwhere
Gii
G k
i
koii
,0
,1)(~,
1)(
11~],|[
01
0
1
1),1(:1
What is Dirichlet Process Prior?
Different alphas
α = 2.8 α = 10ϕi’s
No. of unique ϕi’s = 8 No. of unique ϕi’s = 29
0 2 4 6 8
010
2030
-2 0 2 4 6 8 10
05
1015
Neal Function
Update for in tth iteration of MCMC consists of 2 steps of Metropolis-Hastings algorithm. STEP 1) Updating Clustering
STEP 2) Updating the unique values of
Ex) if K=3 clusters in tth iteration
After step 1,K=4 clusters in (t+1)th iteration
From step 1,
We are updating,
After updating, say
So in (t+1)th iteration
Neal Function
STEP 1) Assume in tth iteration
Draw a candidate for (t+1)th iteration by using the DPPCalculate the MH ratio for each i.
Thus we have
Neal Function
STEP 2) We update
Draw a candidate from
Calculate the MH ratio r for each i
Then
Results
Parameter Mean SD 2.5th Median 97.5th-0.295 0.093 -0.477 -0.293 -0.1140.602 0.069 0.481 0.597 0.742
Parameter Mean SD 2.5th Median 97.5th-0.304 0.128 -0.559 -0.302 -0.0590.613 0.098 0.451 0.605 0.833
K 51.541 13.364 22 53 72
Model without DPP
Model with DPP