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