An Empirical Hierarchical Bayes Approach to

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An Empirical Hierarchical Bayes Approach
to Capture-Recapture Abundance Estimation
Tomo Eguchi Environmental Statistics
Group Department of Ecology Montana State
University, Bozeman
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Outline of this presentation

• Performance of the Proposed Method (Simulations)

• Advantages and Limitations of the Method

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Mark-Recapture (Capture-Mark-Recapture) Methods
Very Basic Concept
Time
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Two Broad Categories for CMR Models
1. Open
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2. Closed
Population size is not changing
One parameter of interest (N)
Time
N
N
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Pollocks Robust Design
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Objective
To build a hierarchical empirical Bayes method
for making inference on the abundance of a closed
population using the robust design CMR data.
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Assumptions
1. All individuals are caught with equal capture
probability within a capture occasion.
2. Captures are independent events.
3. At least two secondary occasions exist within
a primary period.
4. Capture probabilities are exchangeable among
all secondary occasions.
5. Capture probabilities are independent for all
secondary occasions and primary periods.
6. Capture probability is a random sample from a
hyperdistribution (beta distribution).
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Notations
t primary period, t 1, 2, , T
kt the number of secondary occasions in the tth
primary period
N(t) population size during the tth primary
period
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Notations
the number of individuals with a particular
capture history ? during the tth primary period
Capture histories (?) with 3 capture
occasions 111, 110, 101, 100, 011, 010, 001, 000
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The assumed structure of hierarchy
p(a, ß)
Hyperdistribution
Capture Probabilities
?1,1
?1,2
?1,3
?2,1

?T,2
?T,3
?T,4
Data
n1,1
n1,2
n1,3
n2,1

nT,1
nT,2
nT,3
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General Approach
Step 1
Step 2
N(T)
t T-1
t 1
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Step 1 Inference on hyperparameters
Likelihood function Recapture processes for
(T-1) primary periods
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p(a, ßn?, u)
Hyper distribution
Capture Probabilities
?1,1
?1,2
?1,3
?2,1

?T-1,1
?T-1,2
?T-1,3
Data (Recapture processes u, n?)
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The joint posterior distribution of p(a, ßn?,u)
is numerically obtained using the
Metropolis-Hastings algorithm.
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General Approach
t T-1
t 1
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Step 2 Inference on abundance
Likelihood function Capture-Recapture Process
within the Tth primary period
Two models
1. Beta-Multinomial Model
Includes capture and recapture data
2. Beta-Binomial Model
Includes only capture data
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Other components
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p(N(T))

• Can be created from
• the knowledge of the population
• past data analyses
• a guess, e.g., Uniform distribution between
  assumed minimum and maximum

Uniform distribution was used in the analysis.
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T 3 k 8 True (?, ?) (2, 20)
Results of Simulation Analysis
Beta-Binomial
Widths 32-1152 Prop. 95
Beta-Multinomial
Widths 28-302 Prop. 95
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Advantages of the proposed method

• Interpretation of results is straightforward.

• Inference procedure works regardless of sample
  size.

• Models are simple and intuitive.

• The method can be applied to any primary period,
  if necessary.

• It can be applied sequentially for an on-going
  study.

• Incorporation of past information is simple.

• May add other parameters if reasonable models can
  be built and enough data can be obtained.

• To use the method, one may have to subscribe to
  the philosophy of Bayesian statistics.

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Limitations of the proposed method

• No canned package for the necessary
  computations.

• Successful application of the method is
  contingent on the validity of the assumptions.

• If the prior distribution of abundance does not
  cover the true abundance, the posterior
  distribution does not include the true abundance.

• To use the method, one may have to subscribe to
  the philosophy of Bayesian statistics.

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Acknowledgments
Robert Boik Department of Statistics, MSU Daniel
Goodman Department of Ecology, MSU