Page 1
Background Model definitions Tools Tests
Estimating demographic parameters fromsamples of unmarked individuals
Ben Bolker
Departments of Mathematics & Statistics and Biology, McMaster University
8 March 2011
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 2
Background Model definitions Tools Tests
Outline
1 BackgroundEcological/statistical motivationsStudy system
2 Model definitions
3 ToolsState-space modelsMarkov chain Monte CarloGibbs/block sampling
4 TestsPerfect observation, trivial dynamicsPerfect observation, non-trivial dynamicsImperfect observation
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 3
Background Model definitions Tools Tests
Ecological/statistical motivations
Cryptic density-dependence
(Wilson & Osenberg 2002, Shima &Osenberg 2003)
Environmental variation
+ correlated variation indensity . . .
= crypticdensity-dependence
best possible estimates ofdemographic parameters?
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 4
Background Model definitions Tools Tests
Ecological/statistical motivations
Data
Repeated samples of individual animals
. . . Multiple times, observers, locations
Presence and size (imperfect)
Estimate demographic parameters:
birth/immigrationdeath/emigrationchange (growth)
. . . as a function of size, (population density)
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 5
Background Model definitions Tools Tests
Ecological/statistical motivations
Data
Repeated samples of individual animals
. . . Multiple times, observers, locations
Presence and size (imperfect)
Estimate demographic parameters:
birth/immigrationdeath/emigrationchange (growth)
. . . as a function of size, (population density)
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 6
Background Model definitions Tools Tests
Study system
Natural history
photo: Leonard Low, Flickr via species.wikimedia.org
Thalassoma hardwicke(sixbar wrasse)
Settlement: ≈ 5–10 mm
Growth: a few mm per month
Death: by predator(competition for refuges)
Immigration/emigration:rare below 40–50 mm
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 7
Background Model definitions Tools Tests
Study system
Where they live
Patch reefs, FrenchPolynesia (andthroughout the southPacific)
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 8
Background Model definitions Tools Tests
Study system
What they look like to me
Date
Est
imat
ed fi
sh s
ize
(mm
)
20
40
60
80
Feb Mar Apr May Jun Jul
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 9
Background Model definitions Tools Tests
Study system
What they look like to me
Date
Est
imat
ed fi
sh s
ize
(mm
)
20
40
60
80
Feb Mar Apr May Jun Jul
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 10
Background Model definitions Tools Tests
Study system
Outline
1 BackgroundEcological/statistical motivationsStudy system
2 Model definitions
3 ToolsState-space modelsMarkov chain Monte CarloGibbs/block sampling
4 TestsPerfect observation, trivial dynamicsPerfect observation, non-trivial dynamicsImperfect observation
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 11
Background Model definitions Tools Tests
Outline
1 BackgroundEcological/statistical motivationsStudy system
2 Model definitions
3 ToolsState-space modelsMarkov chain Monte CarloGibbs/block sampling
4 TestsPerfect observation, trivial dynamicsPerfect observation, non-trivial dynamicsImperfect observation
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 12
Background Model definitions Tools Tests
Demographic parameters as f (size)
Fish size (mm)
Probability(per day)
0.0
0.2
0.4
0.6
0.8
1.0
Growth
l.grow
1020304050607080
Appearance
a.settleb.settle
tot.settle
1020304050607080
Persistence
a.mortc.mort
1020304050607080
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 13
Background Model definitions Tools Tests
Observation model
Fish size (mm)
0.2
0.4
0.6
0.8
Detection(probability)
10 20 30 40 50 60 70 80
1.0
1.5
2.0
2.5
3.0
3.5
Measurement(standard deviation)
10 20 30 40 50 60 70 80
Measurement errorDouble−countZero−count
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 14
Background Model definitions Tools Tests
Outline
1 BackgroundEcological/statistical motivationsStudy system
2 Model definitions
3 ToolsState-space modelsMarkov chain Monte CarloGibbs/block sampling
4 TestsPerfect observation, trivial dynamicsPerfect observation, non-trivial dynamicsImperfect observation
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 15
Background Model definitions Tools Tests
State-space models
State-space models
Problem: estimating parameters of systems with un- orimperfectly-observed states?
Easy if no feedback (measurement error models)
With feedback (dynamic models), brute force approachesbecome infeasible . . .
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 16
Background Model definitions Tools Tests
State-space models
Directed acyclic graph (DAG)
parameters
N1 N2 N3 N4 N5
obs1 obs2 obs3 obs4 obs5
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 17
Background Model definitions Tools Tests
Markov chain Monte Carlo
Bayesian statistics, in 30 seconds
computational (convenience) Bayesian statistics(forget all that stuff about philosophy of statistics, subjectivity, incorporating prior information . . . )
have a likelihood, L(θ) = Prob(data|θ)
want a posterior probability, Ppost(θ) = Prob(θ|data)
Bayes’ rule:
Ppost(θ) =L(θ) · prior(θ)∫∫∫L(θ′) · prior(θ′) dθ′
may want mean, mode, confidence intervals, . . . denominatorvery high-dimensional
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 18
Background Model definitions Tools Tests
Markov chain Monte Carlo
Markov chain Monte CarloP
oste
rior
prob
abili
ty
Parameter value
θAMCMC rule:
ifP(θA)
P(θB)=
J(θB → θA)
J(θA → θB)
then stationary distribution =posterior probability(provided chain is irreducible etc.)
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 19
Background Model definitions Tools Tests
Markov chain Monte Carlo
Markov chain Monte CarloP
oste
rior
prob
abili
ty
Parameter value
θA
● ●
MCMC rule:
ifP(θA)
P(θB)=
J(θB → θA)
J(θA → θB)
then stationary distribution =posterior probability(provided chain is irreducible etc.)
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 20
Background Model definitions Tools Tests
Markov chain Monte Carlo
Markov chain Monte CarloP
oste
rior
prob
abili
ty
Parameter value
θA
● ●●
MCMC rule:
ifP(θA)
P(θB)=
J(θB → θA)
J(θA → θB)
then stationary distribution =posterior probability(provided chain is irreducible etc.)
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 21
Background Model definitions Tools Tests
Markov chain Monte Carlo
Markov chain Monte CarloP
oste
rior
prob
abili
ty
Parameter value
θA
● ●●●
MCMC rule:
ifP(θA)
P(θB)=
J(θB → θA)
J(θA → θB)
then stationary distribution =posterior probability(provided chain is irreducible etc.)
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 22
Background Model definitions Tools Tests
Markov chain Monte Carlo
Markov chain Monte CarloP
oste
rior
prob
abili
ty
Parameter value
θA
● ●●●●●● ●●●
MCMC rule:
ifP(θA)
P(θB)=
J(θB → θA)
J(θA → θB)
then stationary distribution =posterior probability(provided chain is irreducible etc.)
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 23
Background Model definitions Tools Tests
Markov chain Monte Carlo
Markov chain Monte CarloP
oste
rior
prob
abili
ty
Parameter value
θA MCMC rule:
ifP(θA)
P(θB)=
J(θB → θA)
J(θA → θB)
then stationary distribution =posterior probability(provided chain is irreducible etc.)
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 24
Background Model definitions Tools Tests
Markov chain Monte Carlo
Markov chain Monte CarloP
oste
rior
prob
abili
ty
Parameter value
θAθA
θB
MCMC rule:
ifP(θA)
P(θB)=
J(θB → θA)
J(θA → θB)
then stationary distribution =posterior probability(provided chain is irreducible etc.)
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 25
Background Model definitions Tools Tests
Markov chain Monte Carlo
Metropolis-(Hastings) updatingP
oste
rior
prob
abili
ty
Parameter value
θA
Candidate distribution
Metropolis rule:
accept B with probability
min
(1,
P(θB)
P(θA)
)satisfies MCMC rule . . . Don’tneed to know denominator!
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 26
Background Model definitions Tools Tests
Gibbs/block sampling
Gibbs sampling
0.5
0.5
1
1
1.5
1.5
2
2
2.5
2.5
3
3
3.5
3.5 4
4
4.5
Joint distribution ofparameters often difficult
Condition each elementon “known” (i.e. imputed)values of all otherelements: P(a, b, c) =P(a|b, c)P(b, c)
Gibbs sampling: do thisrepeatedly for eachelement, or block ofelements
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 27
Background Model definitions Tools Tests
Gibbs/block sampling
Gibbs sampling
0.5
0.5
1
1
1.5
1.5
2
2
2.5
2.5
3
3
3.5
3.5 4
4
4.5
●
● ●
● ●
●
●●
●
●●●
●●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
Joint distribution ofparameters often difficult
Condition each elementon “known” (i.e. imputed)values of all otherelements: P(a, b, c) =P(a|b, c)P(b, c)
Gibbs sampling: do thisrepeatedly for eachelement, or block ofelements
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 28
Background Model definitions Tools Tests
Gibbs/block sampling
Gibbs sampling
0.5
0.5
1
1
1.5
1.5
2
2
2.5
2.5
3
3
3.5
3.5 4
4
4.5
●
● ●
● ●
●
●●
●
●●●
●●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
Joint distribution ofparameters often difficult
Condition each elementon “known” (i.e. imputed)values of all otherelements: P(a, b, c) =P(a|b, c)P(b, c)
Gibbs sampling: do thisrepeatedly for eachelement, or block ofelements
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 29
Background Model definitions Tools Tests
Gibbs/block sampling
Back to the DAG
parameters
N1 N2 N3 N4 N5
obs1 obs2 obs3 obs4 obs5
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 30
Background Model definitions Tools Tests
Outline
1 BackgroundEcological/statistical motivationsStudy system
2 Model definitions
3 ToolsState-space modelsMarkov chain Monte CarloGibbs/block sampling
4 TestsPerfect observation, trivial dynamicsPerfect observation, non-trivial dynamicsImperfect observation
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 31
Background Model definitions Tools Tests
Perfect observation, trivial dynamics
Outline
1 BackgroundEcological/statistical motivationsStudy system
2 Model definitions
3 ToolsState-space modelsMarkov chain Monte CarloGibbs/block sampling
4 TestsPerfect observation, trivial dynamicsPerfect observation, non-trivial dynamicsImperfect observation
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 32
Background Model definitions Tools Tests
Perfect observation, trivial dynamics
DAG
parameters fates
N1 N2
Demographic parameters|fates:standard M-H
Fates|parameters
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 33
Background Model definitions Tools Tests
Perfect observation, trivial dynamics
Scenario
X
Lsurv = (1 − m)(1 − p)S
Lmort = mp(1 − p)S−1
One fish of the same size observedon two consecutive days . . .was it the same individual?
P(surv|m, p) =(1−m)(1− p)
1−m − p + 2mp
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 34
Background Model definitions Tools Tests
Perfect observation, trivial dynamics
Updating probabilities for parameters
probability
probabilitydensity
0.5
1.0
1.5
0123456
mortality (m)
immigration (p)
0.2 0.4 0.6 0.8
typepriorposterior
If the fish survived, then ourposterior probabilities for theparameters are:
m ∼ Beta(1, 2)(0 mortalities, 1 survival)
p ∼ Beta(1, S + 1)(0 immigrations, Snon-immigrations)
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 35
Background Model definitions Tools Tests
Perfect observation, trivial dynamics
Trivial example: results
value
density
0
1
2
3
4
mortality (m)
2(m + S(1 − m))S + 1
0.0 0.2 0.4 0.6 0.8 1.0
immigration (p)
Beta(p,1,S)
0.0 0.2 0.4 0.6 0.8 1.0
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 36
Background Model definitions Tools Tests
Perfect observation, trivial dynamics
P(mortality & no immigration) ≈ 0.16 = 1/(S + 1);P(survival & immigration) ≈ 0.83 = S/(S + 1);
results are simple and make sense; closed form solution ispossible but ugly . . .
combination of observations and non-observation of otherindividuals provides information (could fail with non-detection)
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 37
Background Model definitions Tools Tests
Perfect observation, non-trivial dynamics
Outline
1 BackgroundEcological/statistical motivationsStudy system
2 Model definitions
3 ToolsState-space modelsMarkov chain Monte CarloGibbs/block sampling
4 TestsPerfect observation, trivial dynamicsPerfect observation, non-trivial dynamicsImperfect observation
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 38
Background Model definitions Tools Tests
Perfect observation, non-trivial dynamics
Non-trivial dynamics
Multiple individuals, measured over multiple days
Still simple: no density-dependent rates,immigration/emigration of large individuals
observations still error-free
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 39
Background Model definitions Tools Tests
Perfect observation, non-trivial dynamics
Parameter/fate sampling
parameters
fates(1,2) fates(2,3)
N1 N2 N3
Parameter sampling: M-Hupdating with simplecandidate distributions
Fate sampling: ???
Enumeratingpossibilities is tedious. . .
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 40
Background Model definitions Tools Tests
Perfect observation, non-trivial dynamics
M-H fate sampling
6 mm 7 mm dead
5 mm G (5) · (1−M(5)) X M(5)6 mm (1− G (6)) · (1−M(6)) G (6) · (1−M(6)) M(6)absent I (6) I (7) X
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 41
Background Model definitions Tools Tests
Perfect observation, non-trivial dynamics
M-H fate sampling
6 mm 7 mm dead
5 mm G (5) · (1−M(5)) X M(5)6 mm (1− G (6)) · (1−M(6)) G (6) · (1−M(6)) M(6)absent I (6) I (7) X
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 42
Background Model definitions Tools Tests
Perfect observation, non-trivial dynamics
M-H fate sampling
6 mm 7 mm dead
5 mm G (5) · (1−M(5)) X M(5)6 mm (1− G (6)) · (1−M(6)) G (6) · (1−M(6)) M(6)absent I (6) I (7) X
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 43
Background Model definitions Tools Tests
Perfect observation, non-trivial dynamics
M-H fate sampling
6 mm 7 mm dead
5 mm G (5) · (1−M(5)) X M(5)6 mm (1− G (6)) · (1−M(6)) G (6) · (1−M(6)) M(6)absent I (6) I (7) X
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 44
Background Model definitions Tools Tests
Perfect observation, non-trivial dynamics
M-H fate sampling
6 mm 7 mm dead
5 mm G (5) · (1−M(5)) X M(5)6 mm (1− G (6)) · (1−M(6)) G (6) · (1−M(6)) M(6)absent I (6) I (7) X
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 45
Background Model definitions Tools Tests
Perfect observation, non-trivial dynamics
M-H fate sampling
6 mm 7 mm dead5 mm G (5) · (1−M(5)) X M(5)6 mm (1− G (6)) · (1−M(6)) G (6) · (1−M(6)) M(6)absent I (6) I (7) X
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 46
Background Model definitions Tools Tests
Perfect observation, non-trivial dynamics
Sampling test
Probability
Fate
both die
6 grows
5 grows
both grow
6 survives
10−3.5 10−3 10−2.5 10−2 10−1.5 10−1 10−0.5
wSamplingTrueGibbs
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 47
Background Model definitions Tools Tests
Perfect observation, non-trivial dynamics
Testbed
Simulate for 80 days, with a settlement rate (tot.settle) of5% day:
29 distinct individuals, size range 5–30, total of 618 fish-days(observations).
Sample fates only (1000 MCMC steps), fixed demographicparameters
Sampled 1000 unique fates (but only 895 unique likelihoods)
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 48
Background Model definitions Tools Tests
Perfect observation, non-trivial dynamics
Testbed
Simulate for 80 days, with a settlement rate (tot.settle) of5% day:
29 distinct individuals, size range 5–30, total of 618 fish-days(observations).
Sample fates only (1000 MCMC steps), fixed demographicparameters
Sampled 1000 unique fates (but only 895 unique likelihoods)
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 49
Background Model definitions Tools Tests
Perfect observation, non-trivial dynamics
True demography
0 20 40 60 80
510
1520
2530
day
size
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 50
Background Model definitions Tools Tests
Perfect observation, non-trivial dynamics
Fate-only results
Log−likelihood
0.00
0.01
0.02
0.03
0.04
0.05
−520 −500 −480 −460
●
Density
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 51
Background Model definitions Tools Tests
Perfect observation, non-trivial dynamics
Sampling fates, parameters, both . . .
Log−likelihood
Density
0.00
0.05
0.10
0.15
0.20
0.25
−500 −490 −480 −470 −460 −450
typefate onlyparameters onlyboth
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 52
Background Model definitions Tools Tests
Perfect observation, non-trivial dynamics
Parameter estimates 1
density
0.51.01.52.02.53.0
0.10.20.30.40.50.60.7
lgrow
−1.9−1.8−1.7−1.6−1.5−1.4−1.3−1.2a.settle
11 12 13 14
0.2
0.4
0.6
0.8
0.1
0.2
0.3
0.4
0.5
a.mort
−4.0 −3.5 −3.0 −2.5b.settle
1.01.52.02.53.03.54.0
20406080
100120140
10
20
30
40
50
c.mort
0.0100.0150.0200.0250.030tot.settle
0.020.030.040.050.06
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 53
Background Model definitions Tools Tests
Perfect observation, non-trivial dynamics
Parameter estimates 2
Fish size (mm)
Probability(per day)
0.0
0.2
0.4
0.6
0.8
1.0
Growth
l.grow
●
●
●
●
●
●
●
●●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
0 5 10 15 20 25 30
0.00
0.02
0.04
0.06
0.08
Appearance
a.settle
b.settle
tot.settle
●
●
●
●
●
●
●
●
●●●●●●●●●●●●●●●●●●
0 5 10 15 20 25 30
0.92
0.94
0.96
0.98
1.00
Persistence
a.mort
c.mort
●
●
●
●
●
●●
●
●
●
●●●●●●●●●●●●●●●●
0 5 10 15 20 25 30
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 54
Background Model definitions Tools Tests
Perfect observation, non-trivial dynamics
Parameter estimates 3
−4.0
−3.5
−3.0
−2.5
−2.0
0.0050.0100.0150.0200.0250.030
a.mort
●
●
●
●
●
●
●
●
●
●
c.mort
●
●
●
●
●
●
●
●
●
●
10111213141516
−1.8
−1.6
−1.4
−1.2
a.settle
●●
● ● ● ●
●● ●
●
lgrow
●●
●
●
●
●
●
● ●
●
1.01.52.02.53.03.54.0
0.0100.0150.0200.0250.0300.0350.0400.045
b.settle
●
● ●
●
●
●
●
●
●
●
tot.settle
●
● ●
● ●●
●●
●
●
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 55
Background Model definitions Tools Tests
Imperfect observation
Outline
1 BackgroundEcological/statistical motivationsStudy system
2 Model definitions
3 ToolsState-space modelsMarkov chain Monte CarloGibbs/block sampling
4 TestsPerfect observation, trivial dynamicsPerfect observation, non-trivial dynamicsImperfect observation
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 56
Background Model definitions Tools Tests
Imperfect observation
Adding errors
parameters
fates(1,2) fates(2,3)
N1 N2 N3
obs1 obs2 obs3
obs parameters
Back to the more typicalgraph: incorporate informationfrom N(t − 1), N(t + 1), andcurrent observation
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 57
Background Model definitions Tools Tests
Imperfect observation
New algorithm
For each individual:
Pick true state at N(t), from all possible states, conditioningon N(t − 1)
Pick identity at time N(t + 1) from feasible set:update probabilities
Pick corresponding observation from observed(t):update probabilities
Calculate likelihood, M-H update
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 58
Background Model definitions Tools Tests
Imperfect observation
Open questions, caveats
Will it work ??
Add complexities:
Multiple populationsSpatial variation, correlation among parameters
Simplify/generalize code
Speed?
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals
Page 59
Background Model definitions Tools Tests
Imperfect observation
Conclusions & musings
MCMC as powerful technology
. . . can it be democratized?
Data limitations shift over time:hierarchical models are great for “modern” (high-volume,low-quality) data
Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University
Estimating unmarked individuals