Motivation
Methods
Simulation Studies
Real Data Analysis
Conclusions
Bayesian Model Selection For
Partially Observed Epidemic
Models
Panayiota Touloupou
joint work with Simon E.F. Spencer, Bärbel Finkenstädt
Rand, Peter Neal, Trevelyan J. McKinley
P. Touloupou et al.
CRiSM Workshop: Estimating Constants
April 21, 2016
Bayesian Model Selection For Partially Observed Epidemic Models
April 21, 2016
Motivation
Outline
Methods
1
Motivation
3
Simulation Studies
2
4
5
Simulation Studies
Real Data Analysis
Conclusions
Methods
Real Data Analysis
Conclusions
P. Touloupou et al.
Bayesian Model Selection For Partially Observed Epidemic Models
April 21, 2016
Motivation
Outline
Methods
1
Motivation
3
Simulation Studies
2
4
5
Simulation Studies
Real Data Analysis
Conclusions
Methods
Real Data Analysis
Conclusions
P. Touloupou et al.
Bayesian Model Selection For Partially Observed Epidemic Models
April 21, 2016
Motivation
Methods
Simulation Studies
Statistical Epidemic Modelling
Real Data Analysis
Conclusions
Insights into dynamics of infectious diseases
ã Prevention
ã Control spread of the disease
Epidemiological data present several challenges
ã Missing data (typically high dimensional)
ã Diagnostic tests imperfect
Model selection
ã Each model an epidemiologically important hypothesis
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Bayesian Model Selection For Partially Observed Epidemic Models
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Motivation
Methods
Our Setup
Simulation Studies
Real Data Analysis
Conclusions
Longitudinal observations
Individuals form groups (e.g. households)
P. Touloupou et al.
: Individual
Bayesian Model Selection For Partially Observed Epidemic Models
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Motivation
Methods
Our Setup
Simulation Studies
Real Data Analysis
Conclusions
Longitudinal observations
Individuals form groups (e.g. households)
Day 1
: Non Infected
P. Touloupou et al.
Day 4
: Infected
Day T − 2
Bayesian Model Selection For Partially Observed Epidemic Models
Day T
Study Period
2 / 33
Motivation
Methods
Simulation Studies
Our Setup
Real Data Analysis
Conclusions
Longitudinal observations
Individuals form groups (e.g. households)
Day 1
Day 2
: Non Infected
P. Touloupou et al.
Day 3
Day 4
: Infected
Day T − 2 Day T − 1
Bayesian Model Selection For Partially Observed Epidemic Models
Day T
Study Period
2 / 33
Motivation
Methods
Simulation Studies
Our Setup
Real Data Analysis
Conclusions
Longitudinal observations
−
−+−
−
Day 1
−:
Individuals form groups (e.g. households)
−
+−−
−
Day 2
Test Negative
P. Touloupou et al.
Day 3
+:
Day 4
Test Positive
−
−−−
−
Day T − 2 Day T − 1
Bayesian Model Selection For Partially Observed Epidemic Models
−
−−−
−
Day T
Study Period
2 / 33
Motivation
Methods
Objective
Simulation Studies
Real Data Analysis
Conclusions
Analysis of this type of data can be challenging
ã Times of acquiring and clearing infection are unobserved
ã Intractable likelihood - need to know missing times
ã Usual solution: large scale data augmentation MCMC
Bayesian model selection
ã Evidence in favour of candidate models
ã Each model an epidemiologically important hypothesis
Objectives:
P. Touloupou et al.
Develop statistical tools for
comparison of competing
hypotheses
Special attention on missing
data
Bayesian Model Selection For Partially Observed Epidemic Models
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Motivation
Outline
Methods
1
Motivation
3
Simulation Studies
2
4
5
Simulation Studies
Real Data Analysis
Conclusions
Methods
Real Data Analysis
Conclusions
P. Touloupou et al.
Bayesian Model Selection For Partially Observed Epidemic Models
April 21, 2016
Motivation
Methods
Simulation Studies
Model Selection For Epidemics
Real Data Analysis
Conclusions
A lot of epidemiologically interesting questions take
the form of model selection questions
What is the transmission mechanism of the disease?
Do individuals develop immunity over time?
Do water troughs spread E. coli O157?
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Bayesian Model Selection For Partially Observed Epidemic Models
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Motivation
Methods
Simulation Studies
Real Data Analysis
Conclusions
Posterior Probabilities And Marginal Likelihoods
Would like the posterior probability in favour of model i
π(y|Mi )P(Mi )
P(Mi |y) = X
π(y|Mj )P(Mj )
j
P. Touloupou et al.
Bayesian Model Selection For Partially Observed Epidemic Models
5 / 33
Motivation
Methods
Simulation Studies
Real Data Analysis
Conclusions
Posterior Probabilities And Marginal Likelihoods
Would like the posterior probability in favour of model i
π(y|Mi )P(Mi )
P(Mi |y) = X
π(y|Mj )P(Mj )
j
Equivalently, the Bayes factor comparing models i and j
Bij =
P. Touloupou et al.
π(y|Mi )
π(y|Mj )
Bayesian Model Selection For Partially Observed Epidemic Models
5 / 33
Motivation
Methods
Simulation Studies
Real Data Analysis
Conclusions
Posterior Probabilities And Marginal Likelihoods
Would like the posterior probability in favour of model i
π(y|Mi )P(Mi )
P(Mi |y) = X
π(y|Mj )P(Mj )
j
Equivalently, the Bayes factor comparing models i and j
Bij =
π(y|Mi )
π(y|Mj )
All we need is the marginal likelihood,
Z
π(y|Mi ) = π(y|θ, Mi )π(θ|Mi ) dθ
but how can we calculate it?
P. Touloupou et al.
Bayesian Model Selection For Partially Observed Epidemic Models
5 / 33
Motivation
Methods
Simulation Studies
Marginal Likelihood Estimation
Real Data Analysis
Conclusions
Most direct approach: Importance sampling
ã Use asymptotic normality of the posterior to find efficient
proposal
Many existing other approaches:
ã
ã
ã
ã
P. Touloupou et al.
Harmonic mean
Chib’s methods
Power posteriors
Bridge sampling
Bayesian Model Selection For Partially Observed Epidemic Models
6 / 33
Motivation
Methods
Simulation Studies
Real Data Analysis
Importance Sampling1
1
2
3
4
Obtain samples from the posterior π(θ|y) with MCMC
Use MCMC samples to inform the proposal distribution
⇒ q(θ)
Draw N samples from q(θ)
Estimate the marginal likelihood by
bIS (y) =
π
1
Conclusions
N
X
π(y|θ i )π(θ i )
i=1
q(θ i )
Clyde et al. (2007). Current Challenges in Bayesian Model Choice
P. Touloupou et al.
Bayesian Model Selection For Partially Observed Epidemic Models
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Motivation
Methods
Missing Data!
Simulation Studies
Real Data Analysis
Conclusions
But how to deal with the missing data?
P. Touloupou et al.
Bayesian Model Selection For Partially Observed Epidemic Models
8 / 33
Motivation
Methods
Simulation Studies
Missing Data!
Real Data Analysis
Conclusions
But how to deal with the missing data?
#1
#2
#3
P. Touloupou et al.
[1]
[1]
xt−2
xt−1
xt−2
[2]
xt−1
[3]
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yt−2
[2]
yt−2
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yt−2
[2]
[3]
[1]
xt
[1]
yt
[2]
xt
[2]
yt
[3]
xt
[3]
yt
[1]
[1]
xt+1
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[2]
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[2]
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Bayesian Model Selection For Partially Observed Epidemic Models
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8 / 33
Motivation
Methods
Simulation Studies
Real Data Analysis
Importance Sampling With Missing Data
1
2
3
4
5
Conclusions
Obtain samples from the joint posterior π(x, θ|y) with
MCMC
Use MCMC samples to inform the proposal distribution
⇒ q(θ)
Draw N samples from q(θ)
For each sampled θ i draw missing data x i from the full
conditional using Forward Filtering Backward Sampling
Estimate the marginal likelihood by
P. Touloupou et al.
bIS (y) =
π
N
X
π(y|x i , θ i ) π(x i |θ i ) π(θ i )
π(x i |y, θ i ) q(θ i )
i=1
Bayesian Model Selection For Partially Observed Epidemic Models
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Motivation
Methods
Harmonic Mean2
Simulation Studies
Real Data Analysis
Conclusions
The marginal likelihood π(y) can be approximated
"
N
1 X
1
bHM (y) =
π
N
π(y|x i , θ i )
i=1
#−1
based on N draws (x 1 , θ 1 ), (x 2 , θ 2 ), . . . , (x N , θ N ) from the
joint posterior π(x, θ|y).
Can be computed directly from MCMC output
Asymptotically consistent
Exhibit large or even infinite variance
Newton M.A. and Raftery A.E. (1994) Approximate Bayesian inference
with the weighted likelihood bootstrap J. R. Stat. Soc. Ser. B. Stat.
Methodol, 56, 3–48
2
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Bayesian Model Selection For Partially Observed Epidemic Models
10 / 33
Motivation
Methods
Chib’s Methods3
Simulation Studies
Real Data Analysis
Conclusions
Based on the observation that
π(y) =
π(y|x, θ) π(x, θ)
π(x, θ|y)
for fixed θ ∗ , x ∗ (high-density posterior point) the log
marginal likelihood can be estimated by
bChib (y) = log π(y|x ∗ , θ ∗ ) + log π(x ∗ , θ ∗ ) − log π
b (x ∗ , θ ∗ |y)
log π
=⇒ is estimated by breaking the parameter vector into
appropriate blocks
Required a separate MCMC run to calculate each block
Chib S. (1995) Marginal likelihood from the Gibbs output J. Amer.
Statist. Assoc, 90, 1313–1321. Chib S. and Jeliazkov I. (2001) Marginal
likelihood from the MH output J. Amer. Statist. Assoc, 96, 270–281
3
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Bayesian Model Selection For Partially Observed Epidemic Models
11 / 33
Motivation
Methods
Simulation Studies
Power Posteriors4
Real Data Analysis
Conclusions
Power Posterior defined as
π(x, θ|y, t) ∝ π(y|x, θ)t π(x, θ)
where t ∈ [0, 1] is a temperature parameter
The log of the marginal likelihood can be represented by
log π(y) =
Z
0
1
n
o
Ex,θ|y,t log π(y|x, θ) dt
=⇒ is calculated numerically by discretising
0 = t0 < t1 < · · · < tn = 1, and then using trapezium rule.
4
Friel N. and Pettitt A. N. (2008) Marginal likelihood estimation via
power posteriors J. R. Stat. Soc. Ser. B. Stat. Methodol. 70, 589–607
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Bayesian Model Selection For Partially Observed Epidemic Models
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Motivation
Methods
Simulation Studies
Power Posteriors: Example
Real Data Analysis
Conclusions
Obtain samples from the
power posterior at each
temperature ti
Variability depends
ã Number of ti ’s
ã Spacing of ti ’s
ã Number of MCMC
samples
Large number =⇒ more
computational effort
P. Touloupou et al.
Bayesian Model Selection For Partially Observed Epidemic Models
13 / 33
Motivation
Methods
Simulation Studies
Real Data Analysis
Reversible Jump MCMC
1
Posterior Model 1
Posterior Model 2
Prior Model 1
Prior Model 2
1
0.5
0.5
0
Conclusions
M1
M2
Model 1
θ1
k
model indicator
0
model-specific parameters
M1
M2
Model 2
θ2
data
P. Touloupou et al.
y
Bayesian Model Selection For Partially Observed Epidemic Models
14 / 33
Motivation
Outline
Methods
1
Motivation
3
Simulation Studies
2
4
5
Simulation Studies
Real Data Analysis
Conclusions
Methods
Real Data Analysis
Conclusions
P. Touloupou et al.
Bayesian Model Selection For Partially Observed Epidemic Models
April 21, 2016
Motivation
Methods
Simulation Studies
Real Data Analysis
Conclusions
Simulation Study: Pnemonococcal Carriage5
Household based longitudinal study on carriage of
Streptococcus Pneumoniae
Diagnostic tests obtained every 4 weeks
ã 10 months period
ã Classified as Negative / Positive
The population is divided into two age groups:
ã Children
ã Adults
: under 5 years old
: over 5 years old
5
Touloupou et al. (2016) Model comparison with missing data using
MCMC and importance sampling. arXiv 1512.04743
P. Touloupou et al.
Bayesian Model Selection For Partially Observed Epidemic Models
15 / 33
Motivation
Methods
Simulation Studies
Model Details6
Real Data Analysis
Conclusions
Discrete time Susceptible-Infected-Susceptible model
The transition probabilities age group i dependent:
βC i IC (t) + βAi IA (t)
· δt
Pi (S −→ I )δt = 1 − exp − ki +
(z − 1)w
Pi ( I −→ S)δt = 1 − exp (−µi · δt)
66
: 166
2600 Observed Data
: 94
6500 Missing Data
Melegaro et al. (2004) Estimating the transmission parameters of
pneumococcal carriage in households. Epidemiology and Infection, 132,
P. Touloupou
et al.
Bayesian Model Selection For Partially Observed Epidemic Models
433–441
6
16 / 33
Motivation
Methods
Simulation Studies
Real Data Analysis
Results: Marginal Likelihood Estimation
-1238
-1237
ISN1
ISN2
ISN3
ISN4
ISmix
ISt4
ISt6
ISt8
ISt10
-1236
-1237.5
-1235
-1234
-1233
-1237.25
-1232
-1237
-1231
Conclusions
-1230
Chib
PP
HM
-931
-929
-927
-925
-923
Log marginal likelihood
-921
-919
• ISNj : N(µ, j Σ) • IStd : td (µ, Σ) • ISmix : 0.95 × N(θ; µ, Σ) + 0.05π(θ) • µ, Σ: from MCMC
P. Touloupou et al.
Bayesian Model Selection For Partially Observed Epidemic Models
17 / 33
Motivation
Methods
Simulation Studies
Real Data Analysis
Conclusions
Heterogeneity In Community Acquisition Rates
Do adults and children acquire infection at the
same rate?
We compare two models:
ã M1 : kA =
6 kC
ã M2 : kA = kC
P. Touloupou et al.
Bayesian Model Selection For Partially Observed Epidemic Models
18 / 33
Motivation
Methods
Simulation Studies
Real Data Analysis
Results: Bayes Factor Estimation
2
ISmix
3
4
5
Chib
ISmix
-3
-2
-1
0
Chib
RJcor
RJcor
ISmix
ISmix
HM
HM
RJ
RJ
PP
PP
2
5
8
11
15
Log B12
19
23
(a) Data simulated from model M1
P. Touloupou et al.
-4
Conclusions
-22
-16
-10
-4
Log B12
0
4
8
(b) Data simulated from model M2
Bayesian Model Selection For Partially Observed Epidemic Models
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Motivation
Methods
Simulation Studies
Real Data Analysis
Conclusions
Results: Evolution Of The Log Bayes Factor
8
10
12
Initial MCMC run for IS and Chib
RJ pilot + RJcor burn in
Log B12
6
HM
4
PP
RJcor
-2
0
2
Chib
0
P. Touloupou et al.
30
60
90
120
Time (minutes)
150
180
210
240
Bayesian Model Selection For Partially Observed Epidemic Models
IS
20 / 33
Motivation
Outline
Methods
1
Motivation
3
Simulation Studies
2
4
5
Simulation Studies
Real Data Analysis
Conclusions
Methods
Real Data Analysis
Conclusions
P. Touloupou et al.
Bayesian Model Selection For Partially Observed Epidemic Models
April 21, 2016
Motivation
Methods
Study Designs
Simulation Studies
Real Data Analysis
Conclusions
Two longitudinal studies of E. coli O157:H7
Subjects
Study duration
Sampling interval
Dataset 1
160 cattle
14 weeks
2 times/week
Each sampling event included a
ã Faecal pat sample
ã Recto-anal mucosal swab (RAMS)
Dataset 2
168 cattle
22 weeks
14 days
Tests were assumed to have perfect specificity but
imperfect sensitivity
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Motivation
Methods
Simulation Studies
Real Data Analysis
Patterns Of Infection
Conclusions
Cattle in Pen 5
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Animal
Positive RAMS
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15
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85
99
Time (Day)
P. Touloupou et al.
Bayesian Model Selection For Partially Observed Epidemic Models
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Motivation
Methods
Simulation Studies
Real Data Analysis
Conclusions
Application 1: E. Coli O157 In Feedlot Cattle
Do animals develop immunity over time?
We compare two models for infection period:
Geometric: lack of memory.
Negative Binomial: probability of recovery depends on
duration of infection.
The Negative Binomial is a generalisation of the
Geometric:
P. Touloupou et al.
Setting Negative Binomial dispersion parameter κ = 1
leads to Geometric.
Bayesian Model Selection For Partially Observed Epidemic Models
23 / 33
Motivation
Methods
Simulation Studies
Application 1: Results
1.5
IS
RJMCMC
Conclusions
RJMCMC and IS agree on the
estimate of the Bayes factor
1.0
IS estimator: faster
convergence
0.5
Bayes factor supports the
Negative Binomial model
0.0
Log BNG
Real Data Analysis
0
P. Touloupou et al.
30
60
90
120
Time (in minutes)
150
180
The longer the colonization,
the greater the probability of
clearance – may indicate an
immune response in the host
Bayesian Model Selection For Partially Observed Epidemic Models
24 / 33
Motivation
Methods
Simulation Studies
Real Data Analysis
Application 2: Role Of Pen Area/Location
1
North
Pen
Set
Pen Size
6m×17m
Pen 14
Pen 15
Pen 6
Pen 7
Pen 8
Pen 9
Pen 10
Conclusions
North = small
Pen 16
Pen 17
Pen 18
Pen 19
Pen 20
Scale
House
Catch pens
from scale
house
South = big
Supplement and Premix Storage
Pen 11
Pen 12
Pen 13
Pen 5
P. Touloupou et al.
South
Pen
Set
Pen Size
6m×37m
Pen 1
Pen 2
Pen 3
Pen 4
Bayesian Model Selection For Partially Observed Epidemic Models
25 / 33
Motivation
Methods
Simulation Studies
Real Data Analysis
Application 2: Role Of Pen Area/Location
Conclusions
Do north and south pens have different risk of infection?
Allow different external (αs , αn ) and/or within-pen (βs , βn )
transmission rates.
Candidate models:
P. Touloupou et al.
Model
1
2
3
4
External
North South
αn
αs
α
α
αn
αs
α
α
Within-pen
North South
βn
βs
βn
βs
β
β
β
β
Bayesian Model Selection For Partially Observed Epidemic Models
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Motivation
Methods
Simulation Studies
Real Data Analysis
Application 2: Posterior Probabilities
Conclusions
Posterior Probability
0.8
RJMCMC and IS
provide identical
conclusions.
●
0.6
Evidence to support
different within-pen
transmission rates.
0.4
0.2
●
●
0.0
1
2
Method
P. Touloupou et al.
3
Model
IS
4
RJMCMC
Animals in smaller pens
more at risk of
within-pen infection
Bayesian Model Selection For Partially Observed Epidemic Models
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Motivation
Methods
Simulation Studies
Real Data Analysis
Application 3: Investigating Transmission
Between Pens
Conclusions
Dataset 2: pens adjacent in a 12 × 2 rectangular grid.
No direct contact across feed buck.
Shared waterers between pairs of adjacent pens.
Pen 24
Pen 1
Pen 23 Pen 22
Pen 21 Pen 20
Pen 19 Pen 18
Pen 17 Pen 16
Pen 15 Pen 14
Pen 13
Pen 2
Pen 4
Pen 6
Pen 8
Pen 10 Pen 11
Pen 12
P. Touloupou et al.
Pen 3
Pen 5
Pen 7
Pen 9
Bayesian Model Selection For Partially Observed Epidemic Models
28 / 33
Motivation
Methods
Simulation Studies
Real Data Analysis
Application 3: Investigating Transmission
Between Pens
Conclusions
Do waterers spread infection?
(a) Model 1: No contacts between pens
P. Touloupou et al.
(b) Model 2: Transmission via a waterer
(c) Model 3: Transmission via any boundary
Bayesian Model Selection For Partially Observed Epidemic Models
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Motivation
Methods
Simulation Studies
Real Data Analysis
Application 3: Posterior Probabilities
Conclusions
Posterior Probability
0.6
RJMCMC: hard to design
efficient jump mechanism
0.5
Using IS results still possible
0.4
0.3
0.2
●
●
1
P. Touloupou et al.
2
Model
3
Evidence for transmission
between pens sharing a
waterer rather than another
boundary
Bayesian Model Selection For Partially Observed Epidemic Models
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Motivation
Outline
Methods
1
Motivation
3
Simulation Studies
2
4
5
Simulation Studies
Real Data Analysis
Conclusions
Methods
Real Data Analysis
Conclusions
P. Touloupou et al.
Bayesian Model Selection For Partially Observed Epidemic Models
April 21, 2016
Motivation
Methods
Simulation Studies
Concluding Remarks
Real Data Analysis
Conclusions
Show how IS can be used to test epidemiological
questions of interest
In this study the importance sampling estimator
outperformed existing tools
ã Smallest Monte Carlo error
Importance sampling approach very easy to implement and
trivially parallelisable
Bayes factors depend on choice of prior
ã Simulations needed to avoid Lindley’s paradox
P. Touloupou et al.
Bayesian Model Selection For Partially Observed Epidemic Models
31 / 33
Motivation
Methods
Simulation Studies
Variations/Extensions
Real Data Analysis
Conclusions
When the full conditional is not available we use a related
full conditional
ã IS corrects for not using the true full conditional
My collaborator Peter Neal used the particle filtering to
estimate π(x|θ)
We recently applied Bridge Sampling for estimating the
marginal likelihood
ã IS a special case
ã Slightly reduced variances
ã We use IS due to ease of implementation
P. Touloupou et al.
Bayesian Model Selection For Partially Observed Epidemic Models
32 / 33
Motivation
Methods
Simulation Studies
Thanks for Listening!
P. Touloupou et al.
Real Data Analysis
Bayesian Model Selection For Partially Observed Epidemic Models
Conclusions
April 21, 2015
Appendix
Results - Posterior summaries of model parameters
Parameter
External transmission probability
Symbol
1 − e−α
Internal transmission probability
1 − e−β
Shape parameter
κ
Mean period of infection
Initial probability of infection
Sensitivity of RAJ test
Sensitivity of faecal test
m
µ
θR
θF
Geometric
0.0090
[0.0064, 0.0117]
0.0107
[0.0077, 0.0141]
8.9942
[7.7460, 10.4369]
—–
0.1001
[0.0568, 0.1545]
0.7750
[0.7304, 0.8156]
0.4639
[0.4206, 0.5073]
Negative Binomial
0.0081
[0.0057, 0.0109]
0.0102
[0.0073, 0.0137]
9.9740
[7.1977, 10.6487]
1.6245
[0.8361, 2.8972]
0.0997
[0.0557, 0.1546]
0.7771
[0.7311, 0.8203]
0.4657
[0.4213, 0.5097]
Posterior mean of the parameters of each model along
with the 95% credible interval in brackets.
Motivation
Methods
Parameter Estimation
Simulation Studies
0.025
Conclusions
0.025
North
South
0.020
Within Pen Rate
0.020
Within Pen Rate
Real Data Analysis
0.015
0.010
0.005
0.015
0.010
0.005
0.000
0.000
0.000
0.005
0.010
0.015
0.020
External Rate
(d) Model 2 - Posterior Prob 0.77
P. Touloupou et al.
0.000
0.005
0.010
0.015
0.020
External Rate
(e) Model 4 - Posterior Prob 0.16
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Motivation
Methods
Parameter Estimation
Simulation Studies
0.025
Conclusions
0.025
North
South
North
South
0.020
Within Pen Rate
0.020
Within Pen Rate
Real Data Analysis
0.015
0.010
0.005
0.015
0.010
0.005
0.000
0.000
0.000
0.005
0.010
0.015
0.020
External Rate
(f) Model 1 - Posterior Prob 0.06
P. Touloupou et al.
0.000
0.005
0.010
0.015
0.020
External Rate
(g) Model 3 - Posterior Prob 0.01
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Motivation
Methods
Simulation Studies
Real Data Analysis
The Choice of Prior Matters!
Conclusions
Simulation study: Heterogeneity in Transmission Rates Among Pens
Prior of Transmission Rates
Gamma(0.1,0.1)
Data generated from Model 2
Gamma(1,1)
Gamma(1,100)
Data generated from Model 4
Posterior Probability
1.00
0.75
0.50
0.25
0.00
1
2
3
4
1
2
3
4
Model
P. Touloupou et al.
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