Statistical inference for agent-based SIS and SIR models

Statistical inference for agent-based
SIS and SIR models
Jeremy Heng
ESSEC Business School
Joint work with Phyllis Ju and Pierre Jacob (Harvard)
Bayesian Young Statisticians Meeting: Online (BAYSM:O)
18 November 2020
JH Agent-based models 1/ 22

Agent-based models
• Agent-based models specify how a population of agents
interact and evolve over time
• Can render realistic macroscopic phenomena from simple
microscopic rules
Figure: SimCity by Electronic Arts

Calibration of agent-based models
• These models are typically calibrated by matching key features
of simulated and actual data
• Can be computationally intensive and difficult to calibrate
126 CHAPTER 5
Figure 5.3. Simulated and historical settlement patterns, in red, for Long House
Valley in A.D. 1125. North is to the top of the page.
of the 1270–1450 period could have supported a reduced but substantial
population in small settlements dispersed across suitable farming habitats
located primarily in areas of high potential crop production in the
Figure: Simulated and historical settlement patterns in long house valley

Statistical inference for agent-based models
• Given occasional noisy measurements of the population, we
could consider statistical inference for such models
• Few works have addressed this important topic as
likelihood-based inference is computationally challenging
• We propose various Monte Carlo algorithms for some
classical agent-based models
• The general principle is to ‘open the black box’ nature of
these models and exploit its inherent structure

Compartmental models in epidemiology
• A population-level approach assigns the population to
compartments and models the number of people in each
compartment over time
SIR model
Susceptible
Infected
Recovered
SIS model
Susceptible
Infected

Agent-based models in epidemiology
• The agent-based approach assumes agents can take these
states and models the state of each agent n over time
SIR model
Susceptible
Infected
Recovered
n
n
SIS model
Susceptible
Infected
n
n

Agent-based SIS model
• We consider the agent-based SIS model and encode
Susceptible = 0 and Infected = 1
• Let Xt = (Xn
t )n2[1:N] 2 {0, 1}N denote the state of a closed
population of N agents at time t 2 [0 : T]
• Initialization X0 ⇠ µ✓ given by
Xn
0 ⇠ Ber(↵n
0), independently for n 2 [1 : N]
• Markov transition Xt ⇠ f✓(·|Xt 1) at time t 2 [1 : T] is
given by
Xn
t ⇠ Ber(↵n
(Xt 1)), independently for n 2 [1 : N]

• Transition probability specified as
↵n
(Xt 1) =
(
nD(n) 1
P
m2N(n) Xm
t 1, if Xn
t 1 = 0
1 n, if Xn
t 1 = 1
• Interactions specified by an undirected network: D(n) and
N(n) denote the degree and neighbours of agent n
• Infection and recovery rates are modelled using
agent-specific attributes
n
= (1 + exp( >
wn
)) 1
, n
= (1 + exp( >
wn
)) 1
,
where , 2 Rd are parameters and wn 2 Rd are the
covariates of agent n (similarly ↵n
0 depends on 0)

• If the network is fully connected D(n) = N, N(n) = [1 : N]
and the agents are homogeneous n = , n =
• We recover the classical SIS model of Kermack and
McKendrick (1927), which has a deterministic limit as
N ! 1
• These simpler models o↵er dimension reduction which
facilitates inference
• However, one cannot incorporate network information and
agent attributes
• We will use these simplifications to construct efficient SMC
proposal distributions for the agent-based model

• Observations (Yt)t2[0:T] are the number of infections
reported over time
• Modelled as conditionally independent given (Xt)t2[0:T], and
Yt ⇠ g✓(·|Xt) = Bin(I(Xt), ⇢)
• I(Xt) =
PN
n=1 Xn
t is the number of infections and ⇢ 2 (0, 1) is
the reporting rate
• Parameters to be inferred ✓ = ( 0, , , ⇢)

Likelihood of agent-based SIS model
• We have a standard hidden Markov model
p✓(x0:T , y0:T ) = µ✓(x0)
T
Y
t=1
f✓(xt|xt 1)
T
Y
t=0
g✓(yt|xt)
• Computing the marginal likelihood
p✓(y0:T ) =
X
x0:T 2{0,1}N⇥(T+1)
p✓(x0:T , y0:T ),
using the forward algorithm costs O(22NT)
• For large N, we have to rely on sequential Monte Carlo
(SMC) methods to approximate the marginal likelihood

Likelihood estimation
• Efficiency of SMC crucially relies on the choice of proposal
distributions
• The bootstrap particle filter (BPF) can be readily
implemented as simulating the latent process is
straightforward
• Performance of BPF is poor if observations are informative,
e.g. its marginal likelihood estimator can collapse to zero
• We show how to implement the fully adapted auxiliary
particle filter (APF) that accounts for the next observation
• We propose a novel controlled SMC (cSMC) method that
takes the entire observation sequence into account

Auxiliary particle filter
• At time t 2 [1 : T], the APF samples particles from
p✓(xt|xt 1, yt) and weights them according to p✓(yt|xt 1)
• The predictive likelihood is
p✓(yt|xt 1) =
X
xt 2{0,1}N
f✓(xt|xt 1)g✓(yt|xt)
=
X
xt 2{0,1}N
N
Y
n=1
Ber(xn
t ; ↵n
(xt 1))Bin(yt; I(xt), ⇢)
=
N
X
it =yt
PoiBin(it; ↵n
(xt 1))Bin(yt; it, ⇢)
since the sum of independent Bernoulli with non-identical
success probabilities follows a Poisson binomial distribution

Auxiliary particle filter
• Poisson binomial PMF costs O(N2) to compute (Chen and
Liu, 1997)
• To sample, we augment It = I(Xt) as an auxiliary variable
p✓(xt, it|xt 1, yt) = p✓(it|xt 1, yt)p✓(xt|xt 1, it)
• Conditional distribution of the number of infections is
p✓(it|xt 1, yt) =
PoiBin(it; ↵n(xt 1))Bin(yt; it, ⇢)
p✓(yt|xt 1)
• Distribution of agent states conditioned on their sum is a
conditioned Bernoulli
p✓(xt|xt 1, it) = CondBer(xt; ↵(xt 1), it),
which costs O(N2) to sample (Chen and Liu, 1997)

Controlled sequential Monte Carlo
• The optimal proposal that gives a zero variance marginal
likelihood estimator is the smoothing distribution
p✓(x0:T |y0:T ) = p✓(x0|y0:T )
T
Y
t=1
p✓(xt|xt 1, yt:T )
• At time t 2 [1 : T], the transition is
p✓(xt|xt 1, yt:T ) =
f✓(xt|xt 1) ?
t (xt)
f✓( ?
t |xt 1)
• ?
t (xt) = p(yt:T |xt) is the backward information filter (BIF)
and f✓( ?
t |xt 1) =
P
xt 2{0,1}N f✓(xt|xt 1) ?
t (xt)

• BIF satisfies the backward recursion ?
T (xT ) = g✓(yT |xT ),
?
t (xt) = g✓(yt|xt)f✓( ?
t+1|xt), t 2 [0 : T 1]
• This costs O(22NT) to compute, so approximations are
necessary when N is large
• Instead of relying on regression (Guarniero et al., 2017; Heng
et al., 2020), our approach is based on dimensionality
reduction by coarse-graining the agent-based model
• We approximate the model with heterogenous agents by a
model with homogenous agents whose individual infection and
recovery rates given by their population averages, i.e.
n ⇡ ¯ = N 1
PN
n=1
n and n ⇡ ¯ = N 1
PN
n=1
n

• BIF of the approximate model t(I(xt)) can be computed
exactly in O(N3T) cost, and approximately in O(N2T)
• We then define the SMC proposal transition as
q✓(xt|xt 1) =
f✓(xt|xt 1) t(I(xt))
f✓( t|xt 1)
,
which can be sampled and weighted in the same way as APF
• Quality of proposals depend on the coarse-graining
approximation
• Finer-grained approximations can be obtained using clustering
of the infection and recovery rates, at the expense of
increased cost

Numerical illustration
• We simulate data for N = 100 fully connected agents and
T = 90 time steps
• E↵ective sample size of SMC methods
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original
small
large
0 25 50 75
0
25
50
75
100
0
25
50
75
100
0
25
50
75
100
time
ESS%
method ● ● ● ●
BPF APF cSMC1 cSMC2
Figure: Simulated observations (top), when observations at
t 2 {25, 50, 75} are replaced by byt/2c (middle), or min(2yt, N) (bottom)

• SMC log-marginal likelihood estimators with 512 particles as
the reporting rate ⇢ varies and other parameters fixed at DGP
1e−02
1e−01
1e+00
1e+01
1e+02
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ρ
Variance
method BPF APF cSMC1 cSMC2
0
20
40
60
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ρ
Relative
efficiency method BPF APF cSMC1 cSMC2
Figure: Variance (left) and relative efficiency compared to BPF (right)

• Estimated log-likelihood function as the number of
observations increases
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T = 10 T = 30 T = 90
−4 0 4 −4 0 4 −4 0 4
−4
0
4
βλ
1
β
λ
2
−100 −20 −10−5 0
log−likelihood
Figure: MLE (black dot) and DGP (red dot)

• Estimated log-likelihood function as the number of
observations increases
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T = 10 T = 30 T = 90
−4 0 4 −4 0 4 −4 0 4
−4
0
4
βλ
2
β
γ
2
−100 −20 −10−5 0
log−likelihood
Figure: MLE (black dot) and DGP (red dot)

Concluding remarks
• SMC algorithms can be readily deployed within particle
MCMC for parameter and state inference
• We considered APF and cSMC for the agent-based SIR model
• Could reduce cost of evaluating Poisson binomial PMF and
sampling conditioned Bernoulli for small and controllable bias
• A general alternative to SMC methods is MCMC algorithms
to sample from the smoothing distribution
• Article and R package to appear on arXiv soon

Statistical inference for agent-based SIS and SIR models

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