Bi-modality of Posterior Distribution for the In Progress GSE

As an epidemic progresses and more data in the form of removal times becomes available the posterior distribution of the parameters evolves and becomes more informative. The exact calculation of the marginal likelihood that was derived above provides a mechanism by which the evolution of the posterior has been studied. In practical terms there is usually more interest in inference for an epidemic that is in progress than for one that is known to be complete. The results show that for simulated epidemics the likelihood for the “in progress” case is bi-modal, with one mode approaching the mode for the complete epidemic and another corresponding to a very high infection rate, with many cases still infected.

Figure 4.3.2: GSE marginal likelihood for simulated epidemic with np = 50,R0 = 1.2, left hand plot complete epidemic size=25, right hand plot epidemic in progress, x-axis is the infection rate and the y-axis is the recovery rate. The green dot indicates the true parameters, and the red dot the MLE for the completed epidemic, the straight lines indicateR0 = 1.2,2,5.

Figure 4.3.4: GSE marginal likelihood complete and in progress

Figure 4.3.5: GSE marginal likelihood complete and in progress long tail example

Some examples are shown in figures 4.3.2-4.3.5, of contours of the exact marginal log likelihood for simulated data on a population of 50, in each figure the green dot indicates the true parameters, and the red dot the MLE for the completed epidemic, the straight lines indicateR0 = 1.2,2,5, the x-axis is the infection rate and the y-axis is the recovery rate. Figure 4.3.2 shows the difference between a complete and in progress epidemic and has “nice” elliptical contours matching the simulatedR0 value of 1.2, an MCMC algorithm on these data might be expected to

perform well. The example in figure 4.3.3 is more surprising, the final size is 50, all are infected and the marginal estimate of the recovery rate is reasonable, however the posterior for the infection rate covers a wide range of values. Figures 4.3.4 and 4.3.5 also correspond to the whole population being infected, with the first from a small population showing even less information in the posterior on the infection rate.

4.4

Inference for Regularly Observed Epidemics

Exact Bayesian inference for epidemics in which all the removal times are observed exactly has been considered in section 4.3, however usually data on epidemics are only available on a daily basis, or sometimes less frequently. Two approaches to model choice and subsequent inference are possible, either a discrete time model as described in section 2.4 or a continuous time model with discrete observations. The choice between these classes of model will often be guided by epidemiological consid- erations, which are not considered here. It is often easier to obtain analytic results from a continuous time model than from a realistic discrete time model however when numerical computations are used this advantage is reduced considerably.

The Markov representation of the GSE as described in section 2.3 combined with the matrix exponential and the binomial model developed in section 2.4 provide two very similar discrete time, discrete state space hidden Markov models (HMM). Inference for the parameters of an HMM has an extensive literature, in addition to the even larger literature studying estimates of the hidden states with known parameters.

This section considers a general finite state space HMM. We follow the common practice of denoting ranges of vectors or random variables as X1:a =

X1, X2. . . , Xaand usingX−t= (X1:t−1, Xt+1,T) whereT is the size ofX and known

from the context. We frame the description in general terms and consider the case where there arens states, and the state att isXtwe have T observations Y1:T and

the transition matrix is P = (pij) i, , j∈ {1. . . ns} wherepij =P(Xt+1=j|Xt=i) with an initial state distribution _P(X1 =i) = νi, the observation distribution is

P(Yt=j|Xt=i) =gi(j) and is independent of the otherX.

In many applications of HMM the transition matrix is known exactly, and interest is in efficient algorithms to calculate the exact marginal posterior distribu- tions P(Xt|Y1:n) which can be calculated using the forward-backward algorithm a

can be calculated2. These calculations and the inference algorithms below rely on recursive computations of quantities, often known as alpha and beta. In the case of finite state and time, which we are considering, they are the probabilities defined as

αt(i) =P(Y1:t, Xt=i) andβt(i) =P(Yt+1:T|Xt=i) (4.4.1)

in more general cases the definitions are similar, Cappé et al. (2005) gives details, he also describes the scaling necessary to prevent underflow in their calculation. These are calculated asα1(i) =νigi(y1) for 1≤i≤ns and a forward recursion

αt+1(j) =

X

i

αt(i)pijgj(yt+1) for 1≤t < T

andβT(i) = 1 for 1≤i≤ns and a backward recursion

βt(i) =X

j

βt+1(j)pijgj(yt+1) for 1≤t < T

the calculation of both recursions isO(n2_sT).

Figure 4.4.1: Exact marginal and full data log likelihood, for regularly observed GSE. 2 simulations numbered 17 and 87 both with np = 50. In the marginal likelihood the epidemic is assumed complete.

Many authors have considered inference in HMM, the book Cappé et al. (2005) provides a comprehensive treatment, also of note is Fearnhead (2011). The particular case of Markov jump processes, of which the SIR epidemic is an exam- ple, has been considered by Bladt and Sorensen (2005) who compare the EM and a standard MCMC algorithm. Golightly and Wilkinson (2011) describe a Pseudo Marginal Metopolis Hastings (PMMH) algorithm for a rabge of Markov jump pro- cesses. Often in the HMM literature the state space is assumed to be small and the parameter space is large and the noise in the observations is significant, whereas in the models we are considering the state space is large, the parameter space is small and the observations are partial without noise.

This section demonstrates that the exact calculation of the transition matrix using the matrix exponential can be used in standard HMM algorithms to provide exact inference for small population sizes.

We frame the algorithms in general terms and consider the case where there arensstates, and the state attisXtwe haveT observationsY1:T and the transition

matrix isP = (pij)i, j∈ {1. . . ns}wherepij =P(Xt+1 =j|Xt=i) the dependence onθ is often dropped below to ease the notation.

Care is needed when using the term likelihood as it can mean_P(X1:T, Y1:T|θ)

orP(Y1:T|θ), the full data likelihood is readily calculated as

P(X1:T =x1:T|θ)P(Y1:T =y1:T|x1:T, θ) =P(X1=x1|θ) T Y t=2 pxt−1xt T Y t=1 gxt(yt)

while the marginal or observed data likelihood is obtained from the forward recursions as P(y1:n) =P(y1) T Y t=2 P(yt|y1:t−1) (4.4.2)

Consideration of the boundary conditions is important and problem specific, usually the first state is assumed to be drawn from a specified distribution and the final state is unconstrained. The SIR epidemic differs, often the epidemic is assumed complete corresponding to a known final state XT. Although the epidemic model starts with a known number of infectives, which is taken to be 1, the time of the first infection is unknown and therefore so is the number of infectives at the first observation.

Figure 4.4.2: Exact marginal and full data log likelihood, for regularly observed GSE. 2 simulations numbered 17 and 87 both with np = 50. In the marginal

likelihood it is not asssumed that the epidemic is complete.

Given values of the parameters θ the exact transition matrix can be calcu- lated using the matrix exponential as shown previously. The likelihoods for example data can then be calculated using the forward recursions and can be maximised nu- merically to calculate a MLE. Contour plots of likelihoods for simulated epidemics are shown in figure 4.4.1 showing the greater dispersion of the marginal likelihood over the full data likelihood. Thes plots are for ∆t= 1 similar plots (not shown) were obtained for ∆t= 0.1. When the epidemic is complete the likelihood is uni-modal, however when this assumption is not made a bi-modal distribution is possible. The same data are analysed without the assumption of complete data and contours are shown in figure 4.4.2. The two modes corresponds to the true model and a “false” model with a high infection rate with a low recovery rate giving a large number of infectives at the last observation. These calculations can be used in a Bayesian framework by multiplying the likelihood by a prior forθand normalizing by numer- ical integration, this approach was taken by Fearnhead and Meligkotsidou (2004) using a continuous time observation model, and an alternative way of calculating the exact likelihood.

MCMC Algorithms for Regularly Observed Epidemics A variety of MCMC algorithms have been developed for inference in HMM, a recent review is given in Fearnhead (2011). Investigation of the relative performance of the algorithms de- scribed there and other GIMH algorithms using the exact transition matrix for the GSE would be useful topic for future study. These algorithms are described in appendix C.

4.4.1 Concluding Remarks on Inference for Regularly Observed