Important Note: These materials have been superseded by materials provided as part of the as part of the SISMID short course on Simulation-based Inference.

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CC-BY_NC This document has its origins in the SISMID short course on Simulation-based Inference given by Aaron King and Edward Ionides.

Produced with R version 3.3.3 and pomp version 1.12.

Objectives

Students completing this lesson will:

  1. Gain an understanding of the nature of the problem of likelihood computation for POMP models.
  2. Be able to explain the simplest particle filter algorithm.
  3. Gain experience in the visualization and exploration of likelihood surfaces.
  4. Learn to apply standard optimization methods for the maximization of the likelihood.

Theory of the particle filter

The likelihood function

  • The basis for modern frequentist, Bayesian, and information-theoretic inference.
  • Method of maximum likelihood introduced by Fisher (1922).
  • The function itself is a representation of the what the data have to say about the parameters.
  • A good general reference on likelihood is Pawitan (2001).

Definition of the likelihood function

Data are a sequence of \(N\) observations, denoted \(y_{1:N}^*\). A statistical model is a density function \(f(y_{1:N};\theta)\) which defines a probability distribution for each value of a parameter vector \(\theta\). To perform statistical inference, we must decide, among other things, for which (if any) values of \(\theta\) it is reasonable to model \(y^*_{1:N}\) as a random draw from \(f(y_{1:N};\theta)\).

The likelihood function is the density function evaluated at the data. It is usually convenient to work with the log likelihood function, \[{\ell}(\theta)=\log f(y^*_{1:N};\theta)\]

Modeling using discrete and continuous distributions

Recall that the probability distribution \(f(y_{1:N};\theta)\) defines a random variable \(Y_{1:N}\) for which probabilities can be computed as integrals of \(f(y_{1:N};\theta)\). Specifically, for any event \(E\) describing a set of possible outcomes of \(Y_{1:N}\), \[P[Y_{1:N} \in E] = \int_E f(y_{1:N};\theta)\, dy_{1:N}.\] If the model corresponds to a discrete distribution, then the integral is replaced by a sum and the probability density function is called a probability mass function. The definition of the likelihood function remains unchanged. We will use the notation of continuous random variables, but all the methods apply also to discrete models.

Indirect specification of the statistical model via a simulation procedure

  • For simple statistical models, we may describe the model by explicitly writing the density function \(f(y_{1:N};\theta)\). One may then ask how to simulate a random variable \(Y_{1:N}\sim f(y_{1:N};\theta)\).
  • For many dynamic models it is much more convenient to define the model via a procedure to simulate the random variable \(Y_{1:N}\). This implicitly defines the corresponding density \(f(y_{1:N};\theta)\). For a complicated simulation procedure, it may be difficult or impossible to write down or even compute \(f(y_{1:N};\theta)\) exactly.
  • It is important to bear in mind that the likelihood function exists even when we don’t know what it is! We can still talk about the likelihood function, and develop numerical methods that take advantage of its statistical properties.

Likelihood for POMP models

The following schematic shows dependence among variables in a POMP model. Measurements, \(Y_n\), at time \(t_n\) depend on the state, \(X_n\), at that time. State variables depend on state variables at the previous timestep. To be more precise, the distribution of the state \(X_{n+1}\), conditional on \(X_{n}\), is independent of the values of \(X_{k}\), \(k<n\) and \(Y_{k}\), \(k\le n\). Moreover, the distribution of the measurement \(Y_{n}\), conditional on \(X_{n}\), is independent of all other variables.

POMP model notation

  • Write \(X_n=X(t_n)\) and \(X_{0:N}=(X_0,\dots,X_N)\).
  • Let \(Y_n\) be a random variable modeling the observation at time \(t_n\).
  • The one-step transition density, \(f_{X_n|X_{n-1}}(x_n|x_{n-1};\theta)\), together with the measurement density, \(f_{Y_n|X_n}(y_n|x_n;\theta)\) and the initial density, \(f_{X_0}(x_0;\theta)\), specify the entire joint density via \[f_{X_{0:N},Y_{1:N}}(x_{0:N},y_{1:N};\theta) = f_{X_0}(x_0;\theta)\,\prod_{n=1}^N\!f_{X_n | X_{n-1}}(x_n|x_{n-1};\theta)\,f_{Y_n|X_n}(y_n|x_n;\theta).\]
  • The marginal density for sequence of measurements, \(Y_{1:N}\), evaluated at the data, \(y_{1:N}^*\), is \[{\mathcal{L}}(\theta) = f_{Y_{1:N}}(y^*_{1:N};\theta)=\int\!f_{X_{0:N},Y_{1:N}}(x_{0:N},y^*_{1:N};\theta)\, dx_{0:N}.\]

Special case: deterministic unobserved state process

Lets’ begin with a special case. Suppose that the unobserved state process is deterministic. That is, \(X_{n}=x_n(\theta)\) is a known function of \(\theta\) for each \(n\). What is the likelihood?

Since the probability of the observation, \(Y_n\), depends only on \(X_n\) and \(\theta\), and since, in particular \(Y_{m}\) and \(Y_{n}\) are independent given \(X_{m}\) and \(X_{n}\), we have \[{\mathcal{L}}(\theta) = \prod_{n} f_{Y_n|X_n}(y_n^*;x_n(\theta),\theta)\] or \[\ell(\theta) = \log{\mathcal{L}}(\theta) = \sum_{n} \log f_{Y_n|X_n}(y_n^*;x_n(\theta),\theta).\] The following diagram illustrates this.

In this diagram, \(\hat y_n\) refers to the model prediction (\(\hat y_n = {\mathbb{E}\left[{Y_n \vert X_n=x_n(\theta)}\right]}\)) and \(y_n^*\) to the data.

General case: stochastic unobserved state process

For a POMP model, the likelihood takes the form of an integral: \[{\mathcal{L}}(\theta)={\mathbb{P}\left[{y^*_{1:N}|\theta}\right]}=\int\!\prod_{n=1}^{N}\!{\mathbb{P}\left[{y^*_n|x_n,\theta}\right]}\,{\mathbb{P}\left[{x_n|x_{n-1},\theta}\right]}\,dx_{1}\,\cdots\,dx_{N}.\] This integral is high dimensional and, except for the simplest cases, can not be reduced analytically. A general approach to computing integrals, which will be useful here is Monte Carlo integration.

[See here for a brief introduction to Monte Carlo methods.]

Let us investigate a Monte Carlo integration scheme for the POMP likelihood. In particular, if we propose trajectories of the unobserved state process according to some probabilistic rule, we can generate a large number of these and approximate \({\mathcal{L}}(\theta)\) by its Monte Carlo estimate. Specifically, let us randomly generate \(J\) trajectories of length \(N\), \(x_{n,j}\), \(j=1\,\dots,J\), \(n=1,\dots,N\). Let \(w_j\) denote the probability that we propose trajectory \(j\). We compute the likelihood of each trajectory \[{\mathcal{L}}{_j}(\theta)=\prod_{n=1}^{N} {\mathbb{P}\left[{y^*_n|x_{n,j},\theta}\right]}\,{\mathbb{P}\left[{x_{n,j}|x_{n-1,j},\theta}\right]}\] Then by the Monte Carlo theorem, we have \[{\mathcal{L}}(\theta) \approx \frac{1}{J}\,\sum_{j=1}^{J}\!\frac{{\mathcal{L}}_j(\theta)}{w_j}.\]

How shall we choose our trajectories? One idea would be to choose them so as to simplify the computation. If we choose them such that \[w_j=\prod_{n=1}^{N} {\mathbb{P}\left[{x_{n,j}|x_{n-1,j},\theta}\right]},\] then we have \[{\mathcal{L}}(\theta) \approx \frac{1}{J}\,\sum_{j=1}^{J} \frac{{\mathcal{L}}_j(\theta)}{w_j} = \frac{1}{J}\,\sum_{j=1}^{J}\!\frac{\prod_{n=1}^{N} {\mathbb{P}\left[{y^*_n|x_{n,j},\theta}\right]}\,{\mathbb{P}\left[{x_{n,j}|x_{n-1,j},\theta}\right]}}{\prod_{n=1}^{N} {\mathbb{P}\left[{x_{n,j}|x_{n-1,j},\theta}\right]}} = \frac{1}{J}\,\sum_{j=1}^{J}\!\prod_{n=1}^{N} {\mathbb{P}\left[{y^*_n|x_{n,j},\theta}\right]}\]

This implies that if we generate trajectories by simulation, all we have to do is compute the likelihood of the data with given each trajectory and then average.

Let’s go back to the boarding school influenza outbreak to see what this looks like in practice. Let’s reconstruct the toy SIR model we were working with.

read.table("http://kingaa.github.io/clim-dis/stochsim/bsflu_data.txt") -> bsflu

rproc <- Csnippet("
  double N = 763;
  double t1 = rbinom(S,1-exp(-Beta*I/N*dt));
  double t2 = rbinom(I,1-exp(-mu_I*dt));
  double t3 = rbinom(R1,1-exp(-mu_R1*dt));
  double t4 = rbinom(R2,1-exp(-mu_R2*dt));
  S  -= t1;
  I  += t1 - t2;
  R1 += t2 - t3;
  R2 += t3 - t4;
")

init <- Csnippet("
  S = 762;
  I = 1;
  R1 = 0;
  R2 = 0;
")

dmeas <- Csnippet("
  lik = dpois(B,rho*R1+1e-6,give_log);
")

rmeas <- Csnippet("
  B = rpois(rho*R1+1e-6);
")

pomp(subset(bsflu,select=-C),
     times="day",t0=0,
     rprocess=euler.sim(rproc,delta.t=1/5),
     initializer=init,rmeasure=rmeas,dmeasure=dmeas,
     statenames=c("S","I","R1","R2"),
     paramnames=c("Beta","mu_I","mu_R1","mu_R2","rho")) -> flu

Let’s generate a large number of simulated trajectories at some particular point in parameter space.

simulate(flu,params=c(Beta=3,mu_I=1/2,mu_R1=1/4,mu_R2=1/1.8,rho=0.9),
         nsim=5000,states=TRUE) -> x
matplot(time(flu),t(x["R1",1:50,]),type='l',lty=1,
        xlab="time",ylab=expression(R[1]),bty='l',col='blue')
lines(time(flu),obs(flu,"B"),lwd=2,col='black')

We can use the function dmeasure to evaluate the log likelihood of the data given the states, the model, and the parameters:

ell <- dmeasure(flu,y=obs(flu),x=x,times=time(flu),log=TRUE,
                params=c(Beta=3,mu_I=1/2,mu_R1=1/4,mu_R2=1/1.8,rho=0.9))
dim(ell)
## [1] 5000   14

According to the equation above, we should sum up the log likelihoods across time:

ell <- apply(ell,1,sum)
summary(exp(ell))
##       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
##  0.000e+00  0.000e+00  0.000e+00 2.899e-136  0.000e+00 1.448e-132

The variability in the individual likelihoods is high and therefore the likelihood esitmate is imprecise. We will need many simulations to get an estimate of the likelihood sufficiently precise to be of any use in parameter estimation or model selection.

What is the problem? Essentially, very few of the trajectories pass anywhere near the data and therefore almost all have extremely bad likelihoods. Moreover, once a trajectory diverges from the data, it almost never comes back. While the calculation is “correct” in that it will converge to the true likelihood as the number of simulations tends to \(\infty\), we waste a lot of effort investigating trajectories of very low likelihood. This is a consequence of the fact that we are proposing trajectories in a way that is completely unconditional on the data. The problem will get much worse with longer data sets.

The particle filter

We arrive at a more efficient algorithm by factorizing the likelihood in a different way: \[{\mathcal{L}}(\theta)={\mathbb{P}\left[{y^*_{1:N}|\theta}\right]} =\prod_{n}\,{\mathbb{P}\left[{y^*_n|y^*_{1:n-1},\theta}\right]} =\prod_{n}\,\int\!{\mathbb{P}\left[{y^*_n|x_n,\theta}\right]}\,{\mathbb{P}\left[{x_n|y^*_{1:n-1},\theta}\right]}\,dx_{n}.\tag{1}\] Now, the Markov property gives us the Chapman-Kolmogorov equation, \[{\mathbb{P}\left[{x_n|y^*_{1:n-1},\theta}\right]} = \int\!{\mathbb{P}\left[{x_n|x_{n-1},\theta}\right]}\,{\mathbb{P}\left[{x_{n-1}|y^*_{1:n-1},\theta}\right]}\,dx_{n-1},\tag{2}\] and Bayes’ theorem tells us that \[{\mathbb{P}\left[{x_{n}|y^*_{1:n},\theta}\right]} = {\mathbb{P}\left[{x_{n}|y^*_{n},y^*_{1:n-1},\theta}\right]} =\frac{{\mathbb{P}\left[{y^*_{n}|x_{n},\theta}\right]}\,{\mathbb{P}\left[{x_{n}|y^*_{1:n-1},\theta}\right]}}{\displaystyle\int\!{\mathbb{P}\left[{y^*_{n}|x_{n},\theta}\right]}\,{\mathbb{P}\left[{x_{n}|y^*_{1:n-1},\theta}\right]}\,dx_{n}}.\tag{3}\]

This suggests that we keep track of two key distributions. We’ll refer to the distribution of \(X_n | y^*_{1:n-1}\) as the prediction distribution at time \(t_n\) and the distribution of \(X_{n} | y^*_{1:n}\) as the filtering distribution at time \(t_n\).

Let’s use Monte Carlo techniques to estimate the integrals. Suppose \(\left\{x_{n-1,j}^{F}\right\}_{j=1}^J\) is a set of points drawn from the filtering distribution at time \(t_{n-1}\). Eqn. 2 tells us that we obtain a sample \(\left\{x_{n,j}^{P}\right\}\) of points drawn from the prediction distribution at time \(t_n\) by simply simulating the process model: \[X_{n,j}^{P} \sim \mathrm{process}(x_{n-1,j}^{F},\theta), \qquad j=1,\dots,J.\] Having obtained \(\left\{x_{n,j}^{P}\right\}\), we obtain a sample of points from the filtering distribution at time \(t_n\) by resampling from \(\left\{x_{n,j}^{P}\right\}\) with weights proportional to \({\mathbb{P}\left[{y^*_{n}|x_{n},\theta}\right]}\). The Monte Carlo theorem tells us, too, that the conditional likelihood \[{\mathcal{L}}_n(\theta) = {\mathbb{P}\left[{y^*_n|y^*_{1:n-1},\theta}\right]} = \sum_{x_{n}}\,{\mathbb{P}\left[{y^*_{n}|x_{n},\theta}\right]}\,{\mathbb{P}\left[{x_{n}|y^*_{1:n-1},\theta}\right]} \approx \frac{1}{J}\,\sum_j\,{\mathbb{P}\left[{y^*_{n}|x_{n,j}^{P},\theta}\right]}.\] We can iterate this procedure through the data, one step at a time, alternately simulating and resampling, until we reach \(n=N\). The full log likelihood is then approximately \[{\ell}(\theta) = \log{{\mathcal{L}}(\theta)} \approx \sum_n \log{{\mathcal{L}}_n(\theta)}.\] It can be shown that this estimate of the likelihood is unbiased.

This is known as the sequential Monte Carlo algorithm or the particle filter. Key references include Kitagawa (1987), Arulampalam et al. (2002), and the book by Doucet et al. (2001). Pseudocode for the above is provided by King et al. (2016).

Sequential Monte Carlo in pomp

Here, we’ll get some practical experience with the particle filter, and the likelihood function, in the context of our influenza-outbreak case study.

In pomp, the basic particle filter is implemented in the command pfilter. We must choose the number of particles to use by setting the Np argument.

pf <- pfilter(flu,Np=5000,params=c(Beta=3,mu_I=1/2,mu_R1=1/4,mu_R2=1/1.8,rho=0.9))
logLik(pf)
## [1] -220.998

We can run a few particle filters to get an estimate of the Monte Carlo variability:

pf <- replicate(10,
                pfilter(flu,Np=5000,
                        params=c(Beta=3,mu_I=1/2,mu_R1=1/4,mu_R2=1/1.8,rho=0.9)))
ll <- sapply(pf,logLik)
logmeanexp(ll,se=TRUE)
##                    se 
## -213.52555    3.09948

The graph of the likelihood function: The likelihood surface

It is extremely useful to visualize the geometric surface defined by the likelihood function.

To get an idea of what the likelihood surface looks like in the neighborhood of a point in parameter space, we can construct some likelihood slices. We’ll make slices in the \(\beta\) and \(\mu_I\) directions. Both slices will pass through the central point.

sliceDesign(
  center=c(Beta=2,mu_I=1,mu_R1=1/4,mu_R2=1/1.8,rho=0.9),
  Beta=rep(seq(from=0.5,to=4,length=40),each=3),
  mu_I=rep(seq(from=0.5,to=2,length=40),each=3)) -> p

library(foreach)
library(doParallel)
registerDoParallel()

set.seed(108028909,kind="L'Ecuyer")

foreach (theta=iter(p,"row"),.combine=rbind,
         .inorder=FALSE,
         .options.multicore=list(set.seed=TRUE)
) %dopar% {
  library(pomp)
  pfilter(flu,params=unlist(theta),Np=5000) -> pf
  theta$loglik <- logLik(pf)
  theta
} -> p

Note that we’ve used the foreach package with the parallel backend (doParallel) to parallelize these computations. To ensure that we have high-quality random numbers in each parallel R session, we use a parallel random number generator (kind="L'Ecuyer", .options.multicore=list(set.seed=TRUE)).

Exercise: Likelihood slice

Add likelihood slices along the \(\rho\) direction.

Slices offer a very limited perspective on the geometry of the likelihood surface. With just two parameters, we can evaluate the likelihood at a grid of points and visualize the surface directly.

bake(file="flu-grid1.rds",seed=421776444,kind="L'Ecuyer",{
  
  expand.grid(Beta=seq(from=1.5,to=5,length=50),
              mu_I=seq(from=0.7,to=4,length=50),
              mu_R1=1/4,mu_R2=1/1.8,
              rho=0.9) -> p
  
  library(foreach)
  library(doParallel)
  registerDoParallel()
  
  ## Now we do the computation
  foreach (theta=iter(p,"row"),.combine=rbind,.inorder=FALSE,
           .options.multicore=list(set.seed=TRUE)
  ) %dopar% 
  {
    library(pomp)
    pfilter(flu,params=unlist(theta),Np=5000) -> pf
    theta$loglik <- logLik(pf)
    theta
  }
  
})-> p

In the above, all points with log likelihoods less than 50 units below the maximum are shown in grey.

Exercise: 2D likelihood slice

Compute a slice of the likelihood in the \(\beta\)-\(\rho\) plane.

Maximizing the likelihood

Call the whole parameter space \(\Theta\). Let \(\Theta^*\) be a subset of \(\Theta\), constraining parameters according to a scientific hypothesis of interest. For example, in a disease transmission model, \(\Theta^*\) could assert that the probability of a case being reported is \(\rho=0.8\).

Exercise: AIC as a formal statistical test

Determine the size of AIC as a hypothesis test for nested hypotheses with \(d=1\) in a regular parametric situation.

Point estimates for parameters: The maximum likelihood estimate (MLE)

We define maximum likelihood estimates (MLEs) \(\hat\theta\) and \(\hat\theta^*\) such that \[\ell(\hat\theta)=\ell_\mathrm{max},\quad \ell(\hat\theta^*)=\ell_\mathrm{max}^*.\]

  • If the likelihood function has a flat region, or ridge, at its maximum then the MLE is not unique. Alternatively, one can talk about a maximum likelihood surface describing the set of parameter values for which \(\ell(\hat\theta)=\ell_\mathrm{max}\).
  • Flat, or nearly flat, ridges in the likelihood surface are not an idle concern. Many dynamic models have combinations of parameters that are weakly identified: they cannot be well estimated on the basis of the data.

Confidence intervals for parameters: Profile likelihood

The likelihood ratio test with \(d=1\) gives a good way to construct confidence intervals. Suppose we are interested in a specific parameter, \(\theta_k\), and we want to consider whether the data support the possibility that \(\theta_k=\theta_k^*\) in the absence of assumptions on the other parameters. We can then take \(\Theta^*\) to be the subset of \(\Theta\) satisfying \(\theta_k=\theta_k^*\). Using the chi-square approximation to the likelihood ratio statistic, a 95% confidence interval for \(\theta_k\) consists of all the values \(\theta_k^*\) for which \[\ell_\mathrm{max}-\ell_\mathrm{max}^* < 1.92.\]

A way to visualize the information about a specific parameter \(\theta_k\) is via the profile likelihood function, defined as \[\ell_\mathrm{profile}(\theta_k^*) = \max\{\ell(\theta): \theta_k=\theta_k^*\}.\] We then plot \(\ell_\mathrm{profile}(\theta_k)\) against \(\theta_k\).

  • The set of values of \(\theta_k\) for which \(\ell_\mathrm{profile}(\theta_k)\) lies above a horizontal line with \(y\)-axis value \(\ell_\mathrm{max}-c\) gives an approximate confidence interval (according to Wilks’ theorem) with confidence level given by \({\mathbb{P}\left[{\chi^2_1<2c}\right]}\).
  • The maximum of \(\ell_\mathrm{profile}(\theta_k)\) over all values of \(\theta_k\) is \(\ell_\mathrm{max}\).
  • Thus, a profile plot allows us to visualize an entire spectrum of confidence intervals.
  • If the profile plot has two peaks (i.e., \(\ell_\mathrm{profile}(\theta_k)\) is bimodal) then a likelihood ratio test helps us to assess whether or not both peaks provide adequate explanations of the data.

Maximizing the likelihood using the particle filter

In the toy example we’ve been working with, the default parameter set is not particularly close to the MLE. One way to find the MLE is to try optimizing the estimated likelihood directly. There are of course many standard optimization algorithms we might use for this. However, three issues arise immediately:

  1. The particle filter gives us a stochastic estimate of the likelihood. We can reduce this variability by making \(J\) larger, but we cannot make it go away. If we use a deterministic optimizer (i.e., one that assumes the objective function is evaluated deterministically), then we must control this variability somehow. For example, we can fix the seed of the pseudo-random number generator (RNG). A side effect will be that the objective function becomes jagged, marked by many small local knolls and pits. Alternatively, we can use a stochastic optimization algorithm, with which we will be only be able to obtain estimates of our MLE. This is the trade-off between a rough and a noisy objective function.
  2. Because the particle filter gives us just an estimate of the likelihood and no information about the derivative, we must choose an algorithm that is “derivative-free”. There are many such, but we can expect less efficiency than would be possible with derivative information. Note that finite differencing is not an especially promising way of constructing derivatives. The price would be a \(n\)-fold increase in cpu time, where \(n\) is the dimension of the parameter space. Also, since the likelihood is noisily estimated, we would expect the derivative estimates to be even noisier.
  3. Finally, the parameters set we must optimize over is not unbounded. We must have \(\beta,\mu_I>0\) and \(0<\rho<1\). We must therefore select an optimizer that can solve this constrained maximization problem, or find some of way of turning it into an unconstrained maximization problem. For example, we can transform the parameters onto a scale on which there are no constraints.

Here, let’s opt for deterministic optimization of a rough function. We’ll try using optim’s default method: Nelder-Mead, fixing the random-number generator seed to make the likelihood calculation deterministic. Since Nelder-Mead is an unconstrained optimizer, we must transform the parameters. The following Csnippets encode an appropriate transformation and its inverse, and introduce them into the pomp object.

toEst <- Csnippet("
 TBeta = log(Beta);
 Tmu_R1 = log(mu_R1);
 Tmu_I = log(mu_I);
 Trho = logit(rho);
")

fromEst <- Csnippet("
 TBeta = exp(Beta);
 Tmu_I = exp(mu_I);
 Tmu_R1 = exp(mu_R1);
 Trho = expit(rho);
")

pomp(flu,toEstimationScale=toEst,
     fromEstimationScale=fromEst,
     paramnames=c("Beta","mu_I","mu_R1","rho")) -> flu

Let’s fix a reference point in parameter space and insert these parameters into the pomp object:

coef(flu) <- c(Beta=2,mu_I=1,mu_R1=512/sum(bsflu$B),mu_R2=512/sum(bsflu$C),rho=0.9)

The following constructs a function returning the negative log likelihood of the data at a given point in parameter space. The parameters to be estimated are named in the est argument. Note how the freeze function is used to fix the seed of the RNG. Note too, how this function returns a large (and therefore bad) value when the particle filter encounters and error. This behavior makes the objective function more robust.

neg.ll <- function (par, est) {
  allpars <- coef(flu,transform=TRUE)
  allpars[est] <- par
  try(
    freeze(
      pfilter(flu,params=partrans(flu,allpars,dir="fromEst"),
              Np=2000),
      seed=915909831
    )
  ) -> pf
  if (inherits(pf,"try-error")) 1e10 else -logLik(pf)
}

Now we call optim to minimize this function:

## use Nelder-Mead with fixed RNG seed
fit <- optim(
  par=c(log(2), log(1), log(0.9/(1-0.9))),
  est=c("Beta","mu_I","rho"),
  fn=neg.ll,
  method="Nelder-Mead",
  control=list(maxit=400,trace=0)
)

mle <- flu
coef(mle,c("Beta","mu_I","rho"),transform=TRUE) <- fit$par
coef(mle)
##      Beta      mu_I     mu_R1     mu_R2       rho 
## 3.3108940 2.0567896 0.3324675 0.5541126 0.9111065
fit$val
## [1] 69.56317
lls <- replicate(n=5,logLik(pfilter(mle,Np=20000)))
ll <- logmeanexp(lls,se=TRUE); ll
##                      se 
## -73.7312880   0.4793083

We plot some simulations at these parameters.

simulate(mle,nsim=10,as.data.frame=TRUE,include.data=TRUE) -> sims

The data are shown in blue. The 10 simulations are shown in red.

Exercise: More slices

Construct likelihood slices through the MLE we just found.

Exercise: Visualizing the likelihood surface

Evaluate the likelihood at points on a grid lying in a 2D slice through the MLE we found above. Each group should choose a different slice. Afterward, we’ll compare results across groups.

Exercise: Global maximization

The search of parameter space we conducted above was local. It is possible that we found a local maximum, but that other maxima exist with higher likelihoods. Conduct a more thorough search by initializing the Nelder-Mead starting points across a wider region of parameter space. Do you find any other local maxima?

Exercise: Modify the measurement model

The Poisson measurement model used here may not seem easy to interpret. Formulate an alternative measurement model and maximize the likelihood to compare the alternative model.

Exercise: Fit more parameters.

Try to estimate \(\beta\), \(\mu_I\), \(\rho\), and \(\mu_{R1}\) simultaneously. Does your estimate of \(\mu_{R1}\) differ from the value we computed from the raw data? How do you interpret the agreement or lack thereof?

Biological interpretation of parameter estimates

When we write down a mechanistic model for an epidemiological system, we have some idea of what we intend parameters to mean; a reporting rate, a contact rate between individuals, an immigration rate, a duration of immunity, etc.

Back to course homepage

R codes for this document

References

Arulampalam, M. S., S. Maskell, N. Gordon, and T. Clapp. 2002. A tutorial on particle filters for online nonlinear, non-Gaussian Bayesian tracking. IEEE Transactions on Signal Processing 50:174–188.

Doucet, A., N. de Freitas, and N. Gordon, editors. 2001. Sequential Monte Carlo Methods in Practice. Springer-Verlag, New York.

Fisher, R. A. 1922. On the mathematical foundations of theoretical statistics. Philosophical Transactions of the Royal Society of London, Series A 222:309–368.

King, A. A., D. Nguyen, and E. L. Ionides. 2016. Statistical inference for partially observed Markov processes via the R package pomp. Journal of Statistical Software 69:1–43.

Kitagawa, G. 1987. Non-Gaussian state-space modeling of nonstationary time series. Journal of the American Statistical Association 82:1032–1041.

Pawitan, Y. 2001. In All Likelihood: Statistical Modelling and Inference Using Likelihood. Clarendon Press, Oxford.