Tuning IF2

Number of particles

• The initial parameter swarm, $\{ \Theta^0_j, j=1,\dots,J\}$, usually consists of $J$ identical replications of some starting parameter vector.
• $J$ is set to be sufficient for particle filtering. Because the addition of random perturbations acts to combat particle depletion, it is typically possible to take $J$ substantially smaller than the value needed to obtain precise likelihood estimates via pfilter. By the time of the last iteration ($m=M$) one should not have effective sample size close to 1.

Perturbations

• Perturbations are usually chosen to be multivariate normal, with $\sigma_m$ being a scale factor for iteration $m$: $$h_n(\theta|\varphi;\sigma) \sim N[\varphi, \sigma^2_m V_n].$$
• $V_n$ is usually taken to be diagonal, $$V_n = \left( \begin{array}{ccccc} v_{1,n}^2 & 0 & 0 & \cdots & 0 \\ 0 & v_{2,n}^2 & 0 & \cdots & 0 \\ 0 & 0 & v_{3,n}^2 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & v_{p,n}^2 \end{array}\right).$$
• If $\theta_i$ is a parameter that affects the dynamics or observations throughout the time series, it is called a regular parameter, and it is often appropriate to specify $$v_{i,n} = v_i.$$
• If $\theta_j$ is a parameter that affects only the initial conditions of the dynamic model, it is called an initial value parameter (IVP) and it is appropriate to specify $$v_{j,n} = \left\{\begin{array}{ll} v_j & \mbox{if $n=0$} \\0 & \mbox{if $n>0$} \end{array}\right.$$
• If $\theta_k$ is a break-point parameter that models how the system changes at time $t_q$, then $\theta_k$ is like an IVP at time $t_q$ and it is appropriate to specify $$v_{j,n} = \left\{\begin{array}{ll} v_j & \mbox{if $n=q$} \\ 0 & \mbox{if $n\neq q$} \end{array}\right.$$

Cooling schedule

• $\sigma_{1:M}$ is called a cooling schedule, following a thermodynamic analogy popularized by simulated annealing. As $\sigma_m$ becomes small, the system cools toward a “freezing point”. If the algorithm is working successfully, the freezing point should be close to the lowest-energy state of the system, i.e., the MLE. Typical choices of the cooling schedule are geometric, $\sigma_m = \alpha^m$, and hyperbolic, $\sigma_m \propto 1/(1+\alpha\,m)$. In mif2, the cooling schedule is parameterized by $\sigma_{50}$, the cooling fraction after 50 IF2 iterations.

Parameter transformations

• It is generally helpful to transform the parameters so that (on the estimation scale) they are real-valued, unconstrained, and have uncertainty on the order of 1 unit. For example, one typically takes a logarithmic transformation of positive parameters and a logistic transformation of $[0,1]$ valued parameters.
• On such a scale, it is surprisingly often effective to take $$v_i \sim 0.02$$ for regular parameters (RPs) and $$v_j \sim 0.1$$ for initial value parameters (IVPs).

Maximizing the likelihood in stages

• Early on in an investigation, one might take $M=100$ and $\sigma_M=0.1$. This allows for a relatively broad search of the parameter space.
• As the investigation proceeds, and one finds oneself in the heights of the likelihood surface, one can refine the search. In doing so, it helps to examine diagnostic plots.
• In particular, one typically needs to reduce the magnitude of the perturbations (rw.sd) and perhaps adjust the cooling schedule (cooling.fraction.50) to eke out the last few units of log likelihood.
• Profile likelihood computations are not only valuable as a way of obtaining confidence intervals. It is often the case that by profiling, one simplifies the task of finding the MLE. Of course, the price that is paid is that a profile calculation requires multiple parallel IF2 computations.

General remarks

• It is remarkable that useful general advice exists for the choice of algorithmic parameters that should in principle be model- and data-specific. Here is one possible explanation: the precision of interest is often the second significant figure and there are often on the order of 100 observations (10 monthly observations would be too few to fit a mechanistic model; 1000 would be unusual for an epidemiological system).

Exercises

References