Iterated filtering

Iterated filtering algorithms are a tool for maximum likelihood inference on partially observed dynamical systems. Stochastic perturbations to the unknown parameters are used to explore the parameter space. Applying sequential Monte Carlo (the particle filter) to this extended model results in the selection of the parameter values that are more consistent with the data. Appropriately constructed procedures, iterating with successively diminished perturbations, converge to the maximum likelihood estimate. Iterated filtering methods have so far been used most extensively to study infectious disease transmission dynamics. Case studies include cholera, Ebola virus, influenza, malaria, HIV, pertussis, poliovirus and measles. Other areas which have been proposed to be suitable for these methods include ecological dynamics and finance. 
The perturbations to the parameter space play several different roles. Firstly, they smooth out the likelihood surface, enabling the algorithm to overcome small-scale features of the likelihood during early stages of the global search. Secondly, Monte Carlo variation allows the search to escape from local minima. Thirdly, the iterated filtering update uses the perturbed parameter values to construct an approximation to the derivative of the log likelihood even though this quantity is not typically available in closed form. Fourthly, the parameter perturbations help to overcome numerical difficulties that can arise during sequential Monte Carlo.
Overview.
The data are a time series formula_1 collected at times formula_2. The dynamic system is modeled by a Markov process formula_3 which is generated by a function formula_4 in the sense that
where formula_6 is a vector of unknown parameters and formula_7 is some random quantity that is drawn independently each time formula_8 is evaluated. An initial condition formula_9 at some time formula_10 is specified by an initialization function, formula_11. A measurement density formula_12 completes the specification of a partially observed Markov process. We present a basic iterated filtering algorithm (IF1) followed by an iterated filtering algorithm implementing an iterated, perturbed Bayes map (IF2).
Software.
"pomp: statistical inference for observed Markov processes" : R package.