State space models are important tools for time series analysis, particularly when data come from partially observed dynamic systems. Despite their importance, likelihood-based inference with these models is challenging because a closed-form expression of the likelihood function is unavailable except in the simplest cases. This dissertation introduces three projects aimed at advancing likelihood-based inference for state space models.

The first project proposes a novel algorithm for maximum likelihood estimation of the parameters of Auto Regressive Moving Average (ARMA) models, which are formally a special case of linear Gaussian state space models. The proposed algorithm overcomes underrecognized optimization shortcomings of existing parameter estimation methods. The second project presents a likelihood-based analysis of the 2010-2019 cholera outbreak in Haiti. This work demonstrates the effectiveness of recently developed algorithms for performing inference on high-dimensional models. A key focus of this project is to assess the strengths and limitations of using state space models to inform public health policy decisions. The third project, which is the primary focus of my presentation, is a novel simulation-based algorithm called the Marginalized Panel Iterated Filter (MPIF). This algorithm is designed for maximum likelihood estimation of parameters from large collections of independent state space models. Theoretical support for this algorithm is provided through an analysis of iterating marginalized Bayes maps. New theoretical developments for the convergence of iterated filtering algorithms on this class of models are also derived.