Bayesian Optimization of Noisy Log-Likelihoods Evaluated by Particle Filters -- One Parameter Case --

Likelihood functions evaluated using particle filters are typically noisy, computationally expensive, and non-differentiable due to Monte Carlo variability. These characteristics make conventional optimization methods difficult to apply directly or potentially unreliable. This paper investigates the use of Bayesian optimization for maximizing log-likelihood functions estimated by particle filters. By modeling the noisy log-likelihood surface with a Gaussian process surrogate and employing an acquisition function that balances exploration and exploitation, the proposed approach identifies the maximizer using a limited number of likelihood evaluations. Through numerical experiments, we demonstrate that Bayesian optimization provides robust and stable estimation in the presence of observation noise. The results suggest that Bayesian optimization is a promising alternative for likelihood maximization problems where exhaustive search or gradient-based methods are impractical. The estimation accuracy is quantitatively assessed using mean squared error metrics by comparison with the exact maximum likelihood solution obtained via the Kalman filter.