Parameter Estimation with Data-Driven Nonparametric Likelihood Functions

Area of Science:

• Computational Mathematics
• Statistical Modeling
• Machine Learning

Background:

• Surrogate modeling is crucial for complex systems where direct simulation is computationally expensive.
• Traditional parametric likelihood functions often fail to capture the underlying geometry of data.
• Data-driven approaches are needed to construct accurate likelihood functions without prior assumptions.

Purpose of the Study:

• To develop a data-driven nonparametric likelihood function using manifold learning and spectral expansion.
• To demonstrate the robustness and accuracy of this approach for parameter estimation.
• To compare the proposed method against existing parametric and non-parametric surrogate models.

Main Methods:

• Constructed a nonparametric likelihood function on a data manifold using kernel embedding of the conditional distribution.
• Employed diffusion maps to obtain data-driven basis functions that respect data geometry.
• Utilized spectral expansion for representing the likelihood function.

Main Results:

• The data-driven likelihood function's error bound is independent of basis function variance, enabling controlled data requirements.
• The proposed method shows superior performance compared to standard parametric models when data manifold dimension is lower than ambient space.
• Achieved comparable estimation accuracy to direct Markov Chain Monte Carlo (MCMC) with significantly fewer function evaluations (8 vs. 4000).

Conclusions:

• The proposed manifold-based, data-driven likelihood function offers a robust and accurate surrogate modeling approach.
• The method is particularly effective for complex, non-smooth, and unknown data manifolds.
• This approach significantly reduces computational cost for parameter estimation in complex systems.