Bayesian Inference for Semiconductor Metrology Spatial Distribution

The patent changes the approach from using fixed base functions to using a probabilistic model where parameters are inferred from data. The model uses a prior distribution over possible parameter values and updates it with observed metrology data to obtain a posterior distribution, allowing the model to adapt to actual observed characteristics rather than forcing data into predetermined functional forms.

The patent implements feedback by using observed metrology data to update the model parameters through Bayesian inference. The measured overlay errors and other metrology parameters provide feedback that refines the spatial distribution model, allowing the system to learn from actual measurements and improve its accuracy iteratively.

The patent applies preliminary action by using a prior distribution that encapsulates existing knowledge and expectations about the spatial distribution before new measurements are taken. This prior can be pre-computed and stored, avoiding the need to perform complex computational operations from scratch for each new measurement, thus reducing processing time while maintaining reliability.

The patent uses partial action by focusing the Bayesian inference on the specific parameters and regions most relevant to the current measurement task, rather than performing complete re-evaluation of all possible spatial distributions. This selective approach reduces computational burden while maintaining sufficient accuracy for process control.