Bayesian interpolation is a method for modeling data with uncertainty, using probabilistic principles to infer parameters and compare models. The paper introduces a Bayesian framework for regularization and model comparison, emphasizing the role of Occam's razor in selecting the most plausible model. It demonstrates how Bayesian methods automatically penalize overly complex models, ensuring that simpler models are preferred when they provide a good fit to the data. This approach is applied to the problem of interpolating noisy data, where the goal is to find a function that best fits the data while accounting for uncertainty. The paper explains that Bayesian inference involves two levels: first, fitting a model to the data to estimate parameters, and second, comparing models based on their evidence, which is the probability of the data given the model. The evidence is calculated by integrating over the parameter space, and it naturally incorporates Occam's razor by penalizing models with too many parameters. This framework, developed by Gull and Skilling, allows for the objective comparison of different models, priors, and basis functions. The paper also discusses the Bayesian approach to determining the values of regularization constants and noise levels. These values are inferred from the data using the evidence, which accounts for both the fit to the data and the complexity of the model. The evidence is shown to be a measure of the model's plausibility, and it naturally incorporates the trade-off between model complexity and data fit. The paper highlights the importance of Bayesian methods in model comparison, as they provide a principled way to evaluate the evidence for different models, leading to more reliable and objective conclusions. It also discusses the relationship between Bayesian model comparison and other statistical methods, such as maximum likelihood and minimum description length (MDL), and emphasizes the advantages of the Bayesian approach in handling uncertainty and complexity. The paper concludes that Bayesian methods offer a powerful and flexible framework for data modeling, particularly in situations where the data is noisy or the models are complex.Bayesian interpolation is a method for modeling data with uncertainty, using probabilistic principles to infer parameters and compare models. The paper introduces a Bayesian framework for regularization and model comparison, emphasizing the role of Occam's razor in selecting the most plausible model. It demonstrates how Bayesian methods automatically penalize overly complex models, ensuring that simpler models are preferred when they provide a good fit to the data. This approach is applied to the problem of interpolating noisy data, where the goal is to find a function that best fits the data while accounting for uncertainty. The paper explains that Bayesian inference involves two levels: first, fitting a model to the data to estimate parameters, and second, comparing models based on their evidence, which is the probability of the data given the model. The evidence is calculated by integrating over the parameter space, and it naturally incorporates Occam's razor by penalizing models with too many parameters. This framework, developed by Gull and Skilling, allows for the objective comparison of different models, priors, and basis functions. The paper also discusses the Bayesian approach to determining the values of regularization constants and noise levels. These values are inferred from the data using the evidence, which accounts for both the fit to the data and the complexity of the model. The evidence is shown to be a measure of the model's plausibility, and it naturally incorporates the trade-off between model complexity and data fit. The paper highlights the importance of Bayesian methods in model comparison, as they provide a principled way to evaluate the evidence for different models, leading to more reliable and objective conclusions. It also discusses the relationship between Bayesian model comparison and other statistical methods, such as maximum likelihood and minimum description length (MDL), and emphasizes the advantages of the Bayesian approach in handling uncertainty and complexity. The paper concludes that Bayesian methods offer a powerful and flexible framework for data modeling, particularly in situations where the data is noisy or the models are complex.