Antonio Aguirre Data Modelling·Bayesian Statistics·Machine Learning

On Bayesian Methodology. Part 6/6

19 Dec 2024

Final Comments

Different Perspectives on Statistical Modeling and Prediction

Statistical modeling and prediction can be approached from several perspectives, each with unique goals and implications:

  1. Traditional Statistical Perspective:
    • Models are typically predefined, and the goal is to accurately summarize the posterior distribution.
    • Computation continues until approximate convergence is achieved, with approximations used sparingly.
  2. Machine Learning Perspective:
    • The primary focus is on prediction rather than parameter estimation.
    • Computation halts once cross-validation accuracy plateaus, emphasizing scalability and efficiency within a fixed computational budget.
  3. Model Exploration Perspective:
    • Applied statistical work often involves trying out many models, some of which may exhibit poor data fit, low predictive performance, or slow convergence.
    • Approximations are more acceptable here but must faithfully reproduce key posterior features.
Key Insight:
  • The distinction here is not between inference vs. prediction or exploratory vs. confirmatory analysis, but rather how much trust is placed in a model and how computation approximates it.
    • Model Trust:
  • The process of iterative model building highlights how much computation and model development rely on trust in individual models and their approximations.

    • These differing perspectives influence how statistical methods evolve as new challenges emerge in applied settings.

    Justification of Iterative Model Building

    The iterative model-building process is central to modern Bayesian Workflows and represents the next transformative step in data science:

    • Historical Progression:
      • Data Summarization: The foundation of statistics up to 1900.
      • Modeling: Began with Gauss and Laplace, continuing to this day.
      • Computation: The current focus, enabling iterative workflows and complex modeling.
    • Real-World Considerations:
      • A Bayesian Workflow acknowledges the limitations of human and computational resources.
      • The goal is to simplify processes for humans, even in idealized settings where exact computation is automated.
      • Fully automated computation yielding perfect results remains unattainable.
    Simplifying Challenges:
  • Computational challenges are easier to address when fewer "moving parts" exist in the modeling process.

    • Model Selection and Overfitting

    An iterative workflow risks overfitting, as model improvement often involves conditioning on data discrepancies:

    • Double Dipping: Using data multiple times during model iteration can compromise the frequency properties of inferences.
    Warning:
  • Double dipping and post-selection inference can lead to overfitting. Model improvement conditioned on discrepancies must be carefully managed.

    • Garden of Forking Paths:

    • The model-building process often involves paths that depend on the specific data observed.
    • Instead of selecting the best-fitting model, a Bayesian Workflow emphasizes iterative improvements, ensuring each step is justified.
    Example:
  • Suppose model \(M_1\) fails a posterior predictive check, leading to the development of \(M_2\), which incorporates more prior information and better fits the data. Had the data differed, \(M_1\) might have sufficed. This highlights the iterative nature of a Bayesian Workflow.

    • To mitigate post-selection inference issues:

    • Embed multiple models in a larger framework.
    • Use predictive model averaging or incorporate all models simultaneously.
    • Perform severe tests of the assumptions underlying each model.

    Bigger Datasets Demand Bigger Models

    Larger datasets enable the fitting of complex models, such as hierarchical Bayesian models and deep learning approaches:

    • These models facilitate information aggregation and partial pooling across diverse data sources.
    • Effective modeling requires:
      1. Regularization: To stabilize estimates.
      2. Latent Variable Modeling: To address missingness and measurement errors.
    Insight:
  • A model does not exist in isolation; it emerges from engagement with the application and available data.

    • Prediction, Generalization, and Poststratification

    Statistical tasks often involve generalization, which Bayesian methods address effectively:

    1. Generalizing from Sample to Population: Using hierarchical models and partial pooling.
    2. Generalizing from Control to Treatment Groups: Leveraging regularization to handle large nonparametric models.
    3. Generalizing from Observed Data to Underlying Constructs: Applying multilevel modeling for latent variables.
    Key Principle:
  • Just as priors are understood in the context of the likelihood, models should be understood in light of their intended use.

    • Methods in Bayesian Framework:

    • Hierarchical modeling and transportability via Bayesian graph models.
    • Regularization to handle complex, large-scale models.

    The iterative nature of Bayesian Workflow, with its emphasis on model trust, computational efficiency, and careful navigation of overfitting risks, reflects the dynamic and evolving nature of modern statistical practice. By embedding models within a larger framework and embracing the iterative process, a Bayesian Workflow ensures robust and insightful statistical analyses.

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