Antonio Aguirre Data Modelling·Bayesian Statistics·Machine Learning

On Bayesian Methodology. Part 3/6

01 Nov 2024

III. Using Constructed Data to Find and Understand Problems

Fake-Data Simulation

The core idea of fake-data simulation is to test whether a procedure can recover correct parameter values when applied to simulated data. This involves the following steps:

  1. Simulate Fake Data:
    Choose reasonable parameter values and generate a fake dataset that matches the size, structure, and shape of the original data.

  2. Evaluate the Procedure:
    • Information Beyond the Prior: Simulate fake data from the model using fixed, known parameters and check if the observed data adds meaningful information. Analyze point estimates and the coverage of posterior intervals.
    • Parameter Recovery: Assess whether the true parameters are recovered within the uncertainty range implied by the fitted posterior distribution.
    • Behavior Across Parameter Space: Explore how the model behaves in different regions of the parameter space, revealing the various “stories” the model encodes about data generation.
  3. Two-Step Procedure:
    Fit the model to real data, draw parameters from the resulting posterior distribution, and use these parameters for fake-data checking.
Key Insight:
  • If a model cannot make reliable inferences on fake data generated from itself, it’s unlikely to provide reasonable inferences on real data.

    • While fake-data simulations help evaluate a model’s ability to recover parameters, they also highlight potential weaknesses. For example:

    • Creating fake data that causes the procedure to fail can deepen understanding of an inference method.
    • Overparameterized models may yield comparable predictions despite wildly different parameter estimates, limiting the usefulness of predictive checks.

    Simulation-Based Calibration (SBC)

    SBC provides a more comprehensive approach than truth-point benchmarking by fitting the model multiple times and comparing posterior distributions to simulated data. However, SBC has its challenges:

    • Computational Cost: Requires significant resources to fit the model repeatedly.
    • Priors and Modeler Bias:
      • Weakly informative priors, often chosen conservatively, can lead to extreme datasets during SBC.
      • This mismatch can obscure insights about calibration and posterior behavior.
    Open Research Question: How effective is SBC with a limited number of simulations?

    Simulation-based calibration and truth-point benchmarking are complementary, with SBC offering broader insights but at a higher computational expense.

    Experimentation Using Constructed Data

    Simulating data from different scenarios provides valuable insights into models and inference methods. This experimentation allows practitioners to:

    • Understand how a model performs under varying conditions.
    • Explore the limits of inferences by fitting the model to data generated from challenging scenarios.
    • Gain a deeper understanding of both computational issues and the underlying data.
    Important Consideration: Testing a model with a single fake dataset is not sufficient.
  • Even if the computational algorithm works, there’s a 5% chance that a random draw will fall outside a 95% uncertainty interval.
  • Bayesian inference is calibrated only when averaging over the prior.
  • Parameter recovery can fail not because of algorithmic errors but due to insufficient information in the observed data.

    • Simulation of statistical systems under diverse conditions not only addresses computational challenges but also enhances our understanding of data and inference.

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