Notes on Bayesian Inference and Data Analysis

Introduction to Bayesian Inference

• Conceptual introduction without calculations.
• Philosophy of Bayesian inference: counting assumptions.
• Aim: connect Bayesian inference to scientific models.

Misconceptions of Statistics

• Common perception: flowchart representation of data analysis.
• Statistics taught after data is produced, which is misleading.
• Historical context: statistical tests like t-tests arise from agricultural trials.

Statistical Tools and Their Contexts

• Different contexts require different statistical tools.
• Example: observing the position of Mars requires scientific modeling, not t-tests.
• Importance of scientific models for prediction vs. traditional statistical tests.

Learning and AI

• Machines can learn, but they don't understand like animals do.
• Emphasizes the need for a foundational understanding of inference.

Golem Metaphor

• Using "golem" instead of robots to illustrate statistical models.
• Golem is blind to intent, similar to statistical models that follow programming.
• Emphasizes caution when using complex statistical models.

Models and Their Validity

• Models are tools, not inherently true or false.
• All models are wrong but can be useful if appropriately used.
• Importance of understanding the assumptions behind models.

Bayesian Inference Explained

• Bayesian inference: counting the implications of assumptions using probability.
• Probability as a description of uncertainty.
• Bayesian is about ranking assumptions based on how well they explain data observed.

Differences Between Bayesian and Frequentist Approaches

• Frequentist: probability as objective and based on sample variation.
• Bayesian: probability as an internal uncertainty based on assumptions.

Conceptual Example: The Garden of Forking Data

• Uses the example of drawing marbles from a bag to explain Bayesian process.
• Each draw gives information about hypotheses regarding the contents of the bag.

Updating Beliefs in Bayesian Framework

• Bayesian updating involves multiplying prior plausibilities with observed evidence.
• Importance of relative counts of outcomes rather than absolute numbers.

Causal Inference and Statistical Models

• Causal inference goes beyond statistical correlations; requires understanding of underlying causal mechanisms.
• Confounding variables can obscure true causal relationships.
• Importance of using causal diagrams to clarify relationships between variables.

Conclusion

• Bayesian and causal inference frameworks rely on counting implications of assumptions.
• Emphasizes the need for sound theory and assumptions in statistical modeling.
• Both frameworks enhance our understanding of data and its implications for scientific inquiry.