Working definition
Bayesian inference combines a prior distribution and likelihood to produce a posterior distribution. The result is conditional on the model, prior, data-generating assumptions, and observation process; posterior concentration does not protect against misspecification or leakage.
Notation
p(θ|D) ∝ p(D|θ)p(θ)posterior ∝ likelihood × priorAssumptions
- Prior and likelihood are declared before evaluation.
- The observation model reflects sampling and censoring.
- Model comparison accounts for complexity.
Invariants
- The posterior normalizes to one.
- Sequential updating is coherent for conditionally independent batches.
- Changing the prior can change sparse-data conclusions.
Reproducible procedure
- Pre-register parameterization and priors.
- Compute and diagnose the posterior.
- Run prior sensitivity and posterior predictive checks.
Error and boundary controls
- Approximate inference adds computational error.
- Misspecified likelihoods produce misleading certainty.
- Selection bias is not removed by Bayes rule.
What this does not establish
Bayesian updating can measure how evidence changes belief under a model; it cannot turn repeated post-hoc astrological correlations into prospective evidence.
Explicit applications
2 cross-domain bridges
Model-conditioned parameter inference
Update parameter distributions from calibrated observations under explicit physical likelihoods and priors.
Inputs
- calibrated data
- physical likelihood
- parameter priors
Outputs
- posterior estimates
- model checks
- prior sensitivity
Transformation: Compute posterior distributions and predictive diagnostics.
Limit: Posterior precision is conditional on the selected model family.
Open connected system →Bayesian rule-pruning protocol
Update bounded rule-effect estimates while shrinking noisy factors and preserving null results.
Inputs
- pre-registered rule features
- objective outcomes
- hierarchical prior
Outputs
- posterior effect distributions
- shrinkage diagnostics
- retained and pruned candidates
Transformation: Fit a versioned hierarchical model and run prior and posterior predictive checks.
Limit: Pruning on one dataset can overfit; retained rules require prospective replication.
Open connected system →Authoritative references
- [1]NIST/SEMATECH e-Handbook of Statistical Methods · National Institute of Standards and Technology
Establishes: Methods for uncertainty analysis, calibration, time-series modeling, process monitoring, experimental design, reliability, and statistical comparison.
Boundary: Statistical procedures quantify evidence under a design and model; they do not repair biased sampling, outcome leakage, post-hoc hypotheses, or unmeasured confounding.