Learning objectives
- Compare models using residual structure, likelihood, and complexity penalties.
- Explain weighted least squares, residuals, AIC/BIC, and posterior checks with the formal object.
- Separate evidence, transformation, inference, and limitation.
Terms Developed in Context
These terms form the lecture's reasoning chain from observable through model to a bounded physical conclusion.
- Residual
The signed difference between an observation and a model prediction. In this lecture it supports the evidence-to-model chain for weighted least squares, residuals, AIC/BIC, and posterior checks.
Use and relationship: Read it together with Posterior predictive check and Weighted least squares. The governing relation \(BIC=k ln(n)-2 ln(L_max)\) shows where the concept enters quantitatively; the student should state whether its value is measured, calibrated, adopted, or inferred.
Boundary: BIC does not replace residual diagnosis or scientific interpretation. This caveat prevents the term from being treated as a unique explanation or a context-free number.
- Weighted least squares
A fit minimizing squared residuals scaled by their uncertainty or covariance. In this lecture it supports the evidence-to-model chain for weighted least squares, residuals, AIC/BIC, and posterior checks.
Use and relationship: Read it together with Residual and Likelihood. The governing relation \(BIC=k ln(n)-2 ln(L_max)\) shows where the concept enters quantitatively; the student should state whether its value is measured, calibrated, adopted, or inferred.
Boundary: BIC does not replace residual diagnosis or scientific interpretation. This caveat prevents the term from being treated as a unique explanation or a context-free number.
- Likelihood
The probability assigned to observed data as a function of model parameters. In this lecture it supports the evidence-to-model chain for weighted least squares, residuals, AIC/BIC, and posterior checks.
Use and relationship: Read it together with Weighted least squares and BIC. The governing relation \(BIC=k ln(n)-2 ln(L_max)\) shows where the concept enters quantitatively; the student should state whether its value is measured, calibrated, adopted, or inferred.
Boundary: BIC does not replace residual diagnosis or scientific interpretation. This caveat prevents the term from being treated as a unique explanation or a context-free number.
- BIC
The Bayesian information criterion, a likelihood-based score with a sample-size-dependent complexity penalty. In this lecture it supports the evidence-to-model chain for weighted least squares, residuals, AIC/BIC, and posterior checks.
Use and relationship: Read it together with Likelihood and Posterior predictive check. The governing relation \(BIC=k ln(n)-2 ln(L_max)\) shows where the concept enters quantitatively; the student should state whether its value is measured, calibrated, adopted, or inferred.
Boundary: BIC does not replace residual diagnosis or scientific interpretation. This caveat prevents the term from being treated as a unique explanation or a context-free number.
- Posterior predictive check
A comparison between observed structure and replicated data generated from a fitted probabilistic model. In this lecture it supports the evidence-to-model chain for weighted least squares, residuals, AIC/BIC, and posterior checks.
Use and relationship: Read it together with BIC and Residual. The governing relation \(BIC=k ln(n)-2 ln(L_max)\) shows where the concept enters quantitatively; the student should state whether its value is measured, calibrated, adopted, or inferred.
Boundary: BIC does not replace residual diagnosis or scientific interpretation. This caveat prevents the term from being treated as a unique explanation or a context-free number.
Method in context
A lower residual sum is not automatically a better scientific model when flexibility and residual structure are ignored. Ask what a second researcher needs to reproduce the claim: raw input, processing state, software or database version, and inferential step.
Derivation and assumptions
Define every symbol and explain the failure mode. A method becomes scientific when assumptions are visible enough to be tested.
Worked example
BIC_2=23.9915, BIC_4=23.9829; delta BIC=0.00853545.
BIC does not replace residual diagnosis or scientific interpretation. Require an independent recomputation and one perturbation.
Misconceptions and study guidance
Do not confuse a data service with a measurement, a catalog identifier with physical identity, a formal error with total uncertainty, or a polished figure with a validated inference. Reconstruct the evidence chain before interpreting.
Verified reading
Astropy Modeling and Statistics documentation; NASA Exoplanet Archive documentation.
Use the exact documentation anchor in the reference log and record any access mismatch.