Preparation: algebra, introductory mechanics, and the previous module's stated prerequisites. This is a 60-minute instructional module; use the slide timings as a guide and pause at the active check.
Terms Developed in Context
These terms organize the chain from recorded signal to calibrated observable, likelihood, and bounded cosmological claim.
- Likelihood
The probability model for the observed data conditional on parameters and nuisance assumptions. In this module it is used to answer
What does a confidence contour really encode?
Observable/model connection: The term links
Planck, Pantheon+, BOSS, and DES summary results use different data vectors, nuisance models, and parameter spaces. Their error bars cannot simply be combined as independent Gaussians
to Prior and the quantitative relation \(p(theta|d) proportional to L(d|theta) p(theta)\). State which part is measured, calibrated, derived, or model-dependent.Limit: A plotted ellipse in the lecture is a synthetic Gaussian geometry demonstration, not a digitized published posterior contour. That boundary must remain attached when the term is used in a cosmological claim.
- Prior
The parameter distribution imposed before conditioning on the current data. In this module it is used to answer
What does a confidence contour really encode?
Observable/model connection: The term links
Planck, Pantheon+, BOSS, and DES summary results use different data vectors, nuisance models, and parameter spaces. Their error bars cannot simply be combined as independent Gaussians
to Posterior and the quantitative relation \(p(theta|d) proportional to L(d|theta) p(theta)\). State which part is measured, calibrated, derived, or model-dependent.Limit: A plotted ellipse in the lecture is a synthetic Gaussian geometry demonstration, not a digitized published posterior contour. That boundary must remain attached when the term is used in a cosmological claim.
- Posterior
The parameter distribution proportional to likelihood times prior. In this module it is used to answer
What does a confidence contour really encode?
Observable/model connection: The term links
Planck, Pantheon+, BOSS, and DES summary results use different data vectors, nuisance models, and parameter spaces. Their error bars cannot simply be combined as independent Gaussians
to Nuisance parameter and the quantitative relation \(p(theta|d) proportional to L(d|theta) p(theta)\). State which part is measured, calibrated, derived, or model-dependent.Limit: A plotted ellipse in the lecture is a synthetic Gaussian geometry demonstration, not a digitized published posterior contour. That boundary must remain attached when the term is used in a cosmological claim.
- Nuisance parameter
A required model parameter that affects the data but is not the primary scientific target. In this module it is used to answer
What does a confidence contour really encode?
Observable/model connection: The term links
Planck, Pantheon+, BOSS, and DES summary results use different data vectors, nuisance models, and parameter spaces. Their error bars cannot simply be combined as independent Gaussians
to Parameter degeneracy and the quantitative relation \(p(theta|d) proportional to L(d|theta) p(theta)\). State which part is measured, calibrated, derived, or model-dependent.Limit: A plotted ellipse in the lecture is a synthetic Gaussian geometry demonstration, not a digitized published posterior contour. That boundary must remain attached when the term is used in a cosmological claim.
- Parameter degeneracy
A direction in parameter space along which substantially different parameters predict similar data. In this module it is used to answer
What does a confidence contour really encode?
Observable/model connection: The term links
Planck, Pantheon+, BOSS, and DES summary results use different data vectors, nuisance models, and parameter spaces. Their error bars cannot simply be combined as independent Gaussians
to Likelihood and the quantitative relation \(p(theta|d) proportional to L(d|theta) p(theta)\). State which part is measured, calibrated, derived, or model-dependent.Limit: A plotted ellipse in the lecture is a synthetic Gaussian geometry demonstration, not a digitized published posterior contour. That boundary must remain attached when the term is used in a cosmological claim.
Physical and observational argument
A likelihood states the probability of data conditional on parameters; a prior encodes information or regularization before the data; the posterior combines both. For correlated Gaussian data, the quadratic residual term must be accompanied by the covariance determinant unless it is constant over the parameters being varied. A contour depends on whether nuisance parameters are fixed, profiled or marginalized, and on parameter coordinates. Published one- or two-dimensional summaries discard much of the joint likelihood and may share calibration or data. Therefore a sum of inverse variances is not a valid multi-probe analysis without a justified independence model.
Inference begins with a data model, a likelihood, and explicit nuisance parameters. Parameter degeneracies are geometric directions in the likelihood or posterior; priors and parameterization can change marginal summaries.
Planck, Pantheon+, BOSS, and DES summary results use different data vectors, nuisance models, and parameter spaces. Their error bars cannot simply be combined as independent Gaussians.
Derivation and interpretation
\[p(theta|d) proportional to L(d|theta) p(theta)\]
Begin by naming observables, coordinate convention, and model assumptions; each algebraic transformation must preserve units and the distinction between measured and inferred quantities. Test a limiting case, then perturb one input and state which observable changes.
A plotted ellipse in the lecture is a synthetic Gaussian geometry demonstration, not a digitized published posterior contour.
Worked numerical anchor
A Gaussian covariance likelihood is r^T C^-1 r; factorization plus linear solves avoids explicitly constructing C^-1.
Reproduce the value with the course computation module, retain full precision in code, and round only for communication. Report units and all conditional assumptions with the result.
Evidence, systematics, and limits
A plotted ellipse in the lecture is a synthetic Gaussian geometry demonstration, not a digitized published posterior contour.
Planck, Pantheon+, BOSS, and DES summary results use different data vectors, nuisance models, and parameter spaces. Their error bars cannot simply be combined as independent Gaussians.
Separate the directly recorded observable, its calibration, the derived intermediate quantity, and the model-dependent parameter. An uncertainty bar is not a complete likelihood when shared covariance or selection is omitted.
Study and retrieval practice
- Write a likelihood for correlated Gaussian residuals and identify nuisance parameters.
- Distinguish likelihood, prior, posterior, and marginalized constraint.
- Explain why independent-looking published summaries may still be statistically dependent.
Checkpoint: Record parameterization, priors, covariance, sampler diagnostics, and posterior predictive checks.
Answer in words before calculating; then identify one observation or null test that could falsify your interpretation.