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.
- Residual vector
The ordered differences between observed supernova magnitudes and model predictions. In this module it is used to answer
Why is a list of error bars not a likelihood?
Observable/model connection: The term links
The public Pantheon+ release documents a 1701 by 1701 statistical-plus-systematic covariance; the included 18 by 18 block is aligned to the selected source rows
to Covariance matrix and the quantitative relation \(chi^2 = r^T C^{-1} r\). State which part is measured, calibrated, derived, or model-dependent.Limit: Subsetting covariance preserves selected-row correlations but does not preserve the full survey's parameter constraints or selection function. That boundary must remain attached when the term is used in a cosmological claim.
- Covariance matrix
The variances and pairwise correlated uncertainties of the ordered data vector. In this module it is used to answer
Why is a list of error bars not a likelihood?
Observable/model connection: The term links
The public Pantheon+ release documents a 1701 by 1701 statistical-plus-systematic covariance; the included 18 by 18 block is aligned to the selected source rows
to Cholesky factorization and the quantitative relation \(chi^2 = r^T C^{-1} r\). State which part is measured, calibrated, derived, or model-dependent.Limit: Subsetting covariance preserves selected-row correlations but does not preserve the full survey's parameter constraints or selection function. That boundary must remain attached when the term is used in a cosmological claim.
- Cholesky factorization
A stable triangular decomposition of a positive-definite covariance used to whiten residuals. In this module it is used to answer
Why is a list of error bars not a likelihood?
Observable/model connection: The term links
The public Pantheon+ release documents a 1701 by 1701 statistical-plus-systematic covariance; the included 18 by 18 block is aligned to the selected source rows
to Generalized least squares and the quantitative relation \(chi^2 = r^T C^{-1} r\). State which part is measured, calibrated, derived, or model-dependent.Limit: Subsetting covariance preserves selected-row correlations but does not preserve the full survey's parameter constraints or selection function. That boundary must remain attached when the term is used in a cosmological claim.
- Generalized least squares
Least-squares estimation using the full inverse-covariance weighting implied by correlated errors. In this module it is used to answer
Why is a list of error bars not a likelihood?
Observable/model connection: The term links
The public Pantheon+ release documents a 1701 by 1701 statistical-plus-systematic covariance; the included 18 by 18 block is aligned to the selected source rows
to Calibration mode and the quantitative relation \(chi^2 = r^T C^{-1} r\). State which part is measured, calibrated, derived, or model-dependent.Limit: Subsetting covariance preserves selected-row correlations but does not preserve the full survey's parameter constraints or selection function. That boundary must remain attached when the term is used in a cosmological claim.
- Calibration mode
A shared systematic shift that moves multiple supernova measurements coherently. In this module it is used to answer
Why is a list of error bars not a likelihood?
Observable/model connection: The term links
The public Pantheon+ release documents a 1701 by 1701 statistical-plus-systematic covariance; the included 18 by 18 block is aligned to the selected source rows
to Residual vector and the quantitative relation \(chi^2 = r^T C^{-1} r\). State which part is measured, calibrated, derived, or model-dependent.Limit: Subsetting covariance preserves selected-row correlations but does not preserve the full survey's parameter constraints or selection function. That boundary must remain attached when the term is used in a cosmological claim.
Physical and observational argument
Let r be the vector of standardized observed magnitudes minus model magnitudes and C their covariance. Correlated calibration modes mean that a residual in one object predicts part of the residual in another. Cholesky factorization C=LL^T permits whitening by solving Ly=r; then chi-square is the squared norm of y. A common intercept can be profiled analytically using the same covariance-weighted solves. The selected block makes that operation tangible, but omitted survey rows, selection and full nuisance treatment prevent generalizing its intercept or chi-square to the parent analysis.
Shared calibration, photometric, and population uncertainties correlate objects. A diagonal-only fit changes the information content and can bias uncertainty; use factorization and linear solves instead of explicitly inverting a covariance matrix.
The public Pantheon+ release documents a 1701 by 1701 statistical-plus-systematic covariance; the included 18 by 18 block is aligned to the selected source rows.
Derivation and interpretation
\[chi^2 = r^T C^{-1} r\]
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.
Subsetting covariance preserves selected-row correlations but does not preserve the full survey's parameter constraints or selection function.
Worked numerical anchor
For the 18-row subset, GLS gives beta=-0.11997 +/- 0.04017 mag; this is conditional and not a survey fit.
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
Subsetting covariance preserves selected-row correlations but does not preserve the full survey's parameter constraints or selection function.
The public Pantheon+ release documents a 1701 by 1701 statistical-plus-systematic covariance; the included 18 by 18 block is aligned to the selected source rows.
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
- Construct residual vectors and interpret covariance off-diagonal terms.
- Evaluate correlated Gaussian chi-square using Cholesky factorization.
- Explain why the supplied subset cannot stand in for the full Pantheon+ likelihood.
Checkpoint: Compare diagonal-only and full-block results as a diagnostic exercise; do not promote either subset result to publication inference.
Answer in words before calculating; then identify one observation or null test that could falsify your interpretation.