Likelihood
Definition: The probability model for the observed data conditional on parameters and nuisance assumptions.
Why it matters here: It is required to answer What does a confidence contour really encode? and to distinguish direct observation from cosmological inference.
Relationships: Read it with Prior, the model \(p(theta|d) proportional to L(d|theta) p(theta)\), and the evidence statement 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.
Boundary or common confusion: A plotted ellipse in the lecture is a synthetic Gaussian geometry demonstration, not a digitized published posterior contour.
Prior
Definition: The parameter distribution imposed before conditioning on the current data.
Why it matters here: It is required to answer What does a confidence contour really encode? and to distinguish direct observation from cosmological inference.
Relationships: Read it with Posterior, the model \(p(theta|d) proportional to L(d|theta) p(theta)\), and the evidence statement 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.
Boundary or common confusion: A plotted ellipse in the lecture is a synthetic Gaussian geometry demonstration, not a digitized published posterior contour.
Posterior
Definition: The parameter distribution proportional to likelihood times prior.
Why it matters here: It is required to answer What does a confidence contour really encode? and to distinguish direct observation from cosmological inference.
Relationships: Read it with Nuisance parameter, the model \(p(theta|d) proportional to L(d|theta) p(theta)\), and the evidence statement 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.
Boundary or common confusion: A plotted ellipse in the lecture is a synthetic Gaussian geometry demonstration, not a digitized published posterior contour.
Nuisance parameter
Definition: A required model parameter that affects the data but is not the primary scientific target.
Why it matters here: It is required to answer What does a confidence contour really encode? and to distinguish direct observation from cosmological inference.
Relationships: Read it with Parameter degeneracy, the model \(p(theta|d) proportional to L(d|theta) p(theta)\), and the evidence statement 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.
Boundary or common confusion: A plotted ellipse in the lecture is a synthetic Gaussian geometry demonstration, not a digitized published posterior contour.
Parameter degeneracy
Definition: A direction in parameter space along which substantially different parameters predict similar data.
Why it matters here: It is required to answer What does a confidence contour really encode? and to distinguish direct observation from cosmological inference.
Relationships: Read it with Likelihood, the model \(p(theta|d) proportional to L(d|theta) p(theta)\), and the evidence statement 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.
Boundary or common confusion: A plotted ellipse in the lecture is a synthetic Gaussian geometry demonstration, not a digitized published posterior contour.