Cosmology · Cosmological parameters and statistical inference

Covariance, correlated data and parameter degeneracy

Follow four exact correlated noise states through a sum-and-difference response, keeping covariance and identifiability distinct.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

Back to the chapter

After reading, you can

Calculate covariance from a fully declared finite sampling law.
Propagate correlated errors into two parameter estimates.
Separate a response degeneracy from the shape of an error distribution.

Shared errors and similar parameter responses are different issues

Two observations can share an error source. A common calibration shift, for example, can move both readouts in the same direction even when the physical signals differ.

Covariance summarizes how centered variations co-vary under a specified probability law. Parameter degeneracy instead concerns whether different parameter choices give the same or similar predicted observations.

These issues can coexist, but one does not define the other. A nonzero covariance alone does not prove a parameter degeneracy, and an invertible covariance matrix does not establish identifiability.

Historical supernova methods illustrate the need to retain shared calibration correlations. Here we use an original four-state construction, with no survey covariance or cosmological parameter fit supplied.

State the entire distribution before using its covariance

Let θ and φ be signed response amplitudes with units U, an arbitrary fixed readout unit. Supply x₁ = θ+φ+s+u and x₂ = θ−φ+s−u.

Choose independent latent errors s = ±2U and u = ±U, each sign equally likely. The shared error s moves both readouts together; u moves them oppositely.

Hold the teaching truth fixed at θ = 10U and φ = 2U. There are exactly four possible pairs of readout errors, each with probability 1/4.

This distribution has four discrete points. No Gaussian density, continuous contour, empirical data or unspecified probability mass between those points is assumed.

Complete supplied sampling space at fixed (θ,φ) = (10,2)U; each row has probability 1/4
s/U	u/U	x₁/U	x₂/U	θ̂/U	φ̂/U
−2	−1	9	7	8	1
−2	1	11	5	8	3
2	−1	13	11	12	1
2	1	15	9	12	3

Sum and difference retain different amounts of noise

Form θ̂ = (x₁+x₂)/2 = θ+s and φ̂ = (x₁−x₂)/2 = φ+u. The corresponding estimator errors recover the two independent latent errors exactly.

Each raw error has variance 4U²+U² = 5U². Their covariance is Var(s)−Var(u) = 3U², so the raw covariance matrix is C = [[5,3],[3,5]]U².

The correlation coefficient is 3/5. In contrast, the estimator covariance is diag(4,1)U²: the sum-mode variance is 4U² and the difference-mode variance is U².

The independence of the estimator errors follows here from the explicitly independent s and u. Zero covariance by itself would not establish independence for a general distribution.

Discarding correlation can misstate uncertainty in either direction

For the sum estimate, Var(θ̂) = (C₁₁+C₂₂+2C₁₂)/4 = (5+5+6)U²/4 = 4U².

For the difference estimate, Var(φ̂) = (C₁₁+C₂₂−2C₁₂)/4 = U². The same shared covariance term increases one mode and decreases the other.

If the off-diagonal terms are incorrectly set to zero while both diagonal variances remain 5U², propagation would report 2.5U² for each estimate. This understates the sum variance and overstates the difference variance.

Optional matrix propagation writes C_est = BCBᵀ for the stated linear sum-and-difference matrix B. This identity for second moments does not require Gaussian errors or produce automatic 68% intervals.

Identifiability belongs to the mean response

Without noise, the first channel alone fixes θ+φ. A change in θ can be canceled by an opposite change in φ, giving the same predicted x₁.

Adding the second mean response θ−φ gives the coefficient matrix [[1,1],[1,−1]], with determinant −2. The two equations then separate θ and φ in this model.

That rank statement uses the forward response, independently of the covariance calculation. Real analyses may have nearly aligned responses, nonlinear dependencies, nuisance parameters and limited sensitivity over a stated parameter region.

The four-point example supports exact covariance arithmetic and a transparent rank test. It supplies neither a survey likelihood ellipse nor a significance level or posterior contour for cosmological parameters.

Four discrete error states illustrate covariance separately from parameter-response identifiability. Independent errors s = ±2U and u = ±U define x₁ = θ+φ+s+u and x₂ = θ−φ+s−u at supplied truth (θ,φ) = (10,2)U. Each sign pair has probability 1/4. Panel A plots raw readout errors (e₁,e₂) = (s+u,s−u), each axis in units U from −4 to 4. Numbered blue circles 1–4 are (−3,−1), (−1,−3), (1,3), (3,1). Their covariance is [[5,3],[3,5]]U² and correlation is 3/5. Panel B plots the corresponding estimator errors (s,u) = (θ̂−θ,φ̂−φ), using orange squares with horizontal s/U from −3 to 3 and vertical u/U from −2 to 2. Points with the same numbers are (−2,−1), (−2,1), (2,−1), (2,1). Their covariance is diag(4,1)U² because the original latent errors are independent. Panel C is a separate noise-free mean-response plot: horizontal θ/U from 0 to 20, vertical φ/U from −5 to 8. Solid green θ+φ = 12U and dashed violet θ−φ = 8U meet at orange diamond (10,2)U. The forward matrix has determinant −2. Covariance rank is not this response-rank test. Incorrectly discarding raw covariance would report variances 2.5U² and 2.5U² instead of 4U² and U². No Gaussian density, continuous probability contour, confidence ellipse or survey dataset is supplied. Original Ark teaching model.

Worked reasoning

Propagate a correlated sum

What happens if C₁₂ = 3U² is discarded when estimating Var(θ̂)?

Use the full covariance: (5+5+2×3)U²/4 = 4U².
Replacing C₁₂ by zero gives (5+5)U²/4 = 2.5U².
For the difference estimate the true value is U², so the same replacement instead overstates its variance.

Ignoring the correlation gives the wrong uncertainties in opposite directions for the two modes.

Try the reasoning

Does the covariance matrix C establish a Gaussian likelihood?
Show the answer and reasoning
No.
The declared law has only four noise states; a covariance is a second-moment summary.
What separates θ and φ in the noise-free example?
Show the answer and reasoning
The second linearly independent mean response.
The determinant −2 of the two-channel response matrix is nonzero.
What are the exact estimator variances?
Show the answer and reasoning
Var(θ̂) = 4U² and Var(φ̂) = U².
Their errors are the supplied s and u, respectively.

What to carry forward

Covariance describes variations under a specified probability law.
Correlated input errors can affect sum and difference uncertainties differently.
Parameter identifiability depends on the forward response and is distinct from covariance or Gaussianity.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.