Cosmology · Cosmological parameters and statistical inference
Physical parameters and nuisance parameters
Separate a response amplitude from a shared calibration shift, then enumerate how readout noise affects both estimates.
Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.
Back to the chapterAfter reading, you can
- Distinguish a parameter of interest from a nuisance parameter by its role.
- Invert two supplied responses to separate amplitude and calibration.
- Interpret exact sampling variation without inventing posterior probabilities.
A nuisance parameter can have a physical cause
A cosmological analysis connects a physical prediction to recorded quantities through an observational response. Parameters may describe the Universe, an instrument, calibration or the selected population.
The parameter of interest is the quantity the analysis asks about. A nuisance parameter is needed to describe the same data but is not the target of that particular question.
These are roles in an analysis. An instrumental offset has physical causes, and a calibration parameter can become the target in a separate calibration study.
Supernova analyses provide a concrete historical example: calibration and standardization quantities enter alongside cosmological parameters. The numerical construction below is supplied mathematics and does not reproduce a supernova dataset.
One response can mix two unknown quantities
Choose an arbitrary readout unit U. Let A ≥ 0 be a response amplitude and b a signed common calibration shift, both in U. Supply y₁ = A+b+e₁ and y₂ = 2A+b+e₂.
The dimensionless response multipliers are 1 and 2. In the absence of noise, a first readout y₁ = 4U permits many pairs satisfying A+b = 4U.
A second noise-free readout y₂ = 7U supplies 2A+b = 7U. Subtracting the two equations gives A = 3U and then b = 1U.
The plotted parameter window is a drawing choice. It supplies no prior distribution, confidence region or probability that one of the allowed pairs is correct.
Canceling a common offset leaves independent noise
Now hold the stipulated truth at A = 3U and b = 1U. Let e₁ and e₂ independently equal −U or +U with probability 1/2 each. There are exactly four equally likely ordered error pairs.
Use the algebraic estimators  = y₂−y₁ and b̂ = 2y₁−y₂. They separate the two mean responses, but each still contains a linear combination of readout errors.
For example, (y₁,y₂) = (3,8)U gives (Â,b̂) = (5,−2)U. The negative offset estimate is allowed because b is signed; this noisy outcome does not establish the supplied true values.
Across the complete four-state sampling space, the estimator means are 3U and 1U. Unbiasedness here is an average over repetitions at fixed truth, not a statement that every estimate equals that truth.
| e₁/U | e₂/U | y₁/U | y₂/U | Â/U | b̂/U |
|---|---|---|---|---|---|
| −1 | −1 | 3 | 6 | 3 | 0 |
| −1 | 1 | 3 | 8 | 5 | −2 |
| 1 | −1 | 5 | 6 | 1 | 4 |
| 1 | 1 | 5 | 8 | 3 | 2 |
Optional mathematical path: propagate the finite errors
The estimator errors are Â−A = e₂−e₁ and b̂−b = 2e₁−e₂. Their means vanish because each input error has mean zero.
Each input error has variance U² and the errors are independent. Therefore Var(Â) = 2U² and Var(b̂) = 5U².
Their covariance is −3U²: the shared e₁ term contributes −2U² and the shared e₂ term contributes −U². Thus the estimator covariance matrix is [[2,−3],[−3,5]]U².
These second moments summarize the declared four-point distribution. They do not turn it into a Gaussian distribution or justify drawing a probability ellipse around an observed estimate.
A nuisance treatment is part of the inference
In a broader inference, nuisance parameters enter the likelihood jointly with the parameters of interest. Their allowed values and relationship to calibration information must be specified.
Profiling varies nuisance parameters to optimize a likelihood at each target value. Bayesian marginalization instead integrates a declared joint posterior over nuisance quantities; any nuisance prior already belongs in that joint posterior.
Neither procedure is performed by this direct inversion exercise. We have supplied a finite sampling law and two estimators, with no prior or posterior attached to A or b.
The practical lesson is to retain observational response and calibration dependencies. A method that cancels one common offset can still carry noise and cannot automatically remove other physical or selection uncertainties.
Worked reasoning
Separate one noisy pair
What estimates follow from y₁ = 3U and y₂ = 8U?
- Subtract the readouts: Â = 8U−3U = 5U.
- Form b̂ = 2×3U−8U = −2U.
- Keep the distinction between this possible outcome and the stipulated sampling truth (3,1)U.
 = 5U and b̂ = −2U; the estimators separate the responses but retain readout noise.
Try the reasoning
Does a single noise-free value y₁ = 4U determine A and b separately?
Show the answer and reasoning
No; it fixes their sum.
Many pairs satisfy A+b = 4U until another independent response or additional information is supplied.
What is Var(Â) under the declared independent sign errors?
Show the answer and reasoning
2U².
The difference e₂−e₁ has the sum of the two input variances.
Does calling b a nuisance parameter mean that its physical causes are unreal?
Show the answer and reasoning
No.
The label identifies its role relative to the current target question.
What to carry forward
- Parameter roles depend on the scientific question.
- Distinct mean responses can separate amplitude and calibration without eliminating noise.
- A finite sampling covariance is not a posterior distribution or an automatic confidence region.
References
Sources supporting the explanations, physical distinctions and examples in this lesson.