Cosmology · Cosmological parameters and statistical inference

Confidence intervals, credible intervals and their interpretation

Compare an explicit repeated-sampling interval rule with central Bayesian intervals for a single hypothetical trial.

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

Identify what varies in a confidence coverage calculation.
Calculate posterior content under a declared prior.
Demonstrate why equal percentage labels do not imply equal intervals or coverage.

An interval label needs its probability interpretation

A frequentist confidence procedure maps possible data to intervals. Its coverage is the probability, under repeated sampling at a fixed true parameter, that the resulting interval contains that truth.

A Bayesian credible interval contains a stated posterior probability under a specified likelihood and prior. After the data are supplied, this is a statement about that conditional parameter distribution.

Both concepts can be studied mathematically for the same sampling experiment. They answer different probability questions, so an identical percentage need not give identical endpoints or repeated-sampling coverage.

Use one hypothetical Bernoulli trial X, with P(X=1|θ) = θ and 0 ≤ θ ≤ 1. No astronomical experiment or physical parameter boundary calibration is supplied.

Evaluate the procedure over all possible observations

Before seeing X, define the custom closed-interval rule C(0) = [0,4/5] and C(1) = [1/5,1]. This is a supplied construction, not a named standard interval recipe.

At fixed θ, X = 0 occurs with probability 1−θ and X = 1 with probability θ. Coverage is (1−θ) times the indicator that θ belongs to C(0), plus θ times the indicator that it belongs to C(1).

For 0 ≤ θ < 1/5, coverage is 1−θ. For 1/5 ≤ θ ≤ 4/5 it is 1, because both intervals contain the truth. For 4/5 < θ ≤ 1 it is θ.

The coverage is at least 4/5 everywhere. Its infimum 4/5 is approached just outside the overlap boundaries; coverage at the included boundary points 1/5 and 4/5 is exactly 1, not 4/5.

A posterior interval requires a prior and a convention

Choose a continuous uniform prior on θ in [0,1]. After X = 0 the posterior density is 2(1−θ); after X = 1 it is 2θ.

For a central equal-tail 80% credible interval, leave posterior probability 0.1 below and 0.1 above. This is an explicit interval convention rather than a highest-density rule.

The resulting intervals are R(0) = [1−√0.9,1−√0.1] ≈ [0.051317,0.683772] and R(1) = [√0.1,√0.9] ≈ [0.316228,0.948683].

Each has posterior content 4/5 under the stated prior and sampling law. That content does not by itself guarantee at least 4/5 coverage at every fixed θ.

Intervals for the two possible observations; every displayed interval includes its endpoints
Observation	Custom confidence output C	Central 80% credible output R	Posterior content of C
X = 0	[0,0.8]	[0.051317,0.683772]	0.96
X = 1	[0.2,1]	[0.316228,0.948683]	0.96

The same intervals can be assessed by both questions

At θ = 0, X is always zero, but R(0) excludes zero. Thus the central credible-interval procedure has zero frequentist coverage there. The corresponding statement holds at θ = 1.

At θ = 1/2, both C outputs and both R outputs contain the truth, so each procedure has coverage 1. At θ = 1/4 the C coverage is 1 while the R coverage is 3/4.

After X = 0, the posterior probability within C(0) is ∫₀⁰·⁸2(1−θ)dθ = 0.96. The confidence rule’s at-least-80% guarantee did not specify this posterior content.

A realized frequentist interval either contains the fixed truth or does not. The coverage label alone does not assign an 80% probability to that fixed truth after this observation.

Repeated-sampling coverage at selected fixed truths; these values are not posterior contents
Fixed θ	Coverage of C	Coverage of R
0	1	0
0.1	0.9	0.9
0.2	1	0.8
0.25	1	0.75
0.5	1	1
0.75	1	0.75
0.8	1	0.8
0.9	0.9	0.9
1	1	0

Keep assumptions and endpoint conventions visible

For this discrete sampling space, the coverage of the custom confidence rule is not exactly 80% at every truth. The complete piecewise calculation establishes its guarantee over the full parameter domain.

For the Bayesian calculation, the uniform prior and central equal-tail rule determine the intervals. A different prior or interval convention can change their endpoints and their coverage.

The endpoint behavior is part of the mathematics. Connecting plotted points across a coverage jump or silently changing an open endpoint to a closed one would change the represented procedure.

When reading cosmological parameter intervals, identify the declared probability meaning and assumptions before comparing percentages. This example proves no general superiority of one inference framework and supplies no empirical confidence claim.

One hypothetical Bernoulli trial has P(X = 1 | θ) = θ on 0 ≤ θ ≤ 1. The upper strips show a custom confidence rule C0 = [0,4/5], C1 = [1/5,1] using solid blue lines and filled circular endpoints. They also show central equal-tail 80% credible outputs R0 = [1−√0.9,1−√0.1] and R1 = [√0.1,√0.9] under a continuous uniform prior, using dashed orange lines and filled square endpoints. Both ends of every output interval are included. The strip axis is θ from 0 to 1; rows are labels, not a numeric ordinate. Two separate lower panels plot repeated-sampling coverage against fixed true θ, with a zero-based vertical coverage scale from 0 to 1.05. The C rule has coverage 1−θ for θ < 1/5, 1 on the closed overlap [1/5,4/5], and θ for θ > 4/5. Its infimum 4/5 is approached just outside the overlap, not attained. A dotted reference marks 0.8. Filled and hollow endpoint markers distinguish included coverage values from excluded one-sided limits, without connecting across jumps. For R, put a₀ = 1−√0.9, a₁ = √0.1, b₀ = 1−√0.1, b₁ = √0.9. Coverage is 0 below a₀, 1−θ on [a₀,a₁), 1 on [a₁,b₀], θ on (b₀,b₁], and 0 above b₁, with endpoint markers drawn exactly. Its coverage is 0 at θ = 0 or 1 and 1 at θ = 1/2. Each R output has posterior content 0.8, while the corresponding C output has posterior content 0.96. Those posterior contents are not the coverage curves. Exact radicals set geometry; this construction supplies no empirical interval or framework-superiority claim. Original Ark teaching model.

Worked reasoning

Ask a posterior question about the confidence output

After X = 0, what posterior probability does C(0) = [0,0.8] contain under the uniform prior?

The posterior density is 2(1−θ).
Integrate to 0.8 using cumulative distribution 2θ−θ²: 1.6−0.64 = 0.96.
Keep this conditional content separate from the procedure’s at-least-0.8 repeated-sampling coverage.

It contains posterior probability 0.96 under this prior.

Try the reasoning

What is the C procedure’s coverage at fixed θ = 1/2?
Show the answer and reasoning
1.
Both possible output intervals contain 1/2, so all possible observations cover that truth.
Does the central 80% credible procedure guarantee 80% coverage at θ = 0?
Show the answer and reasoning
No; its coverage there is zero.
X is always zero, while R(0) starts above zero.
What is the C coverage at θ = 1/5?
Show the answer and reasoning
1.
The explicitly closed endpoints place that truth in both C(0) and C(1).

What to carry forward

Confidence coverage concerns repeated data at fixed truth.
Credible content concerns a posterior conditional on a prior and the observed data.
The same percentage can accompany different intervals, endpoint behavior and coverage.

References

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

Review of Particle Physics: 40. StatisticsG. Cowan; L. ListaTeaching resource · PDFpdg.lbl.govRead source