FR08 · What the data say about an amplitude

Cosmology, Likelihoods and Priors

How can the same sky data support a stable estimate and a changing model preference? A concentrated likelihood can keep an amplitude estimate nearly fixed as its prior widens. The evidence averages that likelihood over the whole prior, so a wider range gives more weight to poorly fitting amplitudes and can reduce support for the free-amplitude model.

Fit a lensing-power amplitude to published ACT DR6 bandpowers. Widen a prior or allow extra response uncertainty, then compare the likelihood, posterior and evidence.

Posterior mean and spread $0.985\pm0.0249$

Central 68 percent interval $[0.96,1.01]$

Minimum chi-square, unrestricted amplitude $13.7$

Log evidence ratio, free / fixed $-1.9$

Change the prior

Trial power amplitude

1

A_\kappa

Prior centre

1

A_{\mathrm{centre}}

Prior half-width

0.25

\Delta A

Keep the observations fixed and change the assumptions. Drag the glowing handle to inspect a trial amplitude.

Data, model and response uncertainty

Extra amplitude uncertainty

0

s_g

ACT DR6 lensing v1.2, baseline multipoles from forty to seven hundred and sixty-three. The published full covariance is used; error bars show its diagonal only.

Extra amplitude uncertainty is a hypothetical Gaussian response offset added to the published covariance. It demonstrates marginalization and is zero in the default analysis.

The free model has a normalized uniform prior. The fixed model has unit amplitude. Both use the same nuisance prior and covariance.

Qu et al., Eq. 45 · NASA LAMBDA

Continue through the cosmology course

Model assumptions

Observed points and the full covariance come from the ACT DR6 lensing v1.2 public release. A fixed supplied fiducial convergence spectrum is binned with the published window matrix. Only its amplitude is varied, with a Gaussian bandpower likelihood and the stated Hartlap correction. Cosmology-dependent reconstruction response and bias corrections are held fixed. The optional extra response-amplitude uncertainty is a teaching nuisance, added to the published covariance. The remapped sky is a synthetic illustration with magnified deflections; it is separate from the observed bandpowers. This conditional amplitude is not the temperature-spectrum lensing parameter or a dark-energy or matter-density measurement.

Published convergence bandpowers

10^7 C_L^{\kappa\kappa}

Normalized prior and posterior; unit-area likelihood

A_\kappa

Blue: likelihoodAmber dashed: priorGreen: posterior

Read the evidence ratio

Positive log evidence ratio favors the free-amplitude model over a fixed unit amplitude, given the displayed prior and frozen spectrum. Negative values favor the fixed model. This comparison does not test a full cosmological extension.

Download the frozen data and provenance

Start with the essentials

Focus question: Why does widening a prior change the evidence more than the fitted amplitude?
One-sentence intuition: The posterior describes amplitudes within a model; the evidence averages the fit across that model’s prior.

Core mathematical model

One amplitude multiplying a fixed spectrum

m_b(A_\kappa)=A_\kappa\sum_L W_{bL}C_{L,\mathrm{fid}}^{\kappa\kappa},\qquad\chi^2=(d-m)^\mathsf{T}\Sigma^{-1}(d-m)

The window matrix bins the fiducial convergence power. The full covariance includes correlations between observed bands. The effective covariance includes the specified finite-simulation precision correction.

Posterior and evidence

p(A_\kappa\mid d,M)=\frac{\mathcal L(d\mid A_\kappa,M)\pi(A_\kappa\mid M)}{Z_M},\qquad Z_M=\int\mathcal L\pi\,dA_\kappa

The uniform prior integrates to one. The fixed model sets the amplitude to one; the free model averages over the selected prior. Their evidence ratio compares precisely these conditional models.

Integrating over an uncertain response

d=(A_\kappa+g)t+\epsilon,\quad g\sim\mathcal N(0,s_g^2),\quad\Sigma=C/h+s_g^2tt^\mathsf T

The optional extra amplitude offset is completely degenerate with the signal amplitude. Marginalizing it broadens the amplitude uncertainty. Its width is an explicit teaching choice, independent of the published ACT uncertainty model.

Common difficulties

A likelihood is not a probability distribution over models

Typical misconceptionIf two models reach the same best fit, the data support them equally.

Better mental modelA model with a wide free parameter can fit well in only a small part of its prior. The evidence counts that fraction; the best fit alone does not.

Run the experiment

01
Let the observations set the width
Choose Data set the scale and compare the likelihood with the posterior.
What to observe: A broad, locally flat prior leaves the concentrated likelihood almost unchanged.
02
Give the free model more room
Choose Much wider prior. Compare the posterior interval and the log evidence ratio.
What to observe: The interval stays nearly fixed, but averaging over more poorly fitting amplitudes reduces the evidence for the free model.
03
Add an uncertain response
Choose Uncertain response, then increase Extra amplitude uncertainty.
What to observe: The same observations now permit a wider signal amplitude because the extra response offset can mimic part of it.

Cosmology, Likelihoods and Priors

One observed spectrum, different prior choices

Separate an estimate from a model comparison

Core mathematical model

One amplitude multiplying a fixed spectrum

Posterior and evidence

Integrating over an uncertain response

Common difficulties

A likelihood is not a probability distribution over models

Run the experiment

Let the observations set the width

Give the free model more room

Add an uncertain response

Related experiments and articles

Expansion, sound, sky statistics and inference