Predictive gains, added freedom and competing explanations

A proposal must connect its successful and tested regimes

An altered gravity or cosmological model should explain a defined set of observations coherently while addressing established tests in its applicable regimes. Improving one selected response does not by itself complete that explanation.

Additional fields, screening or relaxed homogeneity can change predictions in different ways. A credible proposal must derive those changes and state their dynamics, boundaries, conservation and validity conditions.

A nuisance or calibration explanation can also change a fit. Comparing it fairly requires the same relevant data and uncertainty assumptions, rather than assuming the interesting physical interpretation in advance.

Use a fully supplied statistical example to separate fit quality from predictive evidence. Its response directions are algebraic hypotheses, not modified-gravity actions, literal screening transitions or an empirical ranking of candidate theories.

A covariance matrix alone is not a likelihood

Supply four fixed normalized dimensionless channels labeled expansion,growth,Weyl and local force, with synthetic reference vector y = (0,2,0,0). Their normalizations are fixed inputs rather than quantities fitted from a catalogue.

Explicitly assume y|a,M follows a four-dimensional Gaussian with mean ua and known positive-definite covariance C. This Gaussian law is an additional assumption beyond reporting C, and C is held fixed for all templates and amplitudes.

The baseline has fixed zero mean and no fitted parameter. Compare vectors u_U = (0,1,1,1), u_S = (0,1,1,0) and u_N = (0,1,0,0), representing universal-response, screened-response and growth-calibration templates respectively.

Each nonbaseline template has one signed real amplitude a. For evidence calculations give it a proper normalized Gaussian prior N(0,s²), separately choosing s = 1 or s = 3; no prior probabilities for the models themselves are supplied.

A local-regime channel can restrict an otherwise flexible response

Write Q(a) = (y−ua)ᵀC⁻¹(y−ua) = Q₀−2Ba+Aa², where Q₀ = 16/3, A = uᵀC⁻¹u and B = uᵀC⁻¹y. Then â = B/A and Q_min = Q₀−B²/A.

The U template changes the local-force channel as well as growth and Weyl. Its small local variance restricts the fitted amplitude to 1/13, leaving Q_min = 68/13.

S removes that local response but still links growth to Weyl, giving â = 1 and Q_min = 4. N changes only growth and fits the supplied vector exactly at â = 2, with Q_min = 0.

These outcomes expose the role of linked responses and shared covariance. The labels do not demonstrate that a physical theory realizes the assumed vectors, or that an exact calibration-template fit identifies the true cause.

Evidence includes the width as well as the best-fitting region

Integrating the normalized likelihood over the proper amplitude prior gives evidence ratio Z_M/Z_baseline = (1+s²A)^(−1/2)exp[s²B²/(2(1+s²A))]. Shared likelihood normalization cancels because the same C and channel dimension are used.

The exponential includes the region of improved fit, while the prefactor accounts for its prior-averaged normalization. For S with s = 1, the ratio is √(3/7)exp(8/21) ≈ 0.958202 despite its smaller Q_min than the baseline.

Changing s changes the predictive density being averaged. In these supplied examples, going from s = 1 to 3 lowers the U and S evidence ratios but slightly raises N’s; a wider prior is not a universal monotonic penalty.

A Bayes factor is an evidence ratio under the specified hypotheses and priors. Posterior model odds additionally require model prior odds; none of these numbers is a p-value, a Gaussian significance or a probability that GR is true.

An algebraic improvement still needs a physical account

Optional amplitude inference within a template gives posterior variance v = (s⁻²+A)⁻¹ and mean vB. For S at s = 1 these are 3/7 and 4/7, distinct from the maximum-likelihood amplitude 1.

This is parameter inference conditional on S, not the probability that S is the correct model. It also depends on the explicitly supplied Gaussian sampling law and proper prior.

An actual theory comparison must derive its observable links, treat calibration and selection, account for shared information and meet other relevant regime tests. Additional parameter freedom alone does not supply stable dynamics or a causal explanation.

The practical standard is coherent predictive gain under documented assumptions. The toy calculation illustrates how to assess that standard without treating a fit, an evidence ratio or a chosen response name as proof of new physics.

Worked reasoning

Try the reasoning

1. What best amplitude and Q_min follow for N?

   Show the answer and reasoning

   â = 2 and Q_min = 0.

   A = 4/3 and B = 8/3 give the exact fit; a perfect synthetic template fit does not establish its physical truth.

2. Is the evidence invariant when s changes from 1 to 3?

   Show the answer and reasoning

   No.

   The same likelihood is averaged over a different normalized amplitude prior; the resulting predictive density changes.

3. Is an evidence ratio alone a posterior model probability?

   Show the answer and reasoning

   No.

   Posterior model odds also need prior odds for the competing models, and a probability requires the specified set of alternatives.

What to carry forward

• A coherent explanation must connect the regimes and observables it claims to describe.
• Best-fit improvement and prior-averaged evidence answer different questions.
• Statistical template success still needs physical dynamics, shared-assumption checks and independent predictions.