Is the Jeffreys' scale a reliable tool for Bayesian model comparison in cosmology?

Juan Garcia-Bellido; Savvas Nesseris

REVIEW

Is the Jeffreys' scale a reliable tool for Bayesian model comparison in cosmology?

Savvas Nesseris, Juan Garcia-Bellido

Cosmology and Nongalactic Astrophysics · Submitted 29 Oct 2012 · 35 references

Pith's verdict: Correct analytic Bayes-factor formulas and a genuinely useful basis-dependence table, wrapped around a Jeffreys-scale critique whose mock-data demonstration is largely built into the prior choice.

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CitationJuan Garcia-Bellido, Savvas Nesseris. "Is the Jeffreys' scale a reliable tool for Bayesian model comparison in cosmology?." arXiv:1210.7652 (2012). https://pith.science/paper/SPM25OYF

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Bottom line: the analytic Bayes-factor formulas are correct and worth having; the claim that Jeffreys' thresholds aren't universal is right but not new, and the mock-data demonstration that carries the empirical weight is weaker than the framing suggests because the priors are scaled to the fitted errors. Send it out, but expect the referee to push hard on Section II.C. What's actually useful here: Eqs. 2.15 and 2.18 give explicit closed forms for the evidence of linear models under Gaussian and flat priors. These are standard Gaussian integrals, but having them written out for this class of models is a handy reference. Eq. 2.24, ln B12 = -Delta chi2/2 + ln G(M1,M2), makes the correct point that the Bayes factor carries covariance and prior information beyond Delta chi2. And Table II, the FoM ratio across monomial, Legendre and Chebyshev bases, is the most concretely useful thing in the paper and gets the least attention: basis choice moves the FoM by factors of a few, which matters for anyone reporting FoM numbers. Soft spots, in order of how much they matter. First, the (4,14) demonstration. Section II.C sets Delta a_i = n sigma_i. With that scaling each extra parameter contributes erf(n/(2 sqrt2)), about 0.87 for n=3, so ten extra parameters cost only about e^-1.4 in ln B. The near-unity Bayes factor is largely built into the prior choice. The paper varies n (3 vs 7) but never uses priors decoupled from the fitted errors, which is the regime where Jeffreys' thresholds would actually be informative. So the demonstration shows failure for one prior family, not failure in general. The authors do acknowledge prior sensitivity, but acknowledging sensitivity to n isn't the same as testing priors independent of the data. Second, Eq. 2.13 applies the 2x2 determinant identity |A| = (1/2)((tr A)^2 - tr A^2) to M-dimensional matrices. That's only valid for M=2. The trace expansion in Eq. 2.15 is therefore not exact for M > 2, despite the text calling it exact. Fixable, but it should be fixed. Third, the Fig. 4 result, polynomials up to degree 12 favored over the true model, is the Lindley/Jeffreys paradox, and the paper doesn't engage with that literature. That's the standard explanation for what they're seeing and would sharpen the argument. Fourth, no code or data shipped; figures aren't independently reproducible. Minor for this kind of paper. The qualitative conclusion, that Jeffreys' thresholds are ad hoc, is correct and the paper says so itself. It's not a new claim; Kass & Raftery is cited and makes the same point. The real contribution is the explicit formulas and the basis-dependence table, not the headline. Who should read it: anyone doing Bayesian model comparison with linear models who wants the closed forms, and anyone reporting FoM numbers who hasn't thought about basis choice. The Jeffreys critique is a useful discussion piece but won't surprise anyone who's read Kass & Raftery.

Referee report

Major objections (5)

[§II.B.1, Eqs. (2.13)–(2.15)] The determinant identity used in Eq. (2.13), |I_M + B| = M(M−1)/2 + (M−1)tr(B) + |B|, is not valid for general M. It follows from the 2×2 identity |A| = (1/2)(tr(A)² − tr(A²)) and holds only for M = 2; for M = 3 the correct expansion is 1 + tr(B) + (sum of 2×2 principal minors) + |B|, which differs from the stated formula. Since Eq. (2.15) is presented as 'an exact result' and is used to produce Fig. 3 for M up to 14, either the identity must be corrected for general M or the text must state explicitly that the numerical results use the direct determinant |I + H⁻¹F| and that Eq. (2.13) is only a 2×2 illustration.
[§II.C, Figs. 2–3] The headline demonstration sets the prior widths proportional to the fitted parameter errors, Δa_i = nσ_i with n = 3 or 7. This is a data-dependent (empirical-Bayes) prior, and with this scaling each extra parameter contributes a factor erf(n/(2√2)) ≈ 0.87 to the evidence, i.e. only about 0.14 in ln B per parameter, so ten extra parameters cost only ≈ −1.4 in ln B. The (4,14) result of ln B12 ≈ 1.2 is therefore largely built into the prior choice rather than discovered from the data. The paper's own Eq. (2.21) shows that in the small-prior limit the Bayes factor reduces to the likelihood ratio, which is exactly the regime in which the Occam penalty is suppressed. To support the abstract's claim that the scale is not 'one size fits all', the authors should test priors with fixed physical widths independent of the fitted errors, and/or the large-prior limit of Eq. (2.22), where the per-parameter penalty is maximal.
[§II.C and §IV] There is a logical gap between the demonstration and the stated conclusion. If the computed Bayes factor really is ln B12 ≈ 1, then the scale's label 'weak' is the correct mapping from that number to words; the scale is not mislabelling a well-computed evidence ratio. What the demonstration actually shows is that the Bayes factor itself is small, and that this smallness is driven by the prior choice. The paper should either exhibit a case in which a Bayes factor that is robustly computed is mislabelled by the thresholds, or reframe the claim as a statement about the prior-dependence of the Bayes factor rather than about the scale. The Fig. 4 result (the true model rejected in favour of high-order polynomials) is the stronger and more interesting failure mode and should be foregrounded rather than the (4,14) pair.
[§II.B.2, Eqs. (2.17)–(2.18)] The flat-prior evidence in Eq. (2.17) is written with an approximation sign because the prior box is not aligned with the Fisher eigenbasis; the product of error functions is exact only when the box is axis-aligned in that basis. The accuracy of this approximation is never quantified, yet Eq. (2.18) is used to produce Fig. 2 for M up to 14, where the polynomial basis is strongly correlated and the misalignment could be substantial. The authors should either quantify the accuracy of Eq. (2.17) against a direct numerical integration for representative cases, or restrict the quantitative claims to the regime where the approximation is controlled.
[§III, Table II] The FoM ratio across bases in Eq. (3.11) reduces to |Λ|, the Jacobian of a linear reparametrization of the same model, so the statement that 'Chebyshev polynomials provide the best constraints' is a statement about the volume element in parameter space, not about the quality of the fit to the data. This should be phrased explicitly as a basis-dependence of the parameter volume. In addition, Table II is numerically inconsistent with Table III evaluated at x_max = 1.55: the Legendre/Chebyshev entry is 0.750 while the reciprocal Chebyshev/Legendre entry is 1.250, and these are not reciprocals (the correct value is 1.333); the Monomials/Legendre entry 0.296 also differs from the 3.724/12 ≈ 0.310 implied by Table III. These entries need to be recomputed or corrected.

Minor objections (8)

[Abstract] 'Criterium' should be 'criterion'.
[Table I] The second column header reads '0 ≤ B_ij < 1.1', but the entries are values of ln B_ij; the header should read '0 ≤ ln B_ij < 1.1'.
[Fig. 1 caption] 'The best-fit χ² per degree of freedom N−M' should read 'χ²/(N−M)'; as written it denotes a difference rather than a ratio.
[§II.C] The sentence 'we fit these data with two polynomials that have a different number of parameters M1 and M2, e.g. M2 > M1 or M1 > M2' is vacuous as written and should be simplified.
[§IV] The conclusions quote ln B12 ≈ 5.5 for (M1,M2) = (4,14) from Fig. 5, while §II.C quotes ln B12 ≈ 1.2 for the same pair from Fig. 2; the two refer to different generating models and this should be stated explicitly to avoid confusion.
[§II.B.2] The notation 'transverse of the Fisher matrix' should be 'transpose', and the matrix U for M = 2 is a permutation matrix, not a diagonalizer; the construction of F⊥ and D⊥ should be spelled out so that Eq. (2.17) is reproducible.
[§II.A] The statement that including the likelihood normalization 'does not change our conclusions' deserves one line of justification, since the normalization factor (2π)^{−M/2}|F|^{1/2} depends on M and on |F| and therefore does affect comparisons between models with different parameter counts.
[§II] For nested models the Savage–Dickey density ratio is the standard tool and would connect the general formula of Eq. (2.15) to the M2 ⊃ M1 case discussed in the introduction; a brief remark would help the reader.

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The paper

The paper claims

The paper argues that a fixed Bayes-factor threshold cannot be a universal rule for choosing between models, because the evidence ratio for linear models depends on parameter covariances and prior widths, not just on how much better one fit

The reading

Cosmologists routinely decide between competing models by computing a Bayes factor and reading off a verdict from Jeffreys' scale, a fixed ladder of thresholds (roughly $1$–$3$ weak, $3$–$20$ definite, $20$–$150$ strong, above $150$ very strong). This paper asks whether that ladder deserves its authority. The authors restrict attention to models that are linear in their parameters, where the evidence integral can be done exactly, and derive closed-form Bayes factors for both Gaussian and flat priors. They find that the log Bayes factor splits into the usual fit-improvement term, $-Delta chi^2_{min}/2$, plus a second term built from the Fisher matrices and prior widths of the two models, so two models that fit equally well can still be separated or conflated depending on their covariances alone. They then fit mock data with polynomials of different degree and show cases where a 4-parameter and a 14-parameter model come out essentially tied, and where polynomials up to degree about 12 are preferred over the model that actually generated the data. If the argument holds, quoting a Jeffreys threshold as if it settled a model comparison is not defensible, and any such verdict has to be reported together with the priors and covariances that produced it.

What carries the argument

The load-bearing object is the exact evidence integral for a linear model with a Gaussian likelihood. With Gaussian priors of inverse covariance $H$ and a Fisher matrix $F$ from the least-squares fit, completing the square gives $B_1 = e^{-chi^2_{min}/2 - c/2}|I_M + H^{-1}F|^{-1/2}$, where $c$ measures the offset between the prior centre and the best fit; with flat priors of width $Delta a_i$ the evidence becomes a product of error functions of $D^{perp}_{ii} Delta a_i / (2sqrt{2})$, with $D^perp$ from the Cholesky decomposition of the rotated Fisher matrix. Both forms make explicit that the evidence ratio is a product of a fit term and a volume term, and the volume term is what the Jeffreys thresholds ignore. The paper pairs this with an analytic Figure of Merit, the inverse volume of the $n$-$\sigma$ confidence ellipsoid, $mathrm{FoM}(M) = |F|^{1/2} Gamma(M/2+1) pi^{-M/2} (delta chi^2)^{-M/2}$, which lets the authors separate the effect of adding parameters from the effect of changing basis (monomials versus Legendre versus Chebyshev) at fixed parameter count.

Extended reading notes

Core claim

The central claim is that Jeffreys' scale is not a 'one size fits all' tool: its thresholds are stated in terms of the Bayes factor alone, but the Bayes factor for linear models is not a function of model complexity alone. The authors derive the exact evidence for Gaussian priors, $B_1 = e^{-chi^2_{min}/2 - c/2}|I_M + H^{-1}F|^{-1/2}$, and for flat priors an error-function product over the prior widths, and both reduce to the schematic form $ln B_{12} = -Delta chi^2_{min}/2 + ln G(M_1,M_2)$ where $G$ carries all the covariance and prior-volume information. The demonstration uses 31 mock points generated from $f(x) = a + (x-b)e^{-c x^2}$ and fits them with polynomials of degree $M_1$ and $M_2$. Pairs such as $(M_1,M_2)=(4,14)$ come out at $ln B_{12} approx 1.2$, which the scale calls indecisive, and $(4,10)$ with wider priors comes out at $ln B_{12} = 0$, which the scale calls a dead heat, even though ten extra parameters are being added. In the reverse direction, comparing the true generating model against high-degree polynomials yields strong support against the true model for degrees up to $M sim 10$–$12$. The authors describe these as Type I (rejecting a true model) and Type II (failing to reject a false one) failures of the scale.

Load-bearing premise

The headline demonstration sets each prior width proportional to the fitted parameter error, $Delta a_i = n sigma_i$ (or $H_{ii} = n sigma_i^2$), with $n=3$ or $7$; because the prior then scales with the posterior, models whose extra parameters are poorly constrained are guaranteed to come out near a tie, so the failure of the scale is partly built into that choice rather than discovered from the data, and no prior fixed independently of the fitted errors is tested.

Editorial extensions

If this is right

A Bayes factor quoted without its prior specification and Fisher matrices is not a self-contained piece of evidence, since the same fit improvement can move the verdict across several Jeffreys thresholds when the prior widths change.
Adding parameters to a model whose extra directions are poorly constrained by the data need not lower the evidence much, so an Occam penalty read off a fixed scale can understate how little the data actually say.
Comparing a candidate physical model against a flexible polynomial expansion can favor the polynomial even when the physical model is the true generator, so such comparisons need a prior on the expansion coefficients rather than a default width.
Because the Figure of Merit depends on the polynomial basis through a Jacobian determinant, parameter constraints and the associated FoM rankings are basis-dependent at fixed model dimension, and orthogonal bases such as Chebyshev give tighter contours than monomials for the same data.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The failure mode the authors document is structurally the same one that appears whenever an evidence ratio is dominated by the Occam volume rather than the likelihood: the remedy is to report the evidence as a function of prior width, not as a single number compared against a ladder.
A natural follow-up is a calibration study: for a fixed data set, sweep the prior width continuously and record where each Jeffreys threshold is crossed, which would convert the scale from a verdict into a sensitivity curve.
The same decomposition should apply to any model that is linear in a subset of its parameters after profiling the rest, which suggests the criticism extends well beyond polynomial fitting to, for example, linear perturbation spectra with nuisance amplitudes.
If the thresholds are arbitrary in the linear case where everything is computable, the situation can only be less controlled for the non-linear cosmological models the authors mention, where the evidence itself must be estimated numerically.

Original abstract

We are entering an era where progress in cosmology is driven by data, and alternative models will have to be compared and ruled out according to some consistent criterium. The most conservative and widely used approach is Bayesian model comparison. In this paper we explicitly calculate the Bayes factors for all models that are linear with respect to their parameters. We do this in order to test the so called Jeffreys' scale and determine analytically how accurate its predictions are in a simple case where we fully understand and can calculate everything analytically. We also discuss the case of nested models, e.g. one with $M_1$ and another with $M_2\supset M_1$ parameters and we derive analytic expressions for both the Bayes factor and the Figure of Merit, defined as the inverse area of the model parameter's confidence contours. With all this machinery and the use of an explicit example we demonstrate that the threshold nature of Jeffreys' scale is not a "one size fits all" reliable tool for model comparison and that it may lead to biased conclusions. Furthermore, we discuss the importance of choosing the right basis in the context of models that are linear with respect to their parameters and how that basis affects the parameter estimation and the derived constraints.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.32 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.54 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We are entering an era where progress in cosmology is driven by data, and alternative models will have to be compared and ruled out according to some consistent criterium. The most conservative and widely used approach is Bayesian model comparison. In this paper we explicitly calculate the Bayes factors for all models that are linear with respect to their parameters. We do this in order to test the so called Jeffreys' scale and determine analytically how accurate its predictions are in a simple case where we fully understand and can calculate everything analytically. We also discuss the case of nested models, e.g. one with $M_1$ and another with $M_2\supset M_1$ parameters and we derive analytic expressions for both the Bayes factor and the Figure of Merit, defined as the inverse area of the model parameter's confidence contours. With all this machinery and the use of an explicit example we demonstrate that the threshold nature of Jeffreys' scale is not a "one size fits all" reliable tool for model comparison and that it may lead to biased conclusions. Furthermore, we discuss the importance of choosing the right basis in the context of models that are linear with respect to their parameters and how that basis affects the parameter estimation and the derived constraints.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We are entering an era where progress in cosmology is driven by data, and alternative models will have to be compared and ruled out according to some consistent criterium.	background 1.00	0.04	0.02	asserted 0.97
s1	The most conservative and widely used approach is Bayesian model comparison.	background 1.00	0.04	0.02	asserted 1.00
s2	In this paper we explicitly calculate the Bayes factors for all models that are linear with respect to their parameters.	method 0.58	0.96	0.02	asserted 0.98
s3	We do this in order to test the so called Jeffreys' scale and determine analytically how accurate its predictions are in a simple case where we fully understand and can calculate everything analytically.	method 0.66	0.86	0.04	asserted 0.80
s4	We also discuss the case of nested models, e.g. one with $M_1$ and another with $M_2\supset M_1$ parameters and we derive analytic expressions for both the Bayes factor and the Figure of Merit, defined as the inverse area of the model parameter's confidence contours.	contribution 0.63	0.95	0.03	asserted 0.98
s5	With all this machinery and the use of an explicit example we demonstrate that the threshold nature of Jeffreys' scale is not a "one size fits all" reliable tool for model comparison and that it may lead to biased conclusions.	result 0.74	0.95	0.94	asserted 0.84
s6	Furthermore, we discuss the importance of choosing the right basis in the context of models that are linear with respect to their parameters and how that basis affects the parameter estimation and the derived constraints.	contribution 0.90	0.58	0.04	asserted 0.90

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 2.** FIG. 2: Contour plots of the Bayes factor log

**Figure 3.** FIG. 3: Contour plots of the Bayes factor log

**Figure 4.** FIG. 4: The Bayes factor log

**Figure 5.** FIG. 5: Contour plots of the Bayes factor log

**Figure 6.** FIG. 6: Contour plots of the Bayes factor log

**Figure 7.** FIG. 7: The dependence of the function

**Figure 8.** FIG. 8: The dependence of Ratio(

Figures reproduced from arXiv:1210.7652 by the authors.

References

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Generalizing the CPL Parametrization through Dark Sector Interaction
astro-ph.CO 2026-04 unverdicted novelty 5.0 of 10

Generalized interacting dark energy models with constant or dynamical couplings yield analytical density expressions but are not preferred over LambdaCDM by Bayesian evidence from DESI, Pantheon+, and CMB data.
DESI constraints on two-field quintessence with exponential potentials
astro-ph.CO 2025-10 conditional novelty 5.0 of 10

A two-field exponential-potential quintessence model fits DESI+CMB+SnIa data with moderate Bayesian preference over ΛCDM and slope parameters λφ≈λχ≈1.
The $H_0$ and $S_8$ tensions necessitate early and late time changes to $\Lambda$CDM
astro-ph.CO 2021-10 conditional novelty 5.0 of 10

A joint fit of early dark energy plus decaying dark matter, with local H0 and S8 measurements included as Gaussian priors, lands near both local values and is preferred over ΛCDM by ΔAIC = -6.72.
Late-transition vs smooth $H(z)$ deformation models for the resolution of the Hubble crisis
astro-ph.CO 2021-10 conditional novelty 5.0 of 10

A late, sharp transition in supernova absolute magnitude at z about 0.01 fits CMB, BAO, and supernova data better than smooth dark-energy models and can reconcile local and CMB Hubble constant estimates.
The Observational Future of Cosmological Scalar-Tensor Theories
astro-ph.CO 2016-10 conditional novelty 5.0 of 10

Combining Stage IV CMB, LSST and SKA intensity mapping forecasts Horndeski scalar-tensor parameters to roughly 0.07 to 0.16 and the Brans-Dicke parameter to omega > 1.7e4, within a factor of a few of Solar System bounds.
Constraining $f(R)$ gravity and evolving dark energy via large-scale structure and phase-space trajectories
physics.gen-ph 2026-09 conditional novelty 4.0 of 10

Joint constraints from BAO, SNeIa, CMB distance priors, and growth data set tight bounds on the parameters of Hu & Sawicki and Starobinsky f(R) gravity models, with AIC favoring but BIC not confirming a preference over ΛCDM.
Hubble tension or a transition of the Cepheid SnIa calibrator parameters?
astro-ph.CO 2021-09 conditional novelty 4.0 of 10

Allowing the Cepheid Wesenheit color parameter to take two values split at about 16 Mpc lowers the inferred Hubble constant from about 73 to about 68, consistent with Planck.
Beyond $\Lambda$CDM: Problems, solutions, and the road ahead
astro-ph.CO 2015-12 accept novelty 2.0 of 10

A community review concluding that no alternative to Lambda-CDM has yet displaced it, while cataloguing the theoretical and observational problems that keep the search alive.

How the literature uses it

52 arXiv papers cite this in 68 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 60 sentences without a confident label

Uses its method or tool 4 papers

First, we consider the Akaike Information Criterion (AIC) [ref] , defined as AIC ≡-2 lnℒ_max+2 N_tot=χ_min^2 +2 N_tot , where $N_tot$ corresponds to the total number of free parameters of the considered model and $ℒ_max$ corresponds to the maximum likelihood.

Late-transition vs smooth $H(z)$ deformation models for the resolution of the Hubble crisis · Model selection · 0.97
In order to compare two models $ℳ_1$ and $ℳ_2$, we can make use of the ratio of the Bayesian evidences calculated by the Harmonic Mean Approximation (see Ref. [ref] ), given by:

Cosmological Constraints on Higgs-Dilaton Inflation · Bayesian analysis of $w_0w_a$CDM vs. HDM · 0.99
To do this, having two models $M_1$ and $M_2$ the Jeffreys scale with respect of $Δlnε= lnε_M_1- lnε_M_2$ is as the following: [ref] : * for $Δlnε< 1.1$ there is a weak evidence against model $M_2$. * for $1.1<Δlnε< 3$ there is a definite evidence against model $M_2$. * and finally for $3<Δlnε$ there is a strong evidence against model $M_2$.

A Bayesian comparison between $\Lambda$CDM and phenomenologically emergent dark energy models · Observational data and Bayesian inference · 0.88

Cites it as background 1 paper

It can also be found in many other references, as in [ref] , but we add this information here too for completeness.

Quantifying the evidence for the current speed-up of the Universe with low and intermediate-redshift data. A more model-independent approach · Computation of evidences and other quantities of interest · 0.94

What it cites · 5 sentences without a confident label

Uses the method or tool of 1 paper

Then, this gives [ref] , [ref]

Everything you wanted to know about Data Analysis and Fitting but were afraid to ask · Minimization · 0.93

Cites as support 1 paper

It is model predictiveness, not model simplicity, which is "rewarded" in Bayesian model comparison [ref] , [ref] .

Should we doubt the cosmological constant? · Conclusions · 0.99

Cites as background 11 papers

On the other hand, the Bayesian evidence is based on Bayes theorem, see Refs. [ref] , [ref] for in-depth reviews, which expresses the posterior distribution $P(u,M|D)$ for the parameters $u$ of the model $M$ given the data $D$, in terms of the likelihood distribution function $L(D| u,M)$ within a given set of priors $π(u,M)$ [equation] where the likelihood can be obtained from $L(D| u,M)=exp(-χ^2(u)/2)$.

Statistical methods for cosmological parameter selection and estimation · Introduction · 0.96
On the other hand, the Bayesian evidence is based on Bayes theorem, see Refs. [ref] , [ref] for in-depth reviews, which expresses the posterior distribution $P(u,M|D)$ for the parameters $u$ of the model $M$ given the data $D$, in terms of the likelihood distribution function $L(D| u,M)$ within a given set of priors $π(u,M)$ [equation] where the likelihood can be obtained from $L(D| u,M)=exp(-χ^2(u)/2)$.

Bayes in the sky: Bayesian inference and model selection in cosmology · Introduction · 0.96
Several examples of the use of the Jeffreys' scale in cosmology and astronomy can by found in [ref] and references there-in.

Limitations of Bayesian Evidence Applied to Cosmology · The Jeffreys' scale revisited · 0.95

Discussion

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