The Bayesian model comparison approach based on the evaluation of the evidence is being increasingly applied to model building questions such as: are isocurvature contributions to the initial conditions required by the data [46, 58, 60, 182]? Is the Universe flat [183, 58, 70]? What is the best description of the primordial power spectrum for density perturbations [51, 184, 58, 185, 165, 47, 70]? Is dark energy best described as a cosmological constant [186, 50, 51, 187, 188, 189, 190, 191]? In this section we review the status of the field.
6.1. Evidence for the cosmological concordance model
Table 4 is a fairly extensive compilation of recent results regarding possible extensions to (or reduction of) the vanilla CDM concordance cosmological model introduced in section 5.1. We have chosen to compile only results obtained using the full Bayesian evidence, rather than approximate model comparisons obtained via the information criteria because the latter are often not adequate approximations, for the reasons explained in section 4.7. Of course, the outcome depends on the Occam's razor effect brought about by the prior volume (and sometimes, by the choice of parameterization). Where applicable, we have show the sensitivity of the result on the prior assumptions by giving a ballpark range of values for the Bayes factor, as presented in the original studies. The reader ought to refer to the original works for the precise prior and parameter choices and for the justification of the assumed prior ranges.
Competing model | N_{par} | ln B | Ref | Data | Outcome |
Initial conditions | |||||
Isocurvature modes | |||||
CDM isocurvature | +1 | -7.6 | [58] | WMAP3+, LSS | Strong evidence for adiabaticity |
+ arbitrary correlations | +4 | -1.0 | [46] | WMAP1+, LSS, SN Ia | Undecided |
Neutrino entropy | +1 | [-2.5, -6.5]^{p} | [60] | WMAP3+, LSS | Moderate to strong evidence for adiabaticity |
+ arbitrary correlations | +4 | -1.0 | [46] | WMAP1+, LSS, SN Ia | Undecided |
Neutrino velocity | +1 | [-2.5, -6.5]^{p} | [60] | WMAP3+, LSS | Moderate to strong evidence for adiabaticity |
+ arbitrary correlations | +4 | -1.0 | [46] | WMAP1+, LSS, SN Ia | Undecided |
Primordial power spectrum | |||||
No tilt (n_{s} = 1) | -1 | +0.4 | [47] | WMAP1+, LSS | Undecided |
[-1.1, -0.6]^{p} | [51] | WMAP1+, LSS | Undecided | ||
-0.7 | [58] | WMAP1+, LSS | Undecided | ||
-0.9 | [70] | WMAP1+ | Undecided | ||
[-0.7, -1.7]^{p,d} | [185] | WMAP3+ | n_{s} = 1 weakly disfavoured | ||
-2.0 | [184] | WMAP3+, LSS | n_{s} = 1 weakly disfavoured | ||
-2.6 | [70] | WMAP3+ | n_{s} = 1 moderately disfavoured | ||
-2.9 | [58] | WMAP3+, LSS | n_{s} = 1 moderately disfavoured | ||
<-3.9^{c} | [65] | WMAP3+, LSS | Moderate evidence at best against n_{s} ≠ 1 | ||
Running | +1 | [-0.6, 1.0]^{p,d} | [185] | WMAP3+, LSS | No evidence for running |
< 0.2^{c} | [165] | WMAP3+, LSS | Running not required | ||
Running of running | +2 | <0.4^{c} | [165] | WMAP3+, LSS | Not required |
Large scales cut-off | +2 | [1.3, 2.2]^{p,d} | [185] | WMAP3+, LSS | Weak support for a cut-off |
Matter-energy content | |||||
Non-flat Universe | +1 | -3.8 | [70] | WMAP3+, HST | Flat Universe moderately favoured |
-3.4 | [58] | WMAP3+, LSS, HST | Flat Universe moderately favoured | ||
Coupled neutrinos | +1 | -0.7 | [192] | WMAP3+, LSS | No evidence for non-SM neutrinos |
Dark energy sector | |||||
w(z)= w_{eff} ≠ -1 | +1 | [-1.3, -2.7]^{p} | [186] | SN Ia | Weak to moderate support for |
-3.0 | [50] | SN Ia | Moderate support for | ||
-1.1 | [51] | WMAP1+, LSS, SN Ia | Weak support for | ||
[-0.2, -1]^{p} | [187] | SN Ia, BAO, WMAP3 | Undecided | ||
[-1.6, -2.3]^{d} | [188] | SN Ia, GRB | Weak support for | ||
w(z) = w_{0} + w_{1} z | +2 | [-1.5, -3.4]^{p} | [186] | SN Ia | Weak to moderate support for |
-6.0 | [50] | SN Ia | Strong support for | ||
-1.8 | [187] | SN Ia, BAO, WMAP3 | Weak support for | ||
w(z) = w_{0} + w_{a}(1 - a) | +2 | -1.1 | [187] | SN Ia, BAO, WMAP3 | Weak support for |
[-1.2, -2.6]^{d} | [188] | SN Ia, GRB | Weak to moderate support for | ||
Reionization history | |||||
No reionization ( = 0) | -1 | -2.6 | [70] | WMAP3+, HST | ≠ 0 moderately favoured |
No reionization and no tilt | -2 | -10.3 | [70] | WMAP3+, HST | Strongly disfavoured |
^{d} Depending on the choice of datasets. | |||||
^{p} Depending on the choice of priors. | |||||
^{c} Upper bound using Bayesian calibrated
p-values, see section 4.5.
Data sets: WMAP1+ (WMAP3+): WMAP 1st year (3-yr) data and other CMB measurements. LSS: Large scale structures data. SN Ia: supernovae type Ia. BAO: baryonic acoustic oscillations. GRB: gamma ray bursts. |
As anticipated, the 6 parameters CDM concordance model is currently well supported by the data, as the inclusion of extra parameters is not required by the Bayesian evidence. This is shown by the fact that most model comparisons return either an undecided result or they support the CDM model (negative values for ln B in Table 4). The only exception is the support for a cut-off on large scales in the power spectrum reported by [185]. This is clearly driven by the anomalies in the large scale CMB power spectrum, which in this case are interpreted as being a reflection of a lack of power in the primordial power spectrum. Whether such anomalies are of cosmological origin remains however an open question [193, 194]. If extensions of the model are not supported, reduction of CDM to simpler models is not viable, either: recent studies employing WMAP 3-yr data find that a scale invariant spectrum with no spectral tilt is now weakly to moderately disfavoured [184, 70, 58, 65]. Also, a Universe with no reionization is no longer a good description of CMB data, and a non-zero optical depth is indeed required [70].
A few further comments about the results reported in Table 4 are in place:
Let us now turn to models that are not nested within CDM— i.e., alternative theoretical scenarios. Table 5 gives some examples of the outcome of the Bayesian model comparison with the concordance model. As above, we restrict our considerations to studies employing the full Bayesian evidence (there are many other examples in the literature carrying out approximate model comparison using information criteria instead). The model comparison is often more difficult for non-nested models, as priors must be specified for all of the parameters in the alternative model (and in the CDM model, as well), in order to compute the evidence ratio. The usual caveats on prior choice apply in this case. From Table 5 it appears that the data do not seem to require fundamental changes in our underlying theoretical model, either in the form of Bianchi templates representing a violation of cosmic isotropy (see also [206]), or as Lemaitre-Tolman-Bondi models or fractal bubble scenarios with dressed cosmological parameters. The anomalous dipole in the CMB temperature maps is a fine example of Lindley's paradox. When fitting a dipolar template to the CMB maps, the effective chi-square improves by 9 to 11 units (depending on the details of the analysis) for 3 extra parameters [205, 207], which would be deemed a "significant" effect using a standard goodness-of-fit test. However, the Bayesian evidence analysis shows that the odds in favour of an anomalous dipole are 9 to 1 at best (corresponding to ln B < 2.2), which does not reach the "moderate evidence at best" threshold. Hence Bayesian model comparison is conservative, requiring a stronger evidence before deeming an effect to be favoured.
Competing model | N_{par} | ln B | Ref | Data | Outcome |
Alternatives to FRW | |||||
Bianchi VII_{h} | 5 to 8 | [-0.9, 1.2]^{d,p} | [54] | WMAP1, WMAP3 | Weak support (at best) for Bianchi template |
5 to 6 | [-0.1, -1.2]^{p} | [203] | WMAP3 | No evidence after texture correction | |
LTB models | 4 | -3.6 | [204] | WMAP3, BAO, SN Ia | Moderate evidence against LTB |
Fractal bubble model | 2 | 0.3 | [89] | SN Ia | Undecided |
Asymmetry in the CMB | |||||
Anomalous dipole | 3 | 1.8 | [205] | WMAP3 | Weak evidence for anomalous dipole |
< 2.2^{c} | [65] | WMAP3 | Weak evidence at best | ||
^{d} Depending on the choice of datasets. | |||||
^{p} Depending on the choice of priors. | |||||
^{c} Upper bound using Bayesian calibrated p-values, see section 4.5. |
6.2. Other uses of the Bayesian evidence
Beside cosmological model building, the Bayesian evidence can be employed in many other different ways. Here we presents two aspects that are relevant to our topic, namely the applications to the field of multi-model inference and model selection forecasting.
(48) |
where p(, | d, _{i}) is the posterior within each model _{i}, and it is understood that the posterior has non-zero support only along the parameter directions _{i} ⊂ that are relevant for the model, and delta-functions along all other directions. Each term is weighted by the corresponding posterior model probability,
(49) |
The prior model probabilities p(_{i}) are usually set equal, but a model preference can be incorporated here if necessary. The model averaged posterior distribution of Eq. (48) then represents the parameter constraints obtained independently of the model choice, which has been marginalized over. Unless one of the models is overwhelmingly more probable than the others (in which case the model averaging essentially disappears, as all of the weights for the other models go to zero), the model-averaged posterior distribution can be significantly different from the model-specific distribution. A counter-intuitive consequence is that in the case of dark energy, the model-averaged posterior shows tighter constraints around w = -1 than any of the evolving dark energy models by itself. This comes about because CDM is the preferred model and hence much of the weight in the model-averaged posterior is shifted to the point w = -1 [208]. For further details on multi-model inference, see e.g. [209, 210].
The procedure is as follows (see [211] for details and the application to dark energy scenarios). At every point in parameter space, mock data from the future observation are generated and the Bayes factor between the competing models is computed, for example between an evolving dark energy and a cosmological constant. Then one delimits in parameter space the region where the future data would not be able to deliver a clear model comparison verdict, for example | ln B | < 5 (evidence falling short of the "strong" threshold). The experiment with the smallest "model-confusion" volume in parameter space is to be preferred, since it achieves the highest discriminative power between models. An application of a related technique to the spectral index from the Planck satellite is presented in [212, 213].
Alternatively, we can investigate the full probability distribution for the Bayes factor from a future observation. This allows to make probabilistic statements regarding the outcome of a future model comparison, and in particular to quantify the probability that a new observation will be able to achieve a certain level of evidence for one of the models, given current knowledge. This technique is based on the predictive distribution for a future observation, which gives the expected posterior on for an observation with experimental capabilities described by e (this might describe sky coverage, noise levels, target redshift, etc):
(50) |
Here, d are the currently available observations, p(_{i} | d) is the current model posterior, p( | _{i}^{⋆}, e, _{i}) is the posterior on from a future observation e computed assuming _{i}^{⋆} are the correct model parameters, while each term is weighted by the present probability that _{i}^{⋆} is the true value of the parameters, p(_{i}^{⋆} | d, _{i}). The sum over i ensures that the prediction averages over models, as well. From Eq. (50) we can compute the corresponding probability distribution for ln B from experiment e, for example by employing MCMC techniques (further details are given in [59]). This method is called PPOD, for predictive posterior odds distribution and can be useful in the context of experiment design and optimization, when the aim is to determine which choice of e will lead to the best scientific return from the experiment, in this case in terms of model selection capabilities (see [214, 215, 216] for a discussion of performance optimization for parameter constraints). For further details on Bayes factor forecasts and experiment design, see [217].