- Estimation of likelihood contours around the maximum likelihood peak beyond the Fisher matrix approach. We envisage here a programme where simulated mock data will be generated and then used to blindly reconstruct the likelihood surface to sufficient accuracy.
- Estimation of Bayesian posterior distributions and assessment of impact of various priors. Bayesian inference is a mature field in cosmology and we now have at our disposal a number of efficient and reliable numerical algorithms based on Markov Chain Monte Carlo or nested sampling methods.
- Comparison of Bayesian inferences with inferences based on profile likelihoods. Discrepancies might occur in the presence of large “volume effects” arising from insufficiently constraining data sets and highly multi-modal likelihoods [898]. Based on our experience so far, this is unlikely to be a problem for most of the statistical quantities of interest here but we recommend to check this explicitly for the more complicated distributions.
- Investigation of the coverage properties of Bayesian credible and frequentist confidence intervals. Coverage of intervals is a fundamental property in particle physics, but rarely discussed in the cosmological setting. We recommend a careful investigation of coverage from realistically simulated data sets (as done recently in [626]). Fast neural networks techniques might be required to speed up the inference step by several orders of magnitude in order to make this kind of studies computationally feasible [822, 174].
- Computation of the Bayesian evidence to carry out Bayesian model selection [897, 677]. Algorithms based on nested sampling, and in particular, MultiNest [362], seem to be ideally suited to this task, but other approaches are available, as well, such as population Monte Carlo [502] and semi-analytical ones [894, 429]. A robust Bayesian model selection will require a careful assessment of the impact of priors. Furthermore, the outcome of Bayesian model selection is dependent on the chosen parametrization if different nonlinearly related reparametrizations can equally plausibly be chosen from physical consideration (relevant examples include parametrizations of the isocurvature fraction [119], the tensor-to-scalar ratio [710] and the inflaton potential [634]). It will be important to cross check results with frequentist hypothesis testing, as well. The notion of Bayesian doubt, introduced in [624], can also be used to extend the power of Bayesian model selection to the space of unknown models in order to test our paradigm of a CDM cosmological model.
- Bayesian model averaging [571, 709] can also be used to obtain final inferences parameters which take into account the residual model uncertainty. Due to the concentration of probability mass onto simpler models (as a consequence of Occam’s razor), Bayesian model averaging can lead to tighter parameter constraints than non-averaged procedures, for example on the curvature parameter [917].

Living Rev. Relativity 16, (2013), 6
http://www.livingreviews.org/lrr-2013-6 |
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