5.4 Future activities and open challenges
As outlined in the previous sections, several approaches are available to capture the expected scientific
performance of Euclid. As part of future theoretical activities, it will be necessary to build on the above
concepts in order to obtain a realistic assessment of the science return of Euclid. Operationally, this means
that the following tasks will need to be carried out:
- 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
- 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 . 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 ). 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 , seem to be
ideally suited to this task, but other approaches are available, as well, such as population
Monte Carlo  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 , the tensor-to-scalar
ratio  and the inflaton potential ). It will be important to cross check results with
frequentist hypothesis testing, as well. The notion of Bayesian doubt, introduced in , 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 .