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Diffstat (limited to 'docs/source/statistics.rst')
| -rw-r--r-- | docs/source/statistics.rst | 14 |
1 files changed, 7 insertions, 7 deletions
diff --git a/docs/source/statistics.rst b/docs/source/statistics.rst index e2dd85f..b63c9a0 100644 --- a/docs/source/statistics.rst +++ b/docs/source/statistics.rst @@ -100,7 +100,7 @@ Instead, according to Wilks' theorem [3]_, for sufficiently large :math:`\textbf{x}` and provided certain regularity conditions are met (MLE exists and is unique), :math:`-2\ln\lambda\left(\textbf{x}\right)` can be approximated to follow a :math:`\chi^2` distribution. The :math:`\chi^2` -distribution is parameterised by :math:`k`, the *number of degrees of +distribution is parameterized by :math:`k`, the *number of degrees of freedom*, which is defined as the number of independent normally distributed variables that were summed together. When the profile likelihood is used to account for :math:`n` nuisance parameters, the effective number of degrees of @@ -248,7 +248,7 @@ referred to as the *evidence* of a particular model: \pi_j\left(\mathbf{\theta}_j\right)\text{d}\mathbf{\theta}_j -This was seen before as just a normalisation constant above; however, this +This was seen before as just a normalization constant above; however, this quantity is central in Bayesian model selection, which for two models :math:`\mathcal{M}_0` and :math:`\mathcal{M}_1` is realised through the ratio of the posteriors: @@ -354,7 +354,7 @@ mixing parameters are of concern. These parameters are defined in the mixing matrix :math:`U`, in such a way that any valid combination of the mixing angles can be mapped into a unitary matrix. The ideal and most ignorant choice of prior here is one in which there is no distinction among the three neutrino -flavours, compatible with the hypothesis of *neutrino mixing anarchy*, which is +flavors, compatible with the hypothesis of *neutrino mixing anarchy*, which is the hypothesis that :math:`U` can be described as a result of random draws from an unbiased distribution of unitary :math:`3\times3` matrices [11]_, [12]_, [13]_, [14]_. Simply using a flat prior on the mixing angles however, does @@ -368,7 +368,7 @@ is the central assumption of *basis independence* and from this, distributions over the mixing angles are determined by the integration invariant *Haar measure* [13]_. For the group :math:`U(3)` the Haar measure is given by the volume element :math:`\text{d} U`, which can be written using the mixing angles -parameterisation: +parameterization: .. math:: @@ -385,15 +385,15 @@ chosen according to this Haar measure, i.e. in :math:`\sin^2\theta_{12}`, :math:`\cos^4\theta_{13}`, :math:`\sin^2\theta_{23}` and :math:`\delta`. You can see an example on this in action in the :doc:`examples` notebooks. -This also needs to be considered in the case of a flavour composition +This also needs to be considered in the case of a flavor composition measurement using sampling techniques in a Bayesian approach. In this case, the posterior of the measured composition :math:`f_{\alpha,\oplus}` is sampled over as the parameters of interest. Here, the effective number of parameters can be reduced from three to two due to the requirement :math:`\sum_\alpha f_{\alpha,\oplus}=1`. Therefore, the prior on these two parameters must be -determined by Haar measure of the flavour composition volume element, +determined by Haar measure of the flavor composition volume element, :math:`\text{d} f_{e,\oplus}\wedge\text{d} f_{\mu,\oplus}\wedge\text{d} -f_{\tau,\oplus}`. The following *flavour angles* parameterisation is found to +f_{\tau,\oplus}`. The following *flavor angles* parameterization is found to be sufficient: .. math:: |