Likelihood, posterior, prior interpretation and credibility/confidence_level with bayesian/frequentist approaches

Question

This question was originally posted on physics exchange but one advised me to transfer it here.

I try to understand the following article :

testing general relativity from curvature and energy contents at cosmological scale

I don't understand the title of figure 1 :

where it is indicated the prior values for $\omega_{b}, \omega_{\text{cdm}}, \text{h}, ...$ : what do authors mean by "prior ?

1) Does this term "prior"refer to the bayesian formula :

\begin{equation} \text{posterior}= \dfrac{\text{likelihood}\,\times\,\text{prior}}{\text{evidence}}\quad(1) \end{equation}

which, I think, corresponds to the formula :

\begin{equation} p(\theta|d)={\dfrac{p(d|\theta)p(\theta )}{p(d)}}\quad(2) \end{equation}

where $\theta$ is the parameter to estimate and $d$ represent the data

???

So, if this is the case, the prior of parameter $\theta_{i}$ would represent the probability $p(\theta_{i})$, wouldn't it ?

2.1) On the figure 3 :

I don't understand how to get this figure.

Given Likelihood is proportional to posterior (is it right from above equation $(1)$ ?), I have to know the theorical model to compute Likelihood ?

I mean, to get $p(\theta|d)$, I have to generate the probability $p(d|\theta)$ assuming I know the value of $\theta$ parameters, don't I ?

There seems here a paradox : I compute the posterior $p(\theta|d)$ to estimate $\theta$ parameter on one side but I have to know precisely the probability $p(\theta)$ on the other side.

2.2) Moreover, how to compute on this figure the Likelihood of red and black curves which corresponds respectively with parameter $w$ free and $\Omega_{k},\Omega_{dyn}$ with also free ?

I don't know which theorical model (I suppose this is a PDF (probability function)) to use ?

3) Finally, I have a last question about Confidence level (CL with frequentist approach) and Credibility level (Bayesian approach) :

How to make the link between these 2 notions (if this is possible) ? the first is an interval on a random variable and the second is an interval about a parameter, so at first sight, this would't have the same signification.

However, I often see the notion of "Confidence level" for estimation of a parameter, like for example the contours on figure 4 of the article cited above, i.e o this figure :

Any help or explanations are welcome, I am very interested in understanding all these concepts of statistics.

Regards

Answer 1

Disclaimer, I haven't read the article you posted.

1) Yes, most certainly the term prior refers to the prior probability density function (PDF) in the Bayes formula.

2.1) The prior over parameters $\theta_{i}$ would represent the PDF $p(\theta_{i}), \forall i$. A very common assumption would be to consider $\theta_{i}\sim\mathcal{N}(\mu, \sigma^{2}), \forall i$, where $\mu=0$ and $\sigma^{2} = 1$.

(i.e. answers your quesiton on how to use $p(\theta)$ to generate $\theta_{i}, \forall i$)

2.2) The likelihood $p(d|\theta)$ can be pretty much any model, (e.g.) suppose $p(d|\theta) = f(d; \theta)\sim\mathcal{N}(\mu, \sigma^2) = \frac{1}{\sqrt{ 2\pi\sigma}}e^{-\frac{1}{2}\frac{(x-\mu)^{2}}{\sigma}}$

(i.e. should tell you that the posterior amounts to averaging samples from the product of likelihood x prior)

To get the curves form the paper you'll have to know what the authors used as $p(\theta)$ and $p(d|\theta)$. The dashed line represents the true posterior a.k.a the data distribution $d$ and the coloured curves are the different posteriors if slightly change $p(\theta)$.

3) The only thing I can think of is if you can find a correspondence such that you could argue that if a parameter stays within a given range then it induces X PDF or vice versa. (maybe?)