I'll start with a brief introduction of the statistics before addressing the specific questions posed.
Let's say you have conducted an experiment, and from it obtained a data distribution. However, raw data by itself is not exactly interesting; instead, you want to be able to extract some meaningful (i.e. statistical) result from the data.
The process by which this is done is known as statistical inference: given some set of data, $\boldsymbol{X}$, one wishes to test how well it can be described by a model, $M(\boldsymbol{X};\boldsymbol{\Theta})$, where $\boldsymbol\Theta$ are the parameters of the model, whose true values are unknown a priori. To extract likely values of the model parameters given the observed data, one defines the likelihood function: \begin{equation} L(\boldsymbol{\Theta};\boldsymbol{X}) \equiv M(\boldsymbol{X};\boldsymbol{\Theta}), \end{equation} which differs from the model in that the data is fixed, while model parameters are treated as random variables.
A common method of inference is that of maximum likelihood estimation (MLE), which states that the set of parameter values, $\boldsymbol{\hat\Theta}$, that result in maximisation of the likelihood function provides an estimate of the true parameter values. 915
Confidence regions can then be constructed around the $\boldsymbol{\hat\Theta}$ using the likelihood ratio test.
Another method is known as Bayesian inference, which makes use of <PERSON>' theorem: \begin{equation} P(\boldsymbol{\Theta}|\boldsymbol{X},M) = \frac{P(\boldsymbol{X}|\boldsymbol{\Theta},M) P(\boldsymbol{\Theta}|M)}{P(\boldsymbol{X}|M)}\,, \end{equation} where $P(\boldsymbol{\Theta}|\boldsymbol{X},M)$ is the posterior probability distribution, $P(\boldsymbol{X}|\boldsymbol{\Theta},M)=L(\boldsymbol{\Theta};\boldsymbol{X})$ is the likelihood function, $P(\boldsymbol{\Theta}|M)=\pi(\boldsymbol{\Theta})$ is the prior probability distribution, and $P(\boldsymbol{X}|M)=\mathcal{Z}$ is the Bayesian evidence, which normalizes the posterior distribution. Maximum a posteriori (MAP) estimation prescribes a method of estimating model parameters through maximisation of the posterior probability. However, note that it is equivalent to just maximise the likelihood times prior, \begin{equation} L(\boldsymbol{\Theta})\pi(\boldsymbol{\Theta}), \end{equation} since constant factors (such as the evidence) do not shift the location of the maximum in parameter space. So in a practical application of MAP for parameter estimation purposes, one only needs to define the likelihood function and prior distributions: the former according to whichever theoretical model under test, and the latter defining the search region in parameter space.
what do authors mean by "prior"? Does this term "prior" refer to the bayesian formula...So, if this is the case, the prior of parameter $\theta_i$ would represent the probability $p(\theta_i)$, wouldn't it ?
Yes, they are referring to the prior that appears in <PERSON>' theorem, though note that this is a probability distribution.
In this case, what they seem to mean is that they have chosen $\delta$-distributions as priors for the $\omega_b$, $\omega_{cdm}$, $h$, $A_s$, $n_s$ and $\tau_{reio}$ parameters. Put more simply, what they show here is the model distribution when all parameters are fixed to certain values.
Given Likelihood is proportional to posterior (is it right from above equation (1) ?), I have to know the theorical model to compute Likelihood ?
Yes, you need to have a theoretical model before the likelihood function can be defined.
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. 915