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

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 :

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

which, I think, corresponds to the formula :

$$$$p(\theta|d)={\dfrac{p(d|\theta)p(\theta )}{p(d)}}\quad(2)$$$$

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

• I suggest you post these as separate questions. It seems too much for one question. One person might be able to answer one of them perfectly but maybe not all. Commented Aug 11, 2019 at 18:51
• @Curious ok, I am going to split my different issues, regards
– user226073
Commented Aug 11, 2019 at 19:53

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?)