- #1

fab13

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- TL;DR Summary
- I am looking for explanations about an article that I have to study. This concerns differents figures that I try to understand and the underlying method that allows to produce these plots.

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}, ...## :

which, I think, corresponds to the formula :

I don't understand how to get this figure :

How to make the link between these 2 notions (if this is possible since into a previous post, one told me there is no link except both represent an uncertainty on a variable (either random or parameter)) ?

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}

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 of parameter ##p(\theta_{i})##, wouldn't it ?

2.1) On the figure 3 :

\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 of parameter ##p(\theta_{i})##, wouldn't it ?

2.1) On the figure 3 :

I don't understand how to get this figure :

**I compute the posterior ##p(\theta|d)## to estimate ##\theta## parameter on one side but I have to know precisely the probability ##p(\theta)##**

Given Likelihood is proportional to posterior (is it right from above equation ##(1)## ?), I have to know the theoretical 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## parameter, don't I ?

there seems here a paradox :Given Likelihood is proportional to posterior (is it right from above equation ##(1)## ?), I have to know the theoretical 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## parameter, don't I ?

there seems here a paradox :

**I suppose there is a specific PDF (probability function)**

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 theoretical model (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 theoretical model (

**) to use ? but which one ?**

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

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 since into a previous post, one told me there is no link except both represent an uncertainty on a variable (either random or parameter)) ?

##\Rightarrow## the first one is an interval on a random variable and the second one is an interval about the estimation of 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 (i.e so from a bayesian point of view), 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

##\Rightarrow## the first one is an interval on a random variable and the second one is an interval about the estimation of 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 (i.e so from a bayesian point of view), 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

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