Use of Relative Likelihoods in Statistics? I am reading this article here on Confidence Intervals for Binomial Proportions (https://www.scirp.org/pdf/ojs_2021101415023495.pdf).
In this article, the author lists several different methods (e.g. Wald, Wilson, Clopper-Pearson, etc.) that can be used to calculate Confidence Intervals for a Binomial Proportion, and then compares the advantages and disadvantages of these different methods.
One of the approaches that is used to compare the different Confidence Intervals is to use the notion of Relative Likelihoods. As I understand, a Relative Likelihood is a ratio between the actual likelihood function and the likelihood function based on the parameter estimates.
For example, consider the diagram below from this article:

The Relative Likelihood Function is the same in all 4 plots. And in each one of these plots, the horizontal line represents the Confidence Interval calculated using each of these 4 different methods.
I am having difficulty understanding the following point:

*

*Apparently, these plots can be used to show that some of these Confidence Intervals contain "implausible values".

*That is, in instances where the horizontal line extends past the boundaries of the Relative Likelihood Function - values beyond these points are considered "implausible"

I am trying to understand why exactly this is true - can someone please help me understand why are values of the Confidence Interval in regions outside of the Relative Likelihood Function considered as "implausible"?
Thanks!
References:

*

*https://www.eecis.udel.edu/~boncelet/dasp11.pdf

This is an extended comment, to complement the great answers by @ChristianHennig, @SextusEmpiricus and @statmerkur, with background material from related Cross Validated threads. The theory of confidence intervals and likelihood intervals is interesting and complex; reading expositions from multiple points of view might help to understand it better.

The article you cite uses the terms "plausible" / "implausible" to mean that some values for the parameter $p$ have "high" / "low" likelihood — relatively, when compared with other values for $p$. These are not commonly used terms, so I suggest you don't use them either.
Rather than:

The maximum likelihood estimate (MLE) $p$ of $\hat{p}$ is the most plausible value of $p$ in that it makes the observed sample most probable.

It's better to say: The maximum likelihood estimate $\hat{p}$ of the parameter $p$ is the value which maximizes the likelihood. Math notation is clear and precise:
\begin{align}
\hat{p}_{\text{MLE}}:\arg\max_{p}L_x(p)
\end{align}
where $L_x(p)$ is the likelihood function.
Equivalently, you can say that the MLE maximizes the log-likelihood $\log L_x(p)$ or minimizes the negative log-likelihood $-\log L_x(p)$.
And now for the references.

*

*If you'd like to learn more about likelihood-based confidence intervals, see How to create an interval estimate based on being within some percentage of the maximum value of the likelihood function?. 


*If you'd like to learn more about confidence intervals for a binomial proportion, see Confidence interval for Bernoulli sampling and Confidence interval around binomial estimate of 0 or 1.


*Are all values within a 95% confidence interval equally likely? Confidence intervals are defined in terms of long-run frequency: a 100(1-α)% CI has at least 100(1-α)% coverage probability. The (relative) likelihood of the parameter values inside the interval is irrelevant, from a frequentist point of view: to compute the likelihood we fix the data that we observe in an experiment, to compute the coverage we repeat the same experiment (hypothetically). The intervals might be exactly the same, yet their interpretation very different.


*And if you are interested in the theory of constructing confidence intervals using the long-run frequency vs using the likelihood function, I recommend Chapter 2 of In All Likelihood: Statistical Modelling And Inference Using Likelihood. Y. Pawitan (2013).
