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:

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*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:

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


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

These 'implausible values' are considered to be values that are outside the 95% likelihood interval. In the image this occurs when the horizontal line, representing the interval, crosses the normalized likelihood function.
Below we draw the plot and add some extra highlights with red color for the parts that are outside the likelihood interval.


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"?

This is not generally true. It is just an arbitrary definition to set the likelihood interval equal to the boundary of plausible and implausible values.
In the article they made a table where the relative likelihood appears to be the best interval. It is on average the smallest interval and it has an average coverage of 95%.

But this situation only occurs because of the (arbitrary) way that the average is computed. The true value of $p$ is considered as uniform distributed.
This is similar to the credible interval being optimal (Are there any examples where Bayesian credible intervals are obviously inferior to frequentist confidence intervals), and that is only true if the (assumed) prior is true. The likelihood interval is effectively equivalent to a credible interval with uniform prior for the parameter $p$. Then, if you consider the average based on a uniform distribution of $p$, you will get that the likelihood interval performs the best.
So the "good performance" is all based on using the uniform distribution of $p$ to compute the average.

Sidenote: the article is not very clear and consistent with their definition of the 'likelihood interval'. In the text they define it as:

The set of $p$ values for which $R(p) \geq c$ is called a $100c\%$ likelihood interval (LI).

This is the same as the definition used by Wikipedia. But, the percentage does not relate to confidence intervals. A smaller percentage $100c\%$ means a larger interval, and that is opposite to what happens with confidence intervals.
The region that they compute, $[0.7658, 0.9965]$, in the image/example is not a 95% likelihood region. Instead it is a 14.65% likelihood region, which is asymptotically equivalent to a 95% confidence region (based on the likelihood statistic asymptotically approaching a $\chi^2$ distribution according to Wilks' theorem).
The interval that I computed for the image, the bounds $[0.7608, 0.9968]$ are for a $95\%$ highest density region, using the likelihood function as the density.
