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

The relative, or normalized, likelihood (function) $\tilde{\mathcal{L}}$ is defined as
$$
\mathop{\tilde{\mathcal{L}}}\left(\theta\right) = \frac{\mathop{\mathcal L}\left(\theta\right)}{\mathop{\mathcal L}\left(\hat\theta_\text{ML}\right)},
$$
where $\mathcal L$ is the likelihood function and $\hat\theta_\text{ML} = \underset{\theta \,\in\, \Theta}{\mathrm{arg\;max}}\mathop{\mathcal L}\left(\theta\right)$.
Clearly, $\mathop{\tilde{\mathcal{L}}}\left(\theta\right) \in \left[0, \mathop{\tilde{\mathcal{L}}}\left(\hat\theta_\text{ML}\right)\right] = \left[0, 1\right]$.
While there exist pure likelihood approaches based on thresholding $\tilde{\mathcal{L}}$ at certain  cutoff points $c$ (e.g., $\theta$s are: very plausible if $\mathop{\tilde{\mathcal{L}}}\left(\theta\right) \in \left(1/3, 1\right]$, plausible if  $\mathop{\tilde{\mathcal{L}}}\left(\theta\right) \in \left(1/10, 1/3\right]$, less plausible if $\mathop{\tilde{\mathcal{L}}}\left(\theta\right) \in \left(1/100, 1/10\right]$, etc.), deriving confidence intervals with (asympotically) guaranteed coverage probabilities requires calibration of the relative likelihood via probability.
To that end, consider the likelihood ratio statistic $W$ given by
$$
W = 2 \cdot \ln\left(\frac{\mathop{\mathcal L}\left(\hat\theta_\text{ML}\right)}{\mathop{\mathcal L}\left(\theta\right)}\right) = -2 \cdot \ln\left(\mathop{\tilde{\mathcal{L}}}\left(\theta\right)\right),
$$
which, under the Fisher regularity conditions, is asymptotically $\mathop{\chi^2}\left(1\right)$-distributed for a scalar parameter $\theta$.
Hence, an asymptotic $\gamma\cdot 100\%$ confidence set (an interval if the likelihood function is not multimodal) for $\theta$ is given by
$$
\left\{\theta:\mathop{\tilde{\mathcal{L}}}\left(\theta\right) \geq c\right\},\\
c = \exp\left(-\frac{1}{2}\mathop{\chi^2_\gamma}\left(1\right)\right),
$$
where $\mathop{\chi^2_\gamma}$ denotes the $\gamma$-quantile of the $\mathop{\chi^2}\left(1\right)$ distribution.
In Figure 5 all horizontal lines seem to be drawn at the same cutoff $\mathop{\tilde{\mathcal{L}}}\left(p\right) = c$ that determines the likelihood ratio confidence interval. I guess $c = \exp\left(-\frac{1}{2}\mathop{\chi^2_{0.95}}\left(1\right)\right) \approx 0.1465$, which corresponds to an asymptotic $95\%$ likelihood ratio confidence interval for $p$.
If so, the "plausibility judgment" would be entirely based on whether parameter values in the confidence intervals under consideration are contained in the likelihood ratio confidence interval or not. To me, this doesn't seem well-justified.

Reference
Held, L., & Sabanés Bové, D. (2020). Likelihood and Bayesian inference: With applications in biology and medicine (Second edition). Springer.
