According to Williams, D.: Weighing the Odds the p-value of observed data in the likelihood ratio setting is defined as
$$\mathrm{p_{val}}(y^{obs}) := \mathrm{sup}_{\theta \in B_0} \mathbb{P}\big(\mathrm{lr}(Y) \geq \mathrm{lr}(y^{obs}) \vert \theta \big)$$
where
$$ \mathrm{lr}(y) := \frac{\mathrm{sup}_{\theta \in B_A} \mathrm{lhd}(\theta, y)}{\mathrm{sup}_{\theta \in B_0} \mathrm{lhd}(\theta, y)} $$
where $\mathrm{lhd}$ is the likelihood function. He further calls it a 'convoluted concept', which got me thinking what is the reason for this comment. One thing I realized, which I did not realize before, is that in the first equation the supremum can be achieved by a value of $\theta^*$ that could have never generated the $y^{obs}$ (could be $f_{\theta^*}(y^{obs}) = 0).$ So I thought maybe it would make sense to define something like
$$\mathrm{p^{mod}_{val}}(y^{obs}) := \mathrm{sup}_{\theta \in B_0} \bigg (\mathbb{P}\big(\mathrm{lr}(Y) \geq \mathrm{lr}(y^{obs}) \vert \theta \big) \mathbb{P}(y^{obs} \vert \theta) \bigg )$$
Has this been done before? Does it have some name? What about if we change it to
$$\mathrm{p^{mod 2}_{val}}(y^{obs}) := \mathrm{sup}_{\theta \in B_0} \bigg (\mathbb{P}\big(\mathrm{lr}(Y) \geq \mathrm{lr}(y^{obs}) \vert \theta \big) \mathbb{I}_{\mathbb{P}(y^{obs} \vert \theta) > 0} \bigg )$$
