In the context of likelihood-based inference, I've seen some notation concerning the parameter(s) of interest which I've found a little confusing.
For example, notation such as $p_{\theta}(x)$ and ${\mathbb E}_{\theta}\left[S(\theta)\right]$.
What is the significance of the parameter ($\theta$) in subscript notation above? In other words, how should it be read?
My first assumption was that it simply meant "with parameter $\theta$"; for example, for $p_{\theta}(x)$, it would read:
"The probability density of $x$ with parameter $\theta$."
However, this probably isn't correct because $p_{\theta}(x) = L(\theta)$ and, in general, $L(\theta)$ is not a distribution (i.e. it does not integrate to unity); hence it can't be a density, can it?
In addition, in the case of ${\mathbb E}_{\theta}\left[S(\theta)\right]$, I'm not sure what it changes relative to ${\mathbb E}\left[(S(\theta)\right]$ (i.e. with the subscript $\theta$ omitted).
In the above $S(\theta)$ and $L(\theta)$ signify the score function and likelihood function respectively.