# Cramer-Rao Lower Bound for a Conditional Likelihood Function

I'm here looking for assurance that my interpretation is correct. Let the likelihood function under consideration be a conditional likelihood given by

$$p(r|x;\theta)$$

where $$r$$ is some random variable, $$x$$ is a particular realization of another random variable that is given and $$\theta$$ is a parameter we wish to estimate. To find the Cramer-Rao lower bound (CRLB) for any unbiased estimator $$\hat{\theta}$$ of $$\theta$$ we need to compute

$$\text{var}\left(\hat{\theta}\right) \geq \frac{1}{-E_{r|x}\left[\frac{\partial^2 \ln p(r|x;\theta)}{\partial \theta^2} | x\right]}$$

where I've chosen to use a conditional expectation in this step because the original likelihood function was conditioned on $$x$$. In all the standard texts this expectation is always the standard expectation with respect to the data $$r$$. Would someone confirm that the conditional expectation is the correction version to use at this step?