In one of my lectures the professor wrote:
$$p(\theta \mid \bar{x}+\Delta) \approx p(\theta \mid \bar{x})+\nabla_{\bar{x}} p(\theta\mid\bar{x}) \cdot \Delta+O\left(\Delta^{2}\right) $$
Although I know that the first order Taylor expansion is: $$f(x+\Delta) \approx f(x)+\nabla_{x}f(x) \cdot \Delta+O\left(\Delta^{2}\right)$$
What confuses me is the fact that I have never seen a Taylor expansion for a probability, much less for a conditional probability on the term that we condition on. Maybe I just struggle with the general concept of viewing the conditional probability as a function and what role the conditional has in it.
How can a conditional probability be understood as function, what role does the part we condition on play and how can then a Taylor expansion be applied?