I am having some trouble conceptualizing and calculating a conditional RN derivative. When using this definition:
I can see that if $\mathbb{Q} \ll \mathbb{P}$:
$$\mathbb{E}_\mathbb{Q}(g) = \int g(\omega) \mathbb{Q}(\mathsf d\omega) = \int g(\omega) f(\omega) \mathbb{P}(\mathsf d\omega) = \mathbb{E}_\mathbb{P}(gf)$$
where $f$ is the RN derivative defined by:
$$f = \frac{\mathsf d\mathbb{Q}}{\mathsf d\mathbb{P}}$$
Im having trouble with the following relation:
$$\mathbb{E}_\mathbb{Q}(g\mid\mathcal{Y}) \stackrel{?}{=} \mathbb{E}_\mathbb{P}(g \tilde{f}\mid \mathcal{Y})$$
where $\tilde{f}$ is something like a conditional RN derivative:
$$\tilde{f} \stackrel{?}{=} \frac{d\mathbb{Q}_{(\cdot \mid \mathcal{Y})}}{d\mathbb{P}_{(\cdot \mid \mathcal{Y})}} \stackrel{?}{=} \frac{\frac{d\mathbb{Q}}{d\mathbb{P}}}{?}$$
I tried searching for "conditional RN derivative" and the like but couldn't find an appropriate source. If someone could help me conceptually understand as well as mechanically calculate this I'd very much appreciate it!
EDIT:
To clarify, If I start from $\mathbb{P}$ and $\mathbb{Q}$ and form the conditional measures (from Bayes rule) $\mathbb{P}(\cdot | \mathcal{Y})$ and $\mathbb{Q}(\cdot | \mathcal{Y})$ after gathering some evidence/observations, I know I can write:
$$\mathbb{E}_{\mathbb{Q}(\cdot|\mathcal{Y})}(g|\mathcal{Y}) = \int g(\omega) \mathbb{Q}( \cdot | \mathcal{Y})$$
What is the $\tilde{f}$ that makes the following true?
$$\mathbb{E}_{\mathbb{Q}(\cdot|\mathcal{Y})}(g|\mathcal{Y}) = \mathbb{E}_{\mathbb{P}(\cdot | \mathcal{Y})}(g \tilde{f} | \mathcal{Y})$$
Best Answer
Here, \begin{align*} \tilde{f} = \frac{f}{\mathbb{E}_{\mathbb{P}}(f \mid \mathcal{Y})}. \end{align*} That is, \begin{align*} \mathbb{E}_{\mathbb{Q}}(g \mid \mathcal{Y}) = \frac{\mathbb{E}_{\mathbb{P}}(f g\mid \mathcal{Y})}{\mathbb{E}_{\mathbb{P}}(f \mid \mathcal{Y})}. \end{align*} This is the "abstract Bayes formula".