Calculating classification metrics when “true” label is also generated by another classification model

I have a binary classification model $$A$$ and want to calculate its precision and recall for positive and negative classes. The "ground truth" or "true" labels for this model are obtained from another binary classification model $$B$$, whose precision and recall for both classes I know.

I'm almost sure that the metrics for $$A$$ will be probabilistic in nature, but I have no idea on how to calculate them, given the performance metrics of $$B$$. As far as I understand, if I denote the true label by $$y$$, the label predicted by $$B$$ as $$y_B$$ and the label predicted by $$A$$ as $$y_A$$, then I have to calculate $$P(y_A=i\ |\ y=j)$$ where $$i,j\in\{0,1\}$$. And of course, I know the values of $$P(y_A=i\ |\ y_B=j)$$ for all $$i,j$$. So

$$P(y_A=i\ |\ y=j)=\frac{P(y_A=i\ \cap\ y=j)}{P(y=j)}=\frac{P(y_A=i\ \cap\ y=j)}{P(y=j\ \cap\ y_B=1)+P(y=j\ \cap\ y_B=0)}$$

The denominator terms on the RHS can be estimated from the precision-recall metrics of model $$B$$, but I have no idea how to deal with numerator. Maybe an entirely different approach is needed. Would appreciate any help in this regard!