Consider a counting process $N$ and a binary random variable $V \in \{ 0,1 \}$. $N$ has intensity process $\lambda$ with respect to $\mathcal F^{N,V}$, the filtration generated by $N$ and $V$. In particular, $N_\cdot - \int_0^\cdot\lambda_s ds $ is a martingale with respect to $\mathcal F^{N,V}$.
I am interested in calculating the conditional expectation
$$E[V | \mathcal{F}_{t-}^N],$$ where $\mathcal F^{N}$ is the filtration generated by $N$. I'm guessing I need a version of Bayes' rule (e.g. Bremaud 1981, L5 Lemma, p. 171), but I haven't figured out how to use it in practice. Can anyone show/provide a hint on how to perform the calculation in practice?
For simplicity, we can assume that $N$ has at most one jump, i.e. that $\lambda_t = Y_t \cdot \alpha_t$ where $\alpha$ is the hazard function associated with $N$, i.e. $$\alpha_t = \lim_{\Delta \longrightarrow 0+} \frac{P(N_{t+\Delta} -N_t = 1 | N_t = 0,V)}{\Delta},$$ and $Y_t := I(N_{t-} = 0)$. Feel free to assume $\alpha$ has a specific form if needed.